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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms .
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so , can an element with even more orbitals form even more covalent bonds ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) .
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why double bond is more reactive than triple bond ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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are ionic bonds the strongest all of bonds ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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is there any reference page to study coordinate bonds ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) .
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what 's the difference between a polar covalent bond and a covalent bond ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b .
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why does it say that na needs to get rid of an electron to obtain an octet configuration when it only has 1 electron in its outer shell ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding .
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so an `` octet '' configuration is n't always 8 electrons ?
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octet rule - matter always wants to be in the most stable form . for any atom , stability is achieved by following the octet rule , which is to say all atoms ( with a few exceptions ) want 8 electrons in their outermost electron shell ( just like noble gases ) . the electrons present in the outermost shell of an atom are called valence electrons . exceptions to the octet rule include hydrogen ( h ) and helium ( he ) that follow the duet rule instead . they are the first two elements of the periodic table and have a single electron shell which accommodates only 2 electrons . other exceptions include some group 3 elements like boron ( b ) that contain three valence electrons . theoretically , boron can accommodate five more electrons according to the octet rule , but boron is a very small atom and five non-metal atoms ( like hydrogen ) can not pack around the boron nucleus . thus , boron commonly forms three bonds , bh $ \text { } _ { 3 } $ , with a total of six electrons in the outermost shell . this also results in some anomalous properties for boron compounds because they are kind of “ short of electrons ” . it should be thus noted that covalent bonding between non-metals can occur to form compounds with less than an octet on each atom . in general , achieving the octet configuration ( i.e . 8 electrons in the outermost shell ) is the driving force for chemical bonding between atoms . take a look at the outer shell configuration ( i.e . number of valence electrons ) of three atoms – sodium ( na ) , chlorine ( cl ) and neon ( ne ) : ionic and covalent bonds let ’ s look at the following two scenarios a and b . there are two kids , emily and sarah . they both are very good friends . scenario a : scenario b : now let ’ s apply the above analogy to chemical bonding . assume that emily and sarah represent two atoms , and the blanket symbolizes their valence electrons . in scenario a , atom emily is willing to donate her electrons ( blanket ) to atom sarah because by doing so both achieve an octet configuration of 8 electrons in their respective outer shells , making them both happy and stable . this donation of electrons is called ionic bonding . example of an ionic bond in scenario b , both the atoms emily and sarah are equally electronegative . so , neither emily nor sarah is ready to part with her electrons ( blanket ) , and they instead share their valence electrons with each other . this is called a covalent bond . electronegativity is a measure of how strongly an atom attracts electrons from another atom in a chemical bond and this value is governed by where the particular atom is located in the periodic table ( francium is the least electronegative element while fluorine is the most electronegative ) . example of a covalent bond polar and non-polar covalent bond let ’ s go back to emily and sarah : scenario c : scenario d : let ’ s apply the above analogy to a covalent bond formation . in scenario c , both emily and sarah are equally cold ( in our analogy this translates to them having the same electronegativity ) . because they have the same electronegativity , they will share their valence electrons equally with each other . this type of a covalent bond where electrons are shared equally between two atoms is called a non-polar covalent bond . example of a non-polar covalent bond in scenario d , emily is cold but sarah is much colder ( no doubt mild hypothermia from playing outside in the rain too long ) ! together they share the blanket , but sarah has a tendency to keep pulling the blanket from emily in order to warm up more . in the atomic world , one atom ( sarah ) is more electronegative than another atom ( emily ) , and naturally pulls the shared electrons towards itself . this pulling of electrons creates slight polarity in the bond . covalent bonds where electrons are not shared equally between two atoms are called polar covalent bond . example of a polar covalent bond as shown above , the electrons in a covalent bond between two different atoms ( h and cl in this case ) are not equally shared by the atoms . this is due to the electronegativity difference between the two atoms . the more electronegative atom ( cl ) has greater share of the electrons than the less electronegative atom ( h ) . consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom . the electronegativity value for carbon ( c ) and hydrogen ( h ) is 2.55 and 2.1 respectively , so the difference in their electronegativity values is only 0.45 ( & lt ; 0.5 criteria ) ; the electrons are thus equally shared between carbon and hydrogen . so we can conveniently say that a molecule of methane has a total of four non-polar covalent bonds . single and multiple covalent bonds the number of pairs of electrons shared between two atoms determines the type of the covalent bond formed between them . number of electron pairs shared | type of covalent bond formed : - : | : - : 1 | single 2 | double 3 | triple now let ’ s move on to a couple of examples and try to determine the type of covalent bonds formed nitrogen atom can attain an octet configuration by sharing three electrons with another nitrogen atom , forming a triple bond ( three pairs of electrons shared ) consider the molecule carbon dioxide ( co $ \text { } _ { 2 } $ ) . let ’ s determine the type of covalent bonds it forms .
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consequently , the atom that has the greater share of the bonding electrons bears a partial negative charge ( δ- ) and the other atom automatically bears a partial positive charge ( δ+ ) of equal magnitude . properties of non-polar covalent bonds : often occurs between atoms that are the same electronegativity difference between bonded atoms is small ( & lt ; 0.5 pauling units ) electrons are shared equally between atoms properties of polar covalent bond : always occurs between different atoms electronegativity difference between bonded atoms is moderate ( 0.5 and 1.9 pauling units ) electrons are not shared equally between atoms methane ( ch $ \text { } _ { 4 } $ ) is an example of a compound where non-polar covalent bonds are formed between two different atoms . one carbon atom forms four covalent bonds with four hydrogen atoms by sharing a pair of electron between itself and each hydrogen ( h ) atom .
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even if the electronegativity difference is < 0.5 , if the atoms are different and there is some electronegativity difference , would n't the electrons be slightly unequally shared between the two atoms ?
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overview “ black power ” refers to a militant ideology that aimed not at integration and accommodation with white america , but rather preached black self-reliance , self-defense , and racial pride . malcolm x was the most influential thinker of what became known as the black power movement , and inspired others like stokely carmichael of the student nonviolent coordinating committee and huey p. newton and bobby seale of the black panther party . the black panther party in oakland , california , operated as both a black self-defense militia and a provider of services to the black community . the origins of black power though the actual phrase “ black power ” did not come into widespread usage until 1966 , the ideas underlying black power were not new . as early as the 1940s , a. philip randolph , an african american labor activist , called for a march on washington to pressure president franklin d. roosevelt to outlaw racial discrimination in federal employment . randolph envisioned the march as “ an all-negro movement ” that would inculcate “ a sense of self-reliance ” and “ break down the slave psychology and inferiority-complex in negroes which comes and is nourished with negroes relying on white people for direction and support. ” $ ^1 $ though randolph himself eschewed black nationalism , the goals of self-reliance and racial pride would become key components of the black power ideology . the author richard wright had also published a book called black power in 1954 , a non-fiction chronicle of his travels to africa ’ s gold coast , the country that would become ghana. $ ^2 $ wright ’ s journeys underscore the significance of ties between africans and african americans and the centrality of decolonization in black power ideology . in the 1950s and 1960s , african countries were becoming independent after decades of european colonial rule . african american thinkers like richard wright and later , malcolm x , drew a connection between the struggles of africans to overthrow the remaining vestiges of colonial oppression and the struggles of african americans to overcome the white power structure in the united states. $ ^3 $ malcolm x and the nation of islam led by elijah muhammad , born elijah poole , the nation of islam , also known as the black muslims , had existed since the 1930s . malcolm x , born malcolm little , became acquainted with elijah muhammad and the teachings of the nation of islam while serving time for burglary at the norfolk prison colony in massachusetts . after the expiration of his parole , he became involved with the nation of islam , serving as its emissary on a visit to the middle east and africa in 1959 , and becoming the minister of mosque no . 7 in harlem . malcolm x ’ s fiery rhetoric and charismatic presence gained the nation of islam many new adherents in the late 1950s and early 1960s . the nation of islam advocated black self-empowerment and self-reliance , as well as cultural and racial pride . the most famous black muslim was undoubtedly the heavyweight boxer cassius clay , who changed his name to muhammad ali after converting . in 1964 , malcolm x again traveled to the middle east and africa , and made his hajj ( islamic pilgrimage ) to mecca in saudi arabia . upon his return to the united states , he publicly repudiated the nation of islam and the teachings of elijah muhammad , choosing instead to adhere to a more conventional version of sunni islam . he founded the organization of afro-american unity , which embraced the internationalization of the black freedom struggle and continued to emphasize black self-determination and self-defense . on february 21 , 1965 , after months of receiving death threats , malcolm x was assassinated at the audubon ballroom in manhattan by members of the nation of islam. $ ^4 $ his autobiography was published shortly after his death and quickly became a bestseller. $ ^5 $ the black panther party in june 1966 , stokely carmichael of the student nonviolent coordinating committee shouted the words “ black power ” in an address to a freedom rally in greenwood , mississippi. $ ^6 $ the incident reflected the increased militancy of groups like sncc and core , which had previously adhered to nonviolent civil disobedience . the black panther party of self-defense was founded in 1966 in oakland , california , by huey p. newton and bobby seale , who issued a ten-point program demanding , among other things , freedom , employment , and an immediate end to police brutality . the black panthers gained notoriety when in the spring of 1967 , its gun-toting members staged a protest at the state capitol against a gun control bill then being debated by the california state legislature . the black panthers espoused a militant form of black self-defense and functioned as a local militia , taking advantage of open-carry gun laws to patrol black neighborhoods in oakland in order to prevent police harassment and brutality . the panthers also provided community services , such as free breakfasts for children , drug and alcohol rehabilitation programs , self-defense classes , and free medical clinics and childcare centers. $ ^7 $ largely due to the panthers ’ militant rhetoric and armed self-defense , the state of california imposed strictures on open-carry gun laws , and the fbi employed its counter-intelligence program ( cointelpro ) to combat what it perceived as the black panther party ’ s subversive threat to american democracy. $ ^8 $ what do you think ? was black power part of the civil rights movement or was it opposed to the civil rights movement ? how did the goals of the black power movement differ from those of more mainstream civil rights activists ? compare the major demands of the ten-point program with the goals of civil rights campaigns for voting rights and desegregation . why do you think the ideas of black power gained in popularity over the course of the 1960s ?
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overview “ black power ” refers to a militant ideology that aimed not at integration and accommodation with white america , but rather preached black self-reliance , self-defense , and racial pride . malcolm x was the most influential thinker of what became known as the black power movement , and inspired others like stokely carmichael of the student nonviolent coordinating committee and huey p. newton and bobby seale of the black panther party .
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who shot martin luther king ?
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overview “ black power ” refers to a militant ideology that aimed not at integration and accommodation with white america , but rather preached black self-reliance , self-defense , and racial pride . malcolm x was the most influential thinker of what became known as the black power movement , and inspired others like stokely carmichael of the student nonviolent coordinating committee and huey p. newton and bobby seale of the black panther party . the black panther party in oakland , california , operated as both a black self-defense militia and a provider of services to the black community . the origins of black power though the actual phrase “ black power ” did not come into widespread usage until 1966 , the ideas underlying black power were not new . as early as the 1940s , a. philip randolph , an african american labor activist , called for a march on washington to pressure president franklin d. roosevelt to outlaw racial discrimination in federal employment . randolph envisioned the march as “ an all-negro movement ” that would inculcate “ a sense of self-reliance ” and “ break down the slave psychology and inferiority-complex in negroes which comes and is nourished with negroes relying on white people for direction and support. ” $ ^1 $ though randolph himself eschewed black nationalism , the goals of self-reliance and racial pride would become key components of the black power ideology . the author richard wright had also published a book called black power in 1954 , a non-fiction chronicle of his travels to africa ’ s gold coast , the country that would become ghana. $ ^2 $ wright ’ s journeys underscore the significance of ties between africans and african americans and the centrality of decolonization in black power ideology . in the 1950s and 1960s , african countries were becoming independent after decades of european colonial rule . african american thinkers like richard wright and later , malcolm x , drew a connection between the struggles of africans to overthrow the remaining vestiges of colonial oppression and the struggles of african americans to overcome the white power structure in the united states. $ ^3 $ malcolm x and the nation of islam led by elijah muhammad , born elijah poole , the nation of islam , also known as the black muslims , had existed since the 1930s . malcolm x , born malcolm little , became acquainted with elijah muhammad and the teachings of the nation of islam while serving time for burglary at the norfolk prison colony in massachusetts . after the expiration of his parole , he became involved with the nation of islam , serving as its emissary on a visit to the middle east and africa in 1959 , and becoming the minister of mosque no . 7 in harlem . malcolm x ’ s fiery rhetoric and charismatic presence gained the nation of islam many new adherents in the late 1950s and early 1960s . the nation of islam advocated black self-empowerment and self-reliance , as well as cultural and racial pride . the most famous black muslim was undoubtedly the heavyweight boxer cassius clay , who changed his name to muhammad ali after converting . in 1964 , malcolm x again traveled to the middle east and africa , and made his hajj ( islamic pilgrimage ) to mecca in saudi arabia . upon his return to the united states , he publicly repudiated the nation of islam and the teachings of elijah muhammad , choosing instead to adhere to a more conventional version of sunni islam . he founded the organization of afro-american unity , which embraced the internationalization of the black freedom struggle and continued to emphasize black self-determination and self-defense . on february 21 , 1965 , after months of receiving death threats , malcolm x was assassinated at the audubon ballroom in manhattan by members of the nation of islam. $ ^4 $ his autobiography was published shortly after his death and quickly became a bestseller. $ ^5 $ the black panther party in june 1966 , stokely carmichael of the student nonviolent coordinating committee shouted the words “ black power ” in an address to a freedom rally in greenwood , mississippi. $ ^6 $ the incident reflected the increased militancy of groups like sncc and core , which had previously adhered to nonviolent civil disobedience . the black panther party of self-defense was founded in 1966 in oakland , california , by huey p. newton and bobby seale , who issued a ten-point program demanding , among other things , freedom , employment , and an immediate end to police brutality . the black panthers gained notoriety when in the spring of 1967 , its gun-toting members staged a protest at the state capitol against a gun control bill then being debated by the california state legislature . the black panthers espoused a militant form of black self-defense and functioned as a local militia , taking advantage of open-carry gun laws to patrol black neighborhoods in oakland in order to prevent police harassment and brutality . the panthers also provided community services , such as free breakfasts for children , drug and alcohol rehabilitation programs , self-defense classes , and free medical clinics and childcare centers. $ ^7 $ largely due to the panthers ’ militant rhetoric and armed self-defense , the state of california imposed strictures on open-carry gun laws , and the fbi employed its counter-intelligence program ( cointelpro ) to combat what it perceived as the black panther party ’ s subversive threat to american democracy. $ ^8 $ what do you think ? was black power part of the civil rights movement or was it opposed to the civil rights movement ? how did the goals of the black power movement differ from those of more mainstream civil rights activists ? compare the major demands of the ten-point program with the goals of civil rights campaigns for voting rights and desegregation . why do you think the ideas of black power gained in popularity over the course of the 1960s ?
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malcolm x was the most influential thinker of what became known as the black power movement , and inspired others like stokely carmichael of the student nonviolent coordinating committee and huey p. newton and bobby seale of the black panther party . the black panther party in oakland , california , operated as both a black self-defense militia and a provider of services to the black community . the origins of black power though the actual phrase “ black power ” did not come into widespread usage until 1966 , the ideas underlying black power were not new .
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why muhammadali have influnce malcom x. and wat black panther ?
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overview “ black power ” refers to a militant ideology that aimed not at integration and accommodation with white america , but rather preached black self-reliance , self-defense , and racial pride . malcolm x was the most influential thinker of what became known as the black power movement , and inspired others like stokely carmichael of the student nonviolent coordinating committee and huey p. newton and bobby seale of the black panther party . the black panther party in oakland , california , operated as both a black self-defense militia and a provider of services to the black community . the origins of black power though the actual phrase “ black power ” did not come into widespread usage until 1966 , the ideas underlying black power were not new . as early as the 1940s , a. philip randolph , an african american labor activist , called for a march on washington to pressure president franklin d. roosevelt to outlaw racial discrimination in federal employment . randolph envisioned the march as “ an all-negro movement ” that would inculcate “ a sense of self-reliance ” and “ break down the slave psychology and inferiority-complex in negroes which comes and is nourished with negroes relying on white people for direction and support. ” $ ^1 $ though randolph himself eschewed black nationalism , the goals of self-reliance and racial pride would become key components of the black power ideology . the author richard wright had also published a book called black power in 1954 , a non-fiction chronicle of his travels to africa ’ s gold coast , the country that would become ghana. $ ^2 $ wright ’ s journeys underscore the significance of ties between africans and african americans and the centrality of decolonization in black power ideology . in the 1950s and 1960s , african countries were becoming independent after decades of european colonial rule . african american thinkers like richard wright and later , malcolm x , drew a connection between the struggles of africans to overthrow the remaining vestiges of colonial oppression and the struggles of african americans to overcome the white power structure in the united states. $ ^3 $ malcolm x and the nation of islam led by elijah muhammad , born elijah poole , the nation of islam , also known as the black muslims , had existed since the 1930s . malcolm x , born malcolm little , became acquainted with elijah muhammad and the teachings of the nation of islam while serving time for burglary at the norfolk prison colony in massachusetts . after the expiration of his parole , he became involved with the nation of islam , serving as its emissary on a visit to the middle east and africa in 1959 , and becoming the minister of mosque no . 7 in harlem . malcolm x ’ s fiery rhetoric and charismatic presence gained the nation of islam many new adherents in the late 1950s and early 1960s . the nation of islam advocated black self-empowerment and self-reliance , as well as cultural and racial pride . the most famous black muslim was undoubtedly the heavyweight boxer cassius clay , who changed his name to muhammad ali after converting . in 1964 , malcolm x again traveled to the middle east and africa , and made his hajj ( islamic pilgrimage ) to mecca in saudi arabia . upon his return to the united states , he publicly repudiated the nation of islam and the teachings of elijah muhammad , choosing instead to adhere to a more conventional version of sunni islam . he founded the organization of afro-american unity , which embraced the internationalization of the black freedom struggle and continued to emphasize black self-determination and self-defense . on february 21 , 1965 , after months of receiving death threats , malcolm x was assassinated at the audubon ballroom in manhattan by members of the nation of islam. $ ^4 $ his autobiography was published shortly after his death and quickly became a bestseller. $ ^5 $ the black panther party in june 1966 , stokely carmichael of the student nonviolent coordinating committee shouted the words “ black power ” in an address to a freedom rally in greenwood , mississippi. $ ^6 $ the incident reflected the increased militancy of groups like sncc and core , which had previously adhered to nonviolent civil disobedience . the black panther party of self-defense was founded in 1966 in oakland , california , by huey p. newton and bobby seale , who issued a ten-point program demanding , among other things , freedom , employment , and an immediate end to police brutality . the black panthers gained notoriety when in the spring of 1967 , its gun-toting members staged a protest at the state capitol against a gun control bill then being debated by the california state legislature . the black panthers espoused a militant form of black self-defense and functioned as a local militia , taking advantage of open-carry gun laws to patrol black neighborhoods in oakland in order to prevent police harassment and brutality . the panthers also provided community services , such as free breakfasts for children , drug and alcohol rehabilitation programs , self-defense classes , and free medical clinics and childcare centers. $ ^7 $ largely due to the panthers ’ militant rhetoric and armed self-defense , the state of california imposed strictures on open-carry gun laws , and the fbi employed its counter-intelligence program ( cointelpro ) to combat what it perceived as the black panther party ’ s subversive threat to american democracy. $ ^8 $ what do you think ? was black power part of the civil rights movement or was it opposed to the civil rights movement ? how did the goals of the black power movement differ from those of more mainstream civil rights activists ? compare the major demands of the ten-point program with the goals of civil rights campaigns for voting rights and desegregation . why do you think the ideas of black power gained in popularity over the course of the 1960s ?
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african american thinkers like richard wright and later , malcolm x , drew a connection between the struggles of africans to overthrow the remaining vestiges of colonial oppression and the struggles of african americans to overcome the white power structure in the united states. $ ^3 $ malcolm x and the nation of islam led by elijah muhammad , born elijah poole , the nation of islam , also known as the black muslims , had existed since the 1930s . malcolm x , born malcolm little , became acquainted with elijah muhammad and the teachings of the nation of islam while serving time for burglary at the norfolk prison colony in massachusetts . after the expiration of his parole , he became involved with the nation of islam , serving as its emissary on a visit to the middle east and africa in 1959 , and becoming the minister of mosque no .
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why was malcolm x so violent ?
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math is all about relationships . for example , how can we describe the relationship between a person 's height and weight ? or how can we describe the relationship between how much money you make and how many hours you work ? the three main ways to represent a relationship in math are using a table , a graph , or an equation . in this article , we 'll represent the same relationship with a table , graph , and equation to see how this works . example relationship : a pizza company sells a small pizza for $ \ $ 6 $ . each topping costs $ \ $ 2 $ . representing with a table we know that the cost of a pizza with $ 0 $ toppings is $ \ $ 6 $ , the cost of a pizza with $ 1 $ topping is $ \ $ 2 $ more which is $ \ $ 8 $ , and so on . here 's a table showing this : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ : - : | : - : $ 0 $ | $ \ $ 6 $ $ 1 $ | $ \ $ 8 $ $ 2 $ | $ \ $ 10 $ $ 3 $ | $ \ $ 12 $ $ 4 $ | $ \ $ 14 $ of course , this table just shows the total cost for a few of the possible number of toppings . for example , there 's no reason we could n't have $ 7 $ toppings on the pizza . ( other than that it 'd be gross ! ) let 's see how this table makes sense for a small pizza with $ 4 $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of the $ \blued4 $ toppings : $ \blued4 $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ \ $ \goldd8 $ this leads to the total cost of $ \ $ \greend6 + \ $ \goldd8 = \ $ 14 $ . representing with an equation let 's write an equation for the total cost $ y $ of a pizza with $ x $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of $ x $ toppings : $ x $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ x \cdot 2 = \goldd { 2x } $ so here 's the equation for the total cost $ y $ of a small pizza : $ y = \greend6 + \goldd { 2x } $ let 's see how this makes sense for a small pizza with $ 3 $ toppings : $ x = \blued3 $ because there are $ \blued3 $ toppings the total cost is $ 6 + 2 ( \blued3 ) = 6 + 6 = \ $ 12 $ representing with a graph we can create ordered pairs from the $ x $ and $ y $ values : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ | ordered pair $ ( x , y ) $ : - : | : - : | : - : $ 0 $ | $ \ $ 6 $ | $ ( 0 , 6 ) $ $ 1 $ | $ \ $ 8 $ | $ ( 1 , 8 ) $ $ 2 $ | $ \ $ 10 $ | $ ( 2 , 10 ) $ $ 3 $ | $ \ $ 12 $ | $ ( 3 , 12 ) $ $ 4 $ | $ \ $ 14 $ | $ ( 4 , 14 ) $ we can use these order pairs to create a graph : $ $ cool ! notice how the graph helps us easily see that the total cost of the small pizza increases as we add more toppings . we did it ! we represented the situation where a pizza company sells a small pizza for $ \ $ 6 $ , and each topping costs $ \ $ 2 $ using a table , an equation , and a graph . what 's really cool is we used these three methods to represent the same relationship . the table allowed us to see exactly how much a pizza with different number of toppings costs , the equation gave us a way to find the cost of a pizza with any number of toppings , and the graph helped us visually see the relationship . now let 's give you a chance to create a table , an equation , and a graph to represent a relationship . give it a try ! an ice cream shop sells $ 2 $ scoops of ice cream for $ \ $ 3 $ . each additional scoop costs $ \ $ 1 $ . comparing the three different ways we learned that the three main ways to represent a relationship is with a table , an equation , or a graph . what do you think are the advantages and disadvantages of each representation ? for example , why might someone use a graph instead of a table ? why might someone use an equation instead of a graph ? feel free to discuss in the comments below !
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comparing the three different ways we learned that the three main ways to represent a relationship is with a table , an equation , or a graph . what do you think are the advantages and disadvantages of each representation ? for example , why might someone use a graph instead of a table ?
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what do all of the smart khan academy users think is the best way to represent data ?
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math is all about relationships . for example , how can we describe the relationship between a person 's height and weight ? or how can we describe the relationship between how much money you make and how many hours you work ? the three main ways to represent a relationship in math are using a table , a graph , or an equation . in this article , we 'll represent the same relationship with a table , graph , and equation to see how this works . example relationship : a pizza company sells a small pizza for $ \ $ 6 $ . each topping costs $ \ $ 2 $ . representing with a table we know that the cost of a pizza with $ 0 $ toppings is $ \ $ 6 $ , the cost of a pizza with $ 1 $ topping is $ \ $ 2 $ more which is $ \ $ 8 $ , and so on . here 's a table showing this : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ : - : | : - : $ 0 $ | $ \ $ 6 $ $ 1 $ | $ \ $ 8 $ $ 2 $ | $ \ $ 10 $ $ 3 $ | $ \ $ 12 $ $ 4 $ | $ \ $ 14 $ of course , this table just shows the total cost for a few of the possible number of toppings . for example , there 's no reason we could n't have $ 7 $ toppings on the pizza . ( other than that it 'd be gross ! ) let 's see how this table makes sense for a small pizza with $ 4 $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of the $ \blued4 $ toppings : $ \blued4 $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ \ $ \goldd8 $ this leads to the total cost of $ \ $ \greend6 + \ $ \goldd8 = \ $ 14 $ . representing with an equation let 's write an equation for the total cost $ y $ of a pizza with $ x $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of $ x $ toppings : $ x $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ x \cdot 2 = \goldd { 2x } $ so here 's the equation for the total cost $ y $ of a small pizza : $ y = \greend6 + \goldd { 2x } $ let 's see how this makes sense for a small pizza with $ 3 $ toppings : $ x = \blued3 $ because there are $ \blued3 $ toppings the total cost is $ 6 + 2 ( \blued3 ) = 6 + 6 = \ $ 12 $ representing with a graph we can create ordered pairs from the $ x $ and $ y $ values : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ | ordered pair $ ( x , y ) $ : - : | : - : | : - : $ 0 $ | $ \ $ 6 $ | $ ( 0 , 6 ) $ $ 1 $ | $ \ $ 8 $ | $ ( 1 , 8 ) $ $ 2 $ | $ \ $ 10 $ | $ ( 2 , 10 ) $ $ 3 $ | $ \ $ 12 $ | $ ( 3 , 12 ) $ $ 4 $ | $ \ $ 14 $ | $ ( 4 , 14 ) $ we can use these order pairs to create a graph : $ $ cool ! notice how the graph helps us easily see that the total cost of the small pizza increases as we add more toppings . we did it ! we represented the situation where a pizza company sells a small pizza for $ \ $ 6 $ , and each topping costs $ \ $ 2 $ using a table , an equation , and a graph . what 's really cool is we used these three methods to represent the same relationship . the table allowed us to see exactly how much a pizza with different number of toppings costs , the equation gave us a way to find the cost of a pizza with any number of toppings , and the graph helped us visually see the relationship . now let 's give you a chance to create a table , an equation , and a graph to represent a relationship . give it a try ! an ice cream shop sells $ 2 $ scoops of ice cream for $ \ $ 3 $ . each additional scoop costs $ \ $ 1 $ . comparing the three different ways we learned that the three main ways to represent a relationship is with a table , an equation , or a graph . what do you think are the advantages and disadvantages of each representation ? for example , why might someone use a graph instead of a table ? why might someone use an equation instead of a graph ? feel free to discuss in the comments below !
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for example , why might someone use a graph instead of a table ? why might someone use an equation instead of a graph ? feel free to discuss in the comments below !
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can i just use the formula b=- ( mx-y ) to find the initial value of a fuction ?
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math is all about relationships . for example , how can we describe the relationship between a person 's height and weight ? or how can we describe the relationship between how much money you make and how many hours you work ? the three main ways to represent a relationship in math are using a table , a graph , or an equation . in this article , we 'll represent the same relationship with a table , graph , and equation to see how this works . example relationship : a pizza company sells a small pizza for $ \ $ 6 $ . each topping costs $ \ $ 2 $ . representing with a table we know that the cost of a pizza with $ 0 $ toppings is $ \ $ 6 $ , the cost of a pizza with $ 1 $ topping is $ \ $ 2 $ more which is $ \ $ 8 $ , and so on . here 's a table showing this : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ : - : | : - : $ 0 $ | $ \ $ 6 $ $ 1 $ | $ \ $ 8 $ $ 2 $ | $ \ $ 10 $ $ 3 $ | $ \ $ 12 $ $ 4 $ | $ \ $ 14 $ of course , this table just shows the total cost for a few of the possible number of toppings . for example , there 's no reason we could n't have $ 7 $ toppings on the pizza . ( other than that it 'd be gross ! ) let 's see how this table makes sense for a small pizza with $ 4 $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of the $ \blued4 $ toppings : $ \blued4 $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ \ $ \goldd8 $ this leads to the total cost of $ \ $ \greend6 + \ $ \goldd8 = \ $ 14 $ . representing with an equation let 's write an equation for the total cost $ y $ of a pizza with $ x $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of $ x $ toppings : $ x $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ x \cdot 2 = \goldd { 2x } $ so here 's the equation for the total cost $ y $ of a small pizza : $ y = \greend6 + \goldd { 2x } $ let 's see how this makes sense for a small pizza with $ 3 $ toppings : $ x = \blued3 $ because there are $ \blued3 $ toppings the total cost is $ 6 + 2 ( \blued3 ) = 6 + 6 = \ $ 12 $ representing with a graph we can create ordered pairs from the $ x $ and $ y $ values : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ | ordered pair $ ( x , y ) $ : - : | : - : | : - : $ 0 $ | $ \ $ 6 $ | $ ( 0 , 6 ) $ $ 1 $ | $ \ $ 8 $ | $ ( 1 , 8 ) $ $ 2 $ | $ \ $ 10 $ | $ ( 2 , 10 ) $ $ 3 $ | $ \ $ 12 $ | $ ( 3 , 12 ) $ $ 4 $ | $ \ $ 14 $ | $ ( 4 , 14 ) $ we can use these order pairs to create a graph : $ $ cool ! notice how the graph helps us easily see that the total cost of the small pizza increases as we add more toppings . we did it ! we represented the situation where a pizza company sells a small pizza for $ \ $ 6 $ , and each topping costs $ \ $ 2 $ using a table , an equation , and a graph . what 's really cool is we used these three methods to represent the same relationship . the table allowed us to see exactly how much a pizza with different number of toppings costs , the equation gave us a way to find the cost of a pizza with any number of toppings , and the graph helped us visually see the relationship . now let 's give you a chance to create a table , an equation , and a graph to represent a relationship . give it a try ! an ice cream shop sells $ 2 $ scoops of ice cream for $ \ $ 3 $ . each additional scoop costs $ \ $ 1 $ . comparing the three different ways we learned that the three main ways to represent a relationship is with a table , an equation , or a graph . what do you think are the advantages and disadvantages of each representation ? for example , why might someone use a graph instead of a table ? why might someone use an equation instead of a graph ? feel free to discuss in the comments below !
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here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of the $ \blued4 $ toppings : $ \blued4 $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ \ $ \goldd8 $ this leads to the total cost of $ \ $ \greend6 + \ $ \goldd8 = \ $ 14 $ . representing with an equation let 's write an equation for the total cost $ y $ of a pizza with $ x $ toppings . here 's the cost of just the pizza : $ \ $ \greend6 $ here 's the cost of $ x $ toppings : $ x $ toppings $ \cdot $ $ \ $ 2 $ per topping $ = $ $ x \cdot 2 = \goldd { 2x } $ so here 's the equation for the total cost $ y $ of a small pizza : $ y = \greend6 + \goldd { 2x } $ let 's see how this makes sense for a small pizza with $ 3 $ toppings : $ x = \blued3 $ because there are $ \blued3 $ toppings the total cost is $ 6 + 2 ( \blued3 ) = 6 + 6 = \ $ 12 $ representing with a graph we can create ordered pairs from the $ x $ and $ y $ values : toppings on the pizza $ ( x ) $ | total cost $ ( y ) $ | ordered pair $ ( x , y ) $ : - : | : - : | : - : $ 0 $ | $ \ $ 6 $ | $ ( 0 , 6 ) $ $ 1 $ | $ \ $ 8 $ | $ ( 1 , 8 ) $ $ 2 $ | $ \ $ 10 $ | $ ( 2 , 10 ) $ $ 3 $ | $ \ $ 12 $ | $ ( 3 , 12 ) $ $ 4 $ | $ \ $ 14 $ | $ ( 4 , 14 ) $ we can use these order pairs to create a graph : $ $ cool !
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is y=x+1 the correct equation ?
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what does the taj mahal have to do with the tamerlane ? what do persian carpets have to do with turkish tiles ? quite a bit , as it turns out . by the fourteenth century , islam had spread as far east as india and islamic rulers had solidified their power by establishing prosperous cities and a robust trade in decorative arts along the all-important silk road . this is a complex period with competing and overlapping cultures and empires . read below for an introduction to the later islamic dynasties . ottoman ( 1300-1924 ) at its earliest stages , the ottoman state was little more than a group formed as a result of the dissolution of the anatolian seljuq sultanate . however , in 1453 , the ottomans captured the great byzantine capital , constantinople , and in 1517 , they defeated the mamluks and took control of the most significant state in the islamic world . while the ottomans ruled for many centuries , the height of the empire 's cultural and economic prosperity was achieved during süleyman the magnificent 's reign ( r. 1520-1566 ) , a period often referred to as the ottoman 's ‘ golden age. ’ in addition to large-scale architectural projects , the decorative arts flourished , chief among them , ceramics , particularly tiles . iznik tiles , named for the city in anatolia where they were produced , developed a trademark style of curling vines and flowers rendered in beautiful shades of blue and turquoise . these designs were informed by the blue and white floral patterns found in chinese porcelain—similar to earlier mamluk tiles , and timurid art to the east . in addition to iznik , other artistic hubs developed , such as bursa , known for its silks , and cairo for its carpets . the capital , istanbul ( formerly constantinople ) , became a great center for all matters of cultural importance from manuscript illumination to architecture . the architecture of the period , both sacred and secular , incorporates these decorative arts , from the dazzling blue tiles and monumental calligraphy that adorn the walls of topkapi palace ( begun 1459 ) to the carpets that line the floors of the süleymaniye mosque ( 1550-1558 ) . ottoman mosque architecture itself is marked by the use of domes , widely used earlier in byzantium , and towering minarets . the byzantine influence draws primarily from hagia sophia , a former church that was converted into a mosque ( and is now a museum ) . timurid ( 1369-1502 ) this powerful central asian dynasty was named for its founder , tamerlane ( ruled 1370-1405 ) , which is derived from timur the lame . despite his rather pathetic epithet , he claimed to be a descendent of genghis khan and demonstrated some of his supposed ancestor ’ s ruthlessness in conquering neighboring territories . after establishing a vast empire , timur developed a monumental architecture befitting his power , and sought to make samarkand the “ pearl of the world. ” because the capital was situated at a major crossroads of the silk road ( the crucial trade route linking the middle east , central asia , and china ) , and because timur had conquered so widely , the timurids acquired a myriad of artisans and craftspeople from distinct artistic traditions . the resulting style synthesized aesthetic and design principles from as far away as india ( then hindustan ) and the lands in between . the result can be seen in cities filled with buildings created on a lavish scale that exhibited tall , bulbous domes and the finest ceramic tiles . the structures and even the cities themselves are often described foremost by the overwhelming use of blues and golds . while the timurid dynasty itself was short-lived , its legacy survives not only in the grand architecture that it left behind but in its descendents who went on to play significant roles in the ottoman , safavid and mughal empires . safavid ( 1502-1736 ) the safavids , a group with roots in the sufic tradition ( a mystical branch of islam ) , came to power in persia , modern-day iran and azerbaijan . in 1501 the safavid rulers declared shi ’ a islam as its state religion ; and in just ten years the empire came to include all of iran . the art of manuscript illumination was highly prized in the safavid courts , and royal patrons made many large-scale commissions . perhaps the most notable of these is the shahnama ( or ‘ book of kings , ’ a compilation of stories about earlier rulers of iran ) from the 1520s . while painting in this context did not have the same prominent and longstanding tradition as it does in western art , the illustrations exhibit masterful workmanship and an incredible attention to detail . trade in carpets was also important , and even today , people understand the appeal of persian carpets . these large-scale , high-quality pieces were created as luxurious furnishings for royal courts . the most famous—perhaps of all time—is a pair known as the ardabil carpets , created in 1539-1540 . the carpets were nearly identical , perfectly symmetrical and enormous . every inch of space was filled with flowers , scrolling vines , and medallions . the empire began to struggle financially and militarily until the rule of shah abbas ( r. 1587-1629 ) . he moved the capital to isfahan where he built a magnificent new city and established state workshops for textiles , which , along with silk and other goods , were increasingly exported to europe . the mosque architecture made use of earlier persian elements , like the four-iwan plan and building materials of brick and glazed tiles reminiscent of timurid architecture , with its blues and greens and bulbous domes . even in such far-removed lands , the connections between these dynasties are evident in the art they created . mughal ( 1526-1858 ) though islam had been introduced in india centuries before , the mughals were responsible for some of the greatest works of art produced in the canons of both indian and islamic art . the empire established itself when babur , himself a timurid prince of turkish and central asian descent , came to hindustan and defeated the existing islamic sultanate in delhi . tracing their roots to central asia , the mughals produced art , music and poetry that was highly influenced by persian and central asian aesthetics . this is evident in the style and importance given to miniature paintings , created to illustrate manuscripts . the most grandiose of these was the akbarnama , created to record the conquests of akbar , widely regarded as the greatest mughal emperor . the art and architecture created during his reign demonstrate a synthesis of indigenous indian temple architecture with structural and design elements derived from islamic sources farther west . the mughals developed a unique architectural style which , in the years after akbar ’ s reign , began to feature scalloped arches and stylized floral designs in white marble . the most famous example is the taj mahal , constructed by shah jahan from 1632-1653 . the mughal dynasty left a lasting mark on the landscape of india , and remained in power until the british completed their conquest of india in the nineteenth century . although historians generally agree that the major islamic dynasties end in the nineteenth and early twentieth centuries , islamic art and culture have continued to flourish . muslim artists and muslim countries are still producing art . some art historians consider such work as simply modern or contemporary art while others see it within the continuity of islamic art . essay by glenna barlow additional resources : the art of the ottomans before 1600 at the metropolitan museum of art 's timeline of art history the art of the safavids before 1600 at the metropolitan museum of art 's timeline of art history the art of the mughals before 1600 at the metropolitan museum of art 's timeline of art history introduction to the mughal empire from the bbc
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while the ottomans ruled for many centuries , the height of the empire 's cultural and economic prosperity was achieved during süleyman the magnificent 's reign ( r. 1520-1566 ) , a period often referred to as the ottoman 's ‘ golden age. ’ in addition to large-scale architectural projects , the decorative arts flourished , chief among them , ceramics , particularly tiles . iznik tiles , named for the city in anatolia where they were produced , developed a trademark style of curling vines and flowers rendered in beautiful shades of blue and turquoise . these designs were informed by the blue and white floral patterns found in chinese porcelain—similar to earlier mamluk tiles , and timurid art to the east .
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seeing all of the blue in the tiles of these mosques drives me to ask what pigment was used in the glazes that produced it ?
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what does the taj mahal have to do with the tamerlane ? what do persian carpets have to do with turkish tiles ? quite a bit , as it turns out . by the fourteenth century , islam had spread as far east as india and islamic rulers had solidified their power by establishing prosperous cities and a robust trade in decorative arts along the all-important silk road . this is a complex period with competing and overlapping cultures and empires . read below for an introduction to the later islamic dynasties . ottoman ( 1300-1924 ) at its earliest stages , the ottoman state was little more than a group formed as a result of the dissolution of the anatolian seljuq sultanate . however , in 1453 , the ottomans captured the great byzantine capital , constantinople , and in 1517 , they defeated the mamluks and took control of the most significant state in the islamic world . while the ottomans ruled for many centuries , the height of the empire 's cultural and economic prosperity was achieved during süleyman the magnificent 's reign ( r. 1520-1566 ) , a period often referred to as the ottoman 's ‘ golden age. ’ in addition to large-scale architectural projects , the decorative arts flourished , chief among them , ceramics , particularly tiles . iznik tiles , named for the city in anatolia where they were produced , developed a trademark style of curling vines and flowers rendered in beautiful shades of blue and turquoise . these designs were informed by the blue and white floral patterns found in chinese porcelain—similar to earlier mamluk tiles , and timurid art to the east . in addition to iznik , other artistic hubs developed , such as bursa , known for its silks , and cairo for its carpets . the capital , istanbul ( formerly constantinople ) , became a great center for all matters of cultural importance from manuscript illumination to architecture . the architecture of the period , both sacred and secular , incorporates these decorative arts , from the dazzling blue tiles and monumental calligraphy that adorn the walls of topkapi palace ( begun 1459 ) to the carpets that line the floors of the süleymaniye mosque ( 1550-1558 ) . ottoman mosque architecture itself is marked by the use of domes , widely used earlier in byzantium , and towering minarets . the byzantine influence draws primarily from hagia sophia , a former church that was converted into a mosque ( and is now a museum ) . timurid ( 1369-1502 ) this powerful central asian dynasty was named for its founder , tamerlane ( ruled 1370-1405 ) , which is derived from timur the lame . despite his rather pathetic epithet , he claimed to be a descendent of genghis khan and demonstrated some of his supposed ancestor ’ s ruthlessness in conquering neighboring territories . after establishing a vast empire , timur developed a monumental architecture befitting his power , and sought to make samarkand the “ pearl of the world. ” because the capital was situated at a major crossroads of the silk road ( the crucial trade route linking the middle east , central asia , and china ) , and because timur had conquered so widely , the timurids acquired a myriad of artisans and craftspeople from distinct artistic traditions . the resulting style synthesized aesthetic and design principles from as far away as india ( then hindustan ) and the lands in between . the result can be seen in cities filled with buildings created on a lavish scale that exhibited tall , bulbous domes and the finest ceramic tiles . the structures and even the cities themselves are often described foremost by the overwhelming use of blues and golds . while the timurid dynasty itself was short-lived , its legacy survives not only in the grand architecture that it left behind but in its descendents who went on to play significant roles in the ottoman , safavid and mughal empires . safavid ( 1502-1736 ) the safavids , a group with roots in the sufic tradition ( a mystical branch of islam ) , came to power in persia , modern-day iran and azerbaijan . in 1501 the safavid rulers declared shi ’ a islam as its state religion ; and in just ten years the empire came to include all of iran . the art of manuscript illumination was highly prized in the safavid courts , and royal patrons made many large-scale commissions . perhaps the most notable of these is the shahnama ( or ‘ book of kings , ’ a compilation of stories about earlier rulers of iran ) from the 1520s . while painting in this context did not have the same prominent and longstanding tradition as it does in western art , the illustrations exhibit masterful workmanship and an incredible attention to detail . trade in carpets was also important , and even today , people understand the appeal of persian carpets . these large-scale , high-quality pieces were created as luxurious furnishings for royal courts . the most famous—perhaps of all time—is a pair known as the ardabil carpets , created in 1539-1540 . the carpets were nearly identical , perfectly symmetrical and enormous . every inch of space was filled with flowers , scrolling vines , and medallions . the empire began to struggle financially and militarily until the rule of shah abbas ( r. 1587-1629 ) . he moved the capital to isfahan where he built a magnificent new city and established state workshops for textiles , which , along with silk and other goods , were increasingly exported to europe . the mosque architecture made use of earlier persian elements , like the four-iwan plan and building materials of brick and glazed tiles reminiscent of timurid architecture , with its blues and greens and bulbous domes . even in such far-removed lands , the connections between these dynasties are evident in the art they created . mughal ( 1526-1858 ) though islam had been introduced in india centuries before , the mughals were responsible for some of the greatest works of art produced in the canons of both indian and islamic art . the empire established itself when babur , himself a timurid prince of turkish and central asian descent , came to hindustan and defeated the existing islamic sultanate in delhi . tracing their roots to central asia , the mughals produced art , music and poetry that was highly influenced by persian and central asian aesthetics . this is evident in the style and importance given to miniature paintings , created to illustrate manuscripts . the most grandiose of these was the akbarnama , created to record the conquests of akbar , widely regarded as the greatest mughal emperor . the art and architecture created during his reign demonstrate a synthesis of indigenous indian temple architecture with structural and design elements derived from islamic sources farther west . the mughals developed a unique architectural style which , in the years after akbar ’ s reign , began to feature scalloped arches and stylized floral designs in white marble . the most famous example is the taj mahal , constructed by shah jahan from 1632-1653 . the mughal dynasty left a lasting mark on the landscape of india , and remained in power until the british completed their conquest of india in the nineteenth century . although historians generally agree that the major islamic dynasties end in the nineteenth and early twentieth centuries , islamic art and culture have continued to flourish . muslim artists and muslim countries are still producing art . some art historians consider such work as simply modern or contemporary art while others see it within the continuity of islamic art . essay by glenna barlow additional resources : the art of the ottomans before 1600 at the metropolitan museum of art 's timeline of art history the art of the safavids before 1600 at the metropolitan museum of art 's timeline of art history the art of the mughals before 1600 at the metropolitan museum of art 's timeline of art history introduction to the mughal empire from the bbc
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while the timurid dynasty itself was short-lived , its legacy survives not only in the grand architecture that it left behind but in its descendents who went on to play significant roles in the ottoman , safavid and mughal empires . safavid ( 1502-1736 ) the safavids , a group with roots in the sufic tradition ( a mystical branch of islam ) , came to power in persia , modern-day iran and azerbaijan . in 1501 the safavid rulers declared shi ’ a islam as its state religion ; and in just ten years the empire came to include all of iran .
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what is meant by the term `` mystical branch '' of islam ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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will science/medicine ever find a way to make us immune to every virus and bacteria ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein .
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can viruses be considered a separate kingdom from the main five kingdoms of living organisms ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry .
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what are receptors and how can they be present on the host cell ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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why is virus called biological puzzle ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox .
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how is a viral infection treated ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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is a virus technically alive ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce .
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how does the replication of viruses in our cells make us sick ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse .
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do all eukaryotes/prokaryotes permit the viruses to enter their cell membrane ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells !
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are viruses alive and living things ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms .
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also , if they do use the host 's polymerase , in the scenario where the virus has a single stranded dna , can it still be normally coded into rna ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ .
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what happens if we find a virus that can not be cured will we die or become living factories ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid .
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what is the difference between exocytosis and budding ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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assembly . new viral particles are assembled from the genome copies and viral proteins . release .
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what do viral proteins do when they are present in the host cell ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell .
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is it equivalent to the sialic acid released by influenza a virus ( not sure if exclusive to ) to embed in host cell membrane for haemagglutinin attachment and viral entry ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ?
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i know that there are many good bacteria so , is there any such thing as a good virus ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level .
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will we ever get information about all types of viruses and bacteria ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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assembly . new viral particles are assembled from the genome copies and viral proteins . release .
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the text only tells how viral proteins get synthesized , how do the replicate their dna or rna ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts .
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how do you prevent proliferation of viruses ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ?
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how can a virus be helpful or beneficial to its host ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . ''
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i have heard of viroids which cause some disease in potato ... .how are they different from a virus ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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can the same virus infect both plants and humans ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) .
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what are the common viruses that makes the host cell burst ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . ''
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what happens when two different types of viruses come in contact with each other or both come to infect a single cell ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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and why is virus called biological puzzle ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell .
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why is virus called biological puzzle ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein .
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can viruses be considered a separate kingdom from the main five kingdoms of living organisms ?
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key points : a virus is an infectious particle that reproduces by `` commandeering '' a host cell and using its machinery to make more viruses . a virus is made up of a dna or rna genome inside a protein shell called a capsid . some viruses have an internal or external membrane envelope . viruses are very diverse . they come in different shapes and structures , have different kinds of genomes , and infect different hosts . viruses reproduce by infecting their host cells and reprogramming them to become virus-making `` factories . '' introduction scientists estimate that there are roughly $ 10^\text { 31 } $ viruses at any given moment $ ^1 $ . that ’ s a one with $ 31 $ zeroes after it ! if you were somehow able to wrangle up all $ 10^\text { 31 } $ of these viruses and line them end-to-end , your virus column would extend nearly $ 200 $ light years into space . to put it another way , there are over ten million more viruses on earth than there are stars in the entire universe $ ^2 $ . does that mean there are $ 10^\text { 31 } $ viruses just waiting to infect us ? actually , most of these viruses are actually found in oceans , where they attack bacteria and other microbes $ ^3 $ . it may seem odd that bacteria can get a virus , but scientists think that every kind of living organism is probably host to at least one virus ! what is a virus ? a virus is an tiny , infectious particle that can reproduce only by infecting a host cell . viruses `` commandeer '' the host cell and use its resources to make more viruses , basically reprogramming it to become a virus factory . because they ca n't reproduce by themselves ( without a host ) , viruses are not considered living . nor do viruses have cells : they 're very small , much smaller than the cells of living things , and are basically just packages of nucleic acid and protein . still , viruses have some important features in common with cell-based life . for instance , they have nucleic acid genomes based on the same genetic code that 's used in your cells ( and the cells of all living creatures ) . also , like cell-based life , viruses have genetic variation and can evolve . so , even though they do n't meet the definition of life , viruses seem to be in a `` questionable '' zone . ( maybe viruses are actually undead , like zombies or vampires ! ) how are viruses different from bacteria ? even though they can both make us sick , bacteria and viruses are very different at the biological level . bacteria are small and single-celled , but they are living organisms that do not depend on a host cell to reproduce . because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses . the diameter of a typical virus is about $ 20 $ $ \mbox { - } $ $ 300 $ $ \text { nanometers } $ ( $ 1 $ $ \text { nm } $ $ = $ $ 10^\text { -9 } $ $ \text { m } $ ) $ ^4 $ . this is considerably smaller than a typical e. coli bacterium , which has a diameter of roughly $ 1000 $ $ \text { nm } $ ! tens of millions of viruses could fit on the head of a pin . the structure of a virus there are a lot of different viruses in the world . so , viruses vary a ton in their sizes , shapes , and life cycles . if you 're curious just how much , i recommend playing around with the viralzone website . click on a few virus names at random , and see what bizarre shapes and features you find ! viruses do , however , have a few key features in common . these include : a protective protein shell , or capsid a nucleic acid genome made of dna or rna , tucked inside of the capsid a layer of membrane called the envelope ( some but not all viruses ) let 's take a closer look at these features . virus capsids the capsid , or protein shell , of a virus is made up of many protein molecules ( not just one big , hollow one ) . the proteins join to make units called capsomers , which together make up the capsid . capsid proteins are always encoded by the virus genome , meaning that it ’ s the virus ( not the host cell ) that provides instructions for making them . capsids come in many forms , but they often take one of the following shapes ( or a variation of these shapes ) : icosahedral – icosahedral capsids have twenty faces , and are named after the twenty-sided shape called an icosahedron . filamentous – filamentous capsids are named after their linear , thin , thread-like appearance . they may also be called rod-shaped or helical . head-tail –these capsids are kind of a hybrid between the filamentous and icosahedral shapes . they basically consist of an icosahedral head attached to a filamentous tail . virus envelopes in addition to the capsid , some viruses also have a lipid membrane known as an envelope . virus envelopes can be external , surrounding the entire capsid , or internal , found beneath the capsid . viruses with envelopes do not provide instructions for the envelope lipids . instead , they `` borrow '' a patch from the host membranes on their way out of the cell . envelopes do , however , contain proteins that are specified by the virus , which often help viral particles bind to host cells . although envelopes are common , especially among animal viruses , they are not found in every virus ( i.e. , are not a universal virus feature ) . virus genomes all viruses have genetic material ( a genome ) made of nucleic acid . you , like all other cell-based life , use dna as your genetic material . viruses , on the other hand , may use either rna or dna , both of which are types of nucleic acid . we often think of dna as double-stranded and rna as single-stranded , since that 's typically the case in our own cells . however , viruses can have all possible combos of strandedness and nucleic acid type ( double-stranded dna , double-stranded rna , single-stranded dna , or single-stranded rna ) . viral genomes also come in various shapes , sizes , and varieties , though they are generally much smaller than the genomes of cellular organisms . notably , dna and rna viruses always use the same genetic code as living cells . if they did n't , they would have no way to reprogram their host cells ! what is a viral infection ? in everyday life , we tend to think of a viral infection as the nasty collection of symptoms we get when catch a virus , such as the flu or the chicken pox . but what 's actually happening in your body when you have a virus ? at the microscopic scale , a viral infection means that many viruses are using your cells to make more copies of themselves . the viral lifecycle is the set of steps in which a virus recognizes and enters a host cell , `` reprograms '' the host by providing instructions in the form of viral dna or rna , and uses the host 's resources to make more virus particles ( the output of the viral `` program '' ) . for a typical virus , the lifecycle can be divided into five broad steps ( though the details of these steps will be different for each virus ) : attachment . the virus recognizes and binds to a host cell via a receptor molecule on the cell surface . entry . the virus or its genetic material enters the cell . genome replication and gene expression . the viral genome is copied and its genes are expressed to make viral proteins . assembly . new viral particles are assembled from the genome copies and viral proteins . release . completed viral particles exit the cell and can infect other cells . the diagram above shows how these steps might occur for a virus with a single-stranded rna genome . you can see real examples of viral lifecycles in the articles on bacteriophages ( bacteria-infecting viruses ) and animal viruses .
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because of these differences , bacterial and viral infections are treated very differently . for instance , antibiotics are only helpful against bacteria , not viruses . bacteria are also much bigger than viruses .
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why it 's so that bacteria have treatment through antibiotics but viruses a n't ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ .
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also , in example one , when they show what the plane looks like using vector notation , should n't the second vector components be x-8 not x , and y-4 , not y and etc ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing .
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`` ... we consider all inputs to be part of a vector x ... '' should vector x have components x.y and so on instead of x0 , y0 and so on ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function .
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is `` the local linearization '' the same as `` linear approximation '' ( https : //en.wikipedia.org/wiki/linear_approximation ) ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers .
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can this be edited to include an example involving the linear localization of a vector valued function ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) .
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maybe something simple like from $ \mathbb { r } ^2 \to \mathbb { r } ^3 $ ?
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background the gradient what we 're building to local linearization generalizes the idea of tangent planes to any multivariable function . here , i will just talk about the case of scalar-valued multivariable functions . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ . tangent planes as approximations in the previous article , i talked about finding the tangent plane to a two-variable function 's graph . the formula for the tangent plane ended up looking like this . $ \begin { align } \quad t ( x , y ) = f ( x_0 , y_0 ) + f_x ( x_0 , y_0 ) ( x-x_0 ) + { f_y ( x_0 , y_0 ) } ( y-y_0 ) \end { align } $ this function $ t ( x , y ) $ often goes by a different name : the `` local linearization '' of $ f $ at the point $ ( x_0 , y_0 ) $ . you can think about this as the simplest function satisfying two properties : it has the same value of $ f $ at the point $ ( x_0 , y_0 ) $ . it has the same partial derivatives as $ f $ at the point $ ( x_0 , y_0 ) $ . as always in multivariable calculus , it is healthy to contemplate a new concept without relying on graphical intuition . that 's not to say you should not try to think visually . maybe instead think purely about the input space , or think relevant transformation rather than the graph . fundamentally , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . in the case of functions with a two-variable input and a scalar ( i.e . non-vector ) output , this can be visualized as a tangent plane . however , with higher dimensions we do n't have this visual luxury , so we are left to think about it just as an approximation . in real-world applications of multivariable calculus , you almost never care about an actual plane in space . instead , you might have some complicated function , like , oh , i do n't know , air resistance on a parachute as a function of speed and orientation . dealing with the actual function may be tricky or computationally expensive , so it 's helpful to approximate it with something simpler , like a linear function . what do i mean by `` linear function '' ? consider a function with a multidimensional input . $ f ( x_1 , x_2 , \dots , x_n ) $ this function is called linear if in its definition , all the coordinates are just multiplied by constants , with nothing else happening to them . for example , it might look like this : $ f ( x_1 , x_2 , \dots , x_n ) = 2x_1 + 3x_2 + \cdots - 5x_n $ the full story of linearity goes deeper ( hence the existence of the field `` linear algebra '' ) , but for now , this conception will do . typically , instead of writing out all the variable like this , you would treat the input as a vector : $ \textbf { x } = \left [ \begin { array } { c } x_1 \ x_2 \ \vdots \ x_n \end { array } \right ] $ and you would define the function using a dot product : $ f ( \textbf { x } ) = \left [ \begin { array } { c } 2 \ 3 \ \vdots \ -5 \end { array } \right ] \cdot \textbf { x } $ for the purposes of this article , and more generally when you talk about local linearization , you are allowed to add in a constant to this expression : $ f ( \textbf { x } ) = ! ! ! ! ! ! ! \underbrace { c } _ { \text { some constant } } ! ! ! ! ! ! ! + ! ! ! ! ! \overbrace { \textbf { v } } ^ { \text { some vector } } ! ! ! ! ! \cdot \textbf { x } $ if you wanted to be pedantic , this is no longer a linear function . it 's what 's called an `` affine '' function . but most people would say `` whatever , it 's basically linear '' . local linearization now , suppose your function $ f ( \textbf { x } ) $ does not have the luxury of being linear . ( the bolded `` $ \textbf { x } $ '' still represents a multidimensional vector ) . it might be defined by some crazy expression way more wild than a dot product . the idea of a local linearization is to approximate this function near some particular input value , $ \textbf { x } _0 $ , with a function that is linear . specifically , here 's what that new function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ notice , by plugging in $ \textbf { x } = \textbf { x } _0 $ , you can see that both functions $ f $ and $ l_f $ will have the same value at the input $ \textbf { x } _0 $ . the vector dotted against the variable $ \textbf { x } $ is the gradient of $ f $ at the specified input , $ \nabla f ( \textbf { x } _0 ) $ . this ensures that both functions $ f $ and $ l_f $ will have the same gradient at the specified input . in other words , all their partial derivative information will be the same . i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function . which of the following functions will be guaranteed to equal $ f $ at the input $ ( x , y , z ) = ( 8 , 4 , 3 ) $ ? the partial derivatives of $ l_f $ , as you have written it so far , are precisely these constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ . so to force our function to have the same partial derivative information as $ f $ at the point $ ( 8 , 4 , 3 ) $ , we just need to set these constants equal to the corresponding partial derivatives of $ f $ at this point . step 3 : compute each partial derivative of $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ now we evaluate each of these at $ ( 8 , 4 , 3 ) $ . step 4 : replacing the constants $ \blued { a } $ , $ \greene { b } $ and $ \redd { c } $ in the expression of $ l_f $ with these partial derivative values , what do you get ? now notice what this looks like if you write it with vector notation . it is just a specific form of the general formula shown above . $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ example 2 : using local linearization for estimation what follows is by no means a practical application , but working through it will help give a feel for what local linearization is doing . problem : suppose you are on a desert island without a calculator , and you need to estimate $ \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } $ . how would you do it ? solution : we can view this problem as evaluating a certain three-variable function at the point $ ( 2.01 , 0.99 , 9.01 ) $ , namely $ \quad f ( x , y , z ) = \sqrt { x + \sqrt { y + \sqrt { z } } } $ i do n't know about you , but i 'm not sure how to evaluate square roots by hand . if only this function was linear ! then working it out by hand would only involve adding and multiplying numbers . what we could do is find the local linearization at a nearby point where evaluating $ f $ is easier . then we can get close to the right answer by evaluating the linearization at the point $ ( 2.01 , 0.99 , 9.01 ) $ . the point we care about is very close to the much simpler point $ ( 2 , 1 , 9 ) $ , so we find the local linearization of $ f $ near that point . as before , we must find $ f ( 2 , 1 , 9 ) $ all partial derivatives of $ f $ at $ ( 2 , 1 , 9 ) $ the first of these is $ \begin { align } \quad f ( 2 , 1 , 9 ) & amp ; = \sqrt { 2 + \sqrt { 1 + \sqrt { 9 } } } \ & amp ; = \sqrt { 2 + \sqrt { 1 + 3 } } \ & amp ; = \sqrt { 2 + \sqrt { 4 } } \ & amp ; = \sqrt { 2+2 } \ & amp ; = \sqrt { 4 } \ & amp ; = 2 \end { align } $ looks like someone chose a few convenient input values , eh ? on to the partial derivatives ( heavy sigh ) . since the square roots are abundant , let 's write out for ourselves the derivative of $ \sqrt { x } $ . $ \begin { align } \quad \dfrac { d } { dx } \sqrt { x } & amp ; = \dfrac { d } { dx } x^ { \frac { 1 } { 2 } } = \dfrac { 1 } { 2 } x^ { -\frac { 1 } { 2 } } = \dfrac { 1 } { 2\sqrt { x } } \end { align } $ okay , here we go . the simplest partial derivative is $ f_x $ $ \begin { align } \quad f_x & amp ; = \dfrac { \partial } { \partial \blued { x } } \sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { \blued { x } + \sqrt { y + \sqrt { z } } } } \ \end { align } $ since $ y $ is nestled in there , $ f_y $ requires some chain rule action : $ \begin { align } \quad f_y & amp ; = \dfrac { \partial } { \partial \redd { y } } \sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { \redd { y } + \sqrt { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \redd { y } + \sqrt { z } } } \ \end { align } $ nestled even deeper , that tricky $ z $ will require two iterations of the chain rule : $ \begin { align } \quad f_z & amp ; = \dfrac { \partial } { \partial \greend { z } } \sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } = \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { \greend { z } } } } } \cdot \dfrac { 1 } { 2\sqrt { y + \sqrt { \greend { z } } } } \cdot \dfrac { 1 } { 2\sqrt { \greend { z } } } \end { align } $ next , evaluate each one of these at $ ( 2 , 1 , 9 ) $ . this might seem like a lot , but they are all made up of the same three basic components : $ \begin { align } \quad \dfrac { 1 } { 2\sqrt { x + \sqrt { y + \sqrt { z } } } } & amp ; = \dfrac { 1 } { 2\sqrt { 2 + \sqrt { 1+\sqrt9 } } } = \dfrac { 1 } { 2\sqrt { 2+2 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { y + \sqrt { z } } } & amp ; = \dfrac { 1 } { 2\sqrt { 1 + \sqrt { 9 } } } = \dfrac { 1 } { 2\sqrt { 4 } } = \dfrac { 1 } { 4 } \ \dfrac { 1 } { 2\sqrt { z } } & amp ; = \dfrac { 1 } { 2\sqrt { 9 } } = \dfrac { 1 } { 6 } \ \end { align } $ plugging these values into our expressions for the partial derivatives , we have $ \begin { align } \quad f_x ( 2 , 1 , 9 ) & amp ; = \blued { \dfrac { 1 } { 4 } } \ \ f_y ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } = \redd { \dfrac { 1 } { 16 } } \ \ f_z ( 2 , 1 , 9 ) & amp ; = \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 4 } \cdot \dfrac { 1 } { 6 } = \greend { \dfrac { 1 } { 96 } } \ \end { align } $ unraveling the formula for local linearization , we get $ \begin { align } \quad l_f ( \textbf { x } ) & amp ; = f ( \textbf { x } _0 ) + \nabla f ( \textbf { x } _0 ) \cdot ( \textbf { x } - \textbf { x } _0 ) \ \ & amp ; = f ( \textbf { x } _0 ) + \blued { f_x ( \textbf { x } _0 ) ( x - x_0 ) } + \redd { f_y ( \textbf { x } _0 ) } ( y - y_0 ) + \greend { f_z ( \textbf { x } _0 ) } ( z - z_0 ) \ \ & amp ; = \boxed { 2 + \blued { \dfrac { 1 } { 4 } } ( x - 2 ) + \redd { \dfrac { 1 } { 16 } } ( y - 1 ) + \greend { \dfrac { 1 } { 96 } } ( z - 9 ) } \end { align } $ finally , after all this work , we can plug in $ ( x , y , z ) = ( 2.01 , 0.99 , 9.01 ) $ to compute our approximation $ \begin { align } \quad & amp ; \quad 2 + \dfrac { 1 } { 4 } ( 2.01 - 2 ) + \dfrac { 1 } { 16 } ( 0.99 - 1 ) + \dfrac { 1 } { 96 } ( 9.01 - 9 ) \ & amp ; \ & amp ; = 2 + \dfrac { 0.01 } { 4 } + \dfrac { -0.01 } { 16 } + \dfrac { 0.01 } { 96 } \end { align } $ calculating this by hand still is n't easy , but at least it 's doable . when you work it out , the final answer is $ \quad \large \boxed { 2.001979 } $ had we just used a calculator , the answer is $ \quad \sqrt { 2.01 + \sqrt { 0.99 + \sqrt { 9.01 } } } \approx \large \boxed { 2.001978 } $ so our approximation is pretty good ! why do we care ? although it is not common to find yourself estimating square roots on a desert island ( at least where i 'm from ) , what is common in the contexts of math and engineering is wrangling with complicated but differentiable functions . the phrase `` just linearize it '' is tossed around so much that not knowing what it means could be awkward . remember , a local linearization approximates one function near a point based on the information you can get from its derivative ( s ) at that point . even though you can use a computer to evaluate functions , that 's not always enough . you might need to evaluate it many thousands of times per second , and working it out in full takes too long . maybe you do n't even have the function explicitly written out , and you just have a few measurements near a point which you wish to extrapolate . sometimes what you care about is the inverse function , which can be hard or even impossible to find for the function as a whole , whereas inverting linear functions is relatively straight-forward . summary local linearization generalizes the idea of tangent planes to any multivariable function . the idea is to approximate a function near one of its inputs with a simpler function that has the same value at that input , as well as the same partial derivative values . written with vectors , here 's what the approximation function looks like : $ l_f ( \textbf { x } ) = \underbrace { f ( \textbf { x } 0 ) } { \text { constant } } + \underbrace { \nabla f ( \textbf { x } 0 ) } { \text { constant vector } } ! ! ! ! \cdot \overbrace { ( \textbf { x } -\textbf { x } _0 ) } ^ { \textbf { x } \text { is the variable } } $ this is called the local linearization of $ f $ near $ \textbf { x } _0 $ .
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i think the best way to understand this formula is to basically derive it for yourself in the context of a specific function . example 1 : finding a local linearization . problem : have yourself a function : $ f ( x , y , z ) = ze^ { x^2 - y^3 } $ find a linear function $ l_f ( x , y , z ) $ such that the value of $ l_f $ and all its partial derivatives match those of $ f $ at the following point : $ ( x_0 , y_0 , z_0 ) = ( 8 , 4 , 3 ) $ step 1 : evaluate $ f $ at the chosen point step 2 : use this to start writing your function .
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why does n't the partial with respect to z in the last example ( desert island ) have a coefficient of 1/8 ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use .
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why is n't `` n '' upper case ( n ) since this is a population and not a sample ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances .
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also , why use the sample mean symbol vs mu ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population .
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what are the steps to finding the square root of 3.5 ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use .
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is there a way to differentiate when to use the population and when to use the sample ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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why do we use two different types of standard deviation in the first place when the goal of both is the same ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use .
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why does the formula show n and not n-1 ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . }
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if i have a set of data with repeating values , say 2,3,4,6,6,6,9 , would you take the sum of the squared distance for all 7 points or would you only add the 5 different values ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high .
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hi , how do i calculate the standard deviation of bivariate data by hand ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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how would i calculate the standard deviation without given just a random data set ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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what is the unit of the result of standard deviation ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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what are the steps to standard deviation with this data ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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so , we usually weigh the samples after collection , what is the acceptable standard deviation in this kind of research ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points .
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when do we need to remove a sample from the study because of high standard deviation ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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what is that criterion for standard deviation ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ .
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what is standard deviation by definition ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root .
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if i am calculating standard deviation for a problem where i have the number of data points and the frequency , would the n be equal to the number of data points oe the frequency ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ .
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in the formula what do the `` | '' mean ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use .
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what is the difference between population and sample variance ?
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introduction in this article , we 'll learn how to calculate standard deviation `` by hand '' . interestingly , in the real world no statistician would ever calculate standard deviation by hand . the calculations involved are somewhat complex , and the risk of making a mistake is high . also , calculating by hand is slow . very slow . this is why statisticians rely on spreadsheets and computer programs to crunch their numbers . so what 's the point of this article ? why are we taking time to learn a process statisticians do n't actually use ? the answer is that learning to do the calculations by hand will give us insight into how standard deviation really works . this insight is valuable . instead of viewing standard deviation as some magical number our spreadsheet or computer program gives us , we 'll be able to explain where that number comes from . overview of how to calculate standard deviation the formula for standard deviation ( sd ) is $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ where $ \sum $ means `` sum of '' , $ x $ is a value in the data set , $ \mu $ is the mean of the data set , and $ n $ is the number of data points in the population . the standard deviation formula may look confusing , but it will make sense after we break it down . in the coming sections , we 'll walk through a step-by-step interactive example . here 's a quick preview of the steps we 're about to follow : step 1 : find the mean . step 2 : for each data point , find the square of its distance to the mean . step 3 : sum the values from step 2 . step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population . if you 're dealing with a sample , you 'll want to use a slightly different formula ( below ) , which uses $ n-1 $ instead of $ n $ . the point of this article , however , is to familiarize you with the the process of computing standard deviation , which is basically the same no matter which formula you use . $ \text { sd } \text { sample } = \sqrt { \dfrac { \sum\limits { } ^ { } { { \lvert x-\bar { x } \rvert^2 } } } { n-1 } } $ step-by-step interactive example for calculating standard deviation first , we need a data set to work with . let 's pick something small so we do n't get overwhelmed by the number of data points . here 's a good one : $ 6 , 2 , 3 , 1 $ step 1 : finding $ \goldd { \mu } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\goldd { \mu } \rvert^2 } } } { n } } $ in this step , we find the mean of the data set , which is represented by the variable $ \mu $ . step 2 : finding $ \goldd { \lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { \goldd { { \lvert x-\mu } \rvert^2 } } } { n } } $ in this step , we find the distance from each data point to the mean ( i.e. , the deviations ) and square each of those distances . for example , the first data point is $ 6 $ and the mean is $ 3 $ , so the distance between them is $ 3 $ . squaring this distance gives us $ 9 $ . step 3 : finding $ \goldd { \sum\lvert x - \mu \rvert^2 } $ in $ \sqrt { \dfrac { \goldd { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } } { n } } $ the symbol $ \sum $ means `` sum '' , so in this step we add up the four values we found in step 2 . step 4 : finding $ \goldd { \dfrac { \sum\lvert x - \mu \rvert^2 } { n } } $ in $ \sqrt { \goldd { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu } \rvert^2 } } { n } } } $ in this step , we divide our result from step 3 by the variable $ n $ , which is the number of data points . step 5 : finding the standard deviation $ \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $ we 're almost finished ! just take the square root of the answer from step 4 and we 're done . yes ! we did it ! we successfully calculated the standard deviation of a small data set . summary of what we did we broke down the formula into five steps : step 1 : find the mean $ \mu $ . $ \mu = \dfrac { 6+2 + 3 + 1 } { 4 } = \dfrac { 12 } { 4 } = \blued3 $ step 2 : find the square of the distance from each data point to the mean $ \lvert x-\mu\rvert^2 $ . $ x $ | | $ \lvert x - \mu \rvert^2 $ : - : | | : - $ 6 $ || $ \lvert6-\blued { 3 } \rvert^2 = 3^2 = 9 $ $ 2 $ | | $ \lvert2-\blued { 3 } \rvert^2 = 1^2 = 1 $ $ 3 $ | | $ \lvert3-\blued { 3 } \rvert^2 = 0^2 = 0 $ $ 1 $ | | $ \lvert1-\blued { 3 } \rvert^2 = 2^2 = 4 $ steps 3 , 4 , and 5 : $ \begin { align } \text { sd } & amp ; = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } \\\\ & amp ; = \sqrt { \dfrac { 9 + 1 + 0 + 4 } { 4 } } \\\\ & amp ; = \sqrt { \dfrac { { 14 } } { 4 } } ~~~~~~~~\small \text { sum the squares of the distances ( step 3 ) . } \\\\ & amp ; = \sqrt { { 3.5 } } ~~~~~~~~\small \text { divide by the number of data points ( step 4 ) . } \\\\ & amp ; \approx 1.87 ~~~~~~~~\small \text { take the square root ( step 5 ) . } \end { align } $ try it yourself here 's a reminder of the formula : $ \large\text { sd } = \sqrt { \dfrac { \sum\limits_ { } ^ { } { { \lvert x-\mu\rvert^2 } } } { n } } $
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step 4 : divide by the number of data points . step 5 : take the square root . an important note the formula above is for finding the standard deviation of a population .
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why do you have to do the square thing ?
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a prolific career it would be difficult to overestimate the importance of rembrandt harmenszoon van rijn within the history of western art . indeed , rembrandt is considered one of the foremost artists of the dutch baroque period , and even if he had never picked up a paintbrush , he would have been famous both in his day and ours as a printmaker of particular brilliance and as a prolific teacher . in a career that lasted nearly forty years , rembrandt completed approximately 400 paintings , more than 1,000 drawings , and nearly 300 engravings . although he spent his entire life north of the alps , had he been italian and lived a century or so earlier , he likely would have joined his italian brethren—donatello , leonardo , michelangelo , and raphael , as a member of the famed cartoon teenage mutant ninja turtles . patrons : a wealthy , protestant , and expanding middle class but as time and place would have it , rembrandt was neither italian nor a part of the renaissance . instead , rembrandt was born in leiden in 1606 . this place and time—holland during the height of the expansion of the wealthy mercantile class during the middle half of the seventeenth century—served rembrandt well through his long career . the catholic church often commissioned italian artists at this time to undertake large-scale projects to promote religious ideology in support of the counter-reformation . without the catholic church in holland to commission art , rembrandt and his fellow dutch artists were lavishly supported by a wealthy , protestant , and expanding middle class . this group of patrons enthusiastically commissioned works of art with their increasing discretionary income . new subjects ( including group portraits ) many different types of art became popular during the dutch baroque period . genre paintings—small paintings of everyday life—were exceptionally popular with a middle-class clientele , as were still lifes , landscapes , and prints . the majority of these kinds of art were both affordable and small enough to be easily displayed within an average home . larger and more compositionally complicated , group portraiture also became popular in holland during the seventeenth century . this was a mode of painting that was often placed in a public space so that the image could promote a particular organization . relocating to amsterdam although several of rembrandt ’ s most well-known paintings are group portraits—the anatomy lesson of dr. tulp among others—his early education in leiden , first at a latin school and then later at the university , suggest that he was destined for a vocation other than art . however , by the time he was sixteen he decided he wanted to be a painter and a draughtsman . after finding quick success in leiden during the 1620s , rembrandt relocated to amsterdam in 1631 , a wise professional decision , as this was then one of the wealthiest and largest cities in europe . a group portrait for the amsterdam surgeon 's guild just a year after his arrival , rembrandt was offered the commission to complete a group portrait of the amsterdam surgeon ’ s guild , an image that in time has come to be known rather simply as the anatomy lesson of dr. tulp . it is remarkable that rembrandt received this commission as a newcomer to amsterdam when there were other native-born artists available . thomas de keyser ( above ) and nicolaes pickenoy ( below ) , for example , were older and more experienced in the realm of group portraiture . whereas another artist may have simply recreated a previous group image—inserting new heads in place of old ones—rembrandt created something new , and in doing so , completed one of the most recognizable images in the history of painting . dr. nicolaes tulp dr. nicolaes tulp was appointed praelector ( like a professor or lecturer ) of the amsterdam anatomy guild in 1628 . one of the responsibilities of this position was to deliver a yearly public lecture on some aspect of human anatomy . the lecture in 1632 occurred on 16 january , and this is the scene that rembrandt depicts in paint in the anatomy of lesson of dr. tulp . this is a more complicated composition than it at first appears . understandably , the focal point of the image is dr. tulp , the doctor who is shown displaying the flexors of the cadaver ’ s left arm . rembrandt notes the doctor ’ s significance by showing him as the only person who wears a hat . seven colleagues surround dr. tulp , and they look in a variety of directions—some gaze at the cadaver , some stare at the lecturer , and some peek directly at the viewer . each face displays a facial expression that is deeply personal and psychological . the cadaver—a recently executed thief named adriaen adriaenszoon—lies nearly parallel to the picture plane . viewing the illuminated body from his head to his feet brings into focus a book—likely andreas vesalius ’ s de humani corporis fabrica ( fabric of the human body , 1543 ) —propped up in the lower right corner . in all , rembrandt shows nine distinct figures , but does so as if they are a unified group . a comparison comparing the anatomy lesson of dr. tulp to a somewhat similar example , the osteology lesson of dr. sebastiaen egbertszoon ( above ) , shows just how different and novel rembrandt ’ s composition was at the time . the sebastiaen erbertszoon painting is a series of six portraits that surround a single human skeleton ; but neither the heads nor the bodies seem to interact with one another in a real or coherent way . in contrast , the figures in rembrandt ’ s tulp seem to truly be a group , one collection of nine rather than nine individuals . if the composition is different from what rembrandt might have seen in amsterdam , the choice of subject is different than what would have been expected in the parts of europe that were catholic . the catholic tenet of resurrection necessitated that dead bodies be interred in a state of wholeness , and this fact explains why leonardo was forced to dissect human bodies in secret . in protestant holland but 113 years after leonardo ’ s death , however , human dissections were not only common practice , they were often public spectacles , complete with food and wine , music and conversation . artistic license if rembrandt was able to create a truly group portrait—one of a single group rather than a collection of individuals—it is important to note that the artist took some understandable artistic license with some parts of the composition . as any anatomy and physiology student today can attest , a dissection of the human body almost always commences with an exploration of the chest and abdominal areas , parts of the human body most likely to decompose first , and only later does the procedure move onwards to the limbs . moreover , it would have been unlikely that a doctor of tulp ’ s importance would have actually dissected the body ; instead , he would have lectured while the menial task of exposing the inner workings of the body would have been left to others . but in paint , a format without sound , rembrandt put tulp in charge not only in costume , but also in action . as the prominent signature in the upper part of the painting indicates , rembrandt was justifiably proud of this large painting . whereas he had previously signed his works with his monogram rhl ( rembrandt harmenszoon of leiden ) , the anatomy lesson of dr. tulp contains rembrant . f [ ecit ] 1632 . this painting and the latin announcement that “ rembrandt made it ” marks the beginning of the painter ’ s mature career . daring , compositionally innovative , and deeply psychological , the anatomy lesson of dr. tulp launched rembrandt to fame and wealth and influenced generations of artists to come . indeed , without tulp , it seems impossible for thomas eakins to have painted the gross clinic , 1876 almost two and a half centuries later . essay by dr. bryan zygmont additional resources : this painting at the mauritshuis dolores mitchell . `` rembrandt 's `` the anatomy lesson of dr. tulp '' : a sinner among the righteous , '' artibus et historiae , vol . 15 , no . 30 ( nov 1994 ) , p. 145 - 156 aloïs riegl and benjamin binstock . `` excertps from `` the dutch group portrait '' , '' october , vol . 74 , ( autumn 1995 ) , p. 3 - 35
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artistic license if rembrandt was able to create a truly group portrait—one of a single group rather than a collection of individuals—it is important to note that the artist took some understandable artistic license with some parts of the composition . as any anatomy and physiology student today can attest , a dissection of the human body almost always commences with an exploration of the chest and abdominal areas , parts of the human body most likely to decompose first , and only later does the procedure move onwards to the limbs . moreover , it would have been unlikely that a doctor of tulp ’ s importance would have actually dissected the body ; instead , he would have lectured while the menial task of exposing the inner workings of the body would have been left to others .
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after anatomy lessons were complete , did the doctor 's have to pay for burial of the body ?
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a prolific career it would be difficult to overestimate the importance of rembrandt harmenszoon van rijn within the history of western art . indeed , rembrandt is considered one of the foremost artists of the dutch baroque period , and even if he had never picked up a paintbrush , he would have been famous both in his day and ours as a printmaker of particular brilliance and as a prolific teacher . in a career that lasted nearly forty years , rembrandt completed approximately 400 paintings , more than 1,000 drawings , and nearly 300 engravings . although he spent his entire life north of the alps , had he been italian and lived a century or so earlier , he likely would have joined his italian brethren—donatello , leonardo , michelangelo , and raphael , as a member of the famed cartoon teenage mutant ninja turtles . patrons : a wealthy , protestant , and expanding middle class but as time and place would have it , rembrandt was neither italian nor a part of the renaissance . instead , rembrandt was born in leiden in 1606 . this place and time—holland during the height of the expansion of the wealthy mercantile class during the middle half of the seventeenth century—served rembrandt well through his long career . the catholic church often commissioned italian artists at this time to undertake large-scale projects to promote religious ideology in support of the counter-reformation . without the catholic church in holland to commission art , rembrandt and his fellow dutch artists were lavishly supported by a wealthy , protestant , and expanding middle class . this group of patrons enthusiastically commissioned works of art with their increasing discretionary income . new subjects ( including group portraits ) many different types of art became popular during the dutch baroque period . genre paintings—small paintings of everyday life—were exceptionally popular with a middle-class clientele , as were still lifes , landscapes , and prints . the majority of these kinds of art were both affordable and small enough to be easily displayed within an average home . larger and more compositionally complicated , group portraiture also became popular in holland during the seventeenth century . this was a mode of painting that was often placed in a public space so that the image could promote a particular organization . relocating to amsterdam although several of rembrandt ’ s most well-known paintings are group portraits—the anatomy lesson of dr. tulp among others—his early education in leiden , first at a latin school and then later at the university , suggest that he was destined for a vocation other than art . however , by the time he was sixteen he decided he wanted to be a painter and a draughtsman . after finding quick success in leiden during the 1620s , rembrandt relocated to amsterdam in 1631 , a wise professional decision , as this was then one of the wealthiest and largest cities in europe . a group portrait for the amsterdam surgeon 's guild just a year after his arrival , rembrandt was offered the commission to complete a group portrait of the amsterdam surgeon ’ s guild , an image that in time has come to be known rather simply as the anatomy lesson of dr. tulp . it is remarkable that rembrandt received this commission as a newcomer to amsterdam when there were other native-born artists available . thomas de keyser ( above ) and nicolaes pickenoy ( below ) , for example , were older and more experienced in the realm of group portraiture . whereas another artist may have simply recreated a previous group image—inserting new heads in place of old ones—rembrandt created something new , and in doing so , completed one of the most recognizable images in the history of painting . dr. nicolaes tulp dr. nicolaes tulp was appointed praelector ( like a professor or lecturer ) of the amsterdam anatomy guild in 1628 . one of the responsibilities of this position was to deliver a yearly public lecture on some aspect of human anatomy . the lecture in 1632 occurred on 16 january , and this is the scene that rembrandt depicts in paint in the anatomy of lesson of dr. tulp . this is a more complicated composition than it at first appears . understandably , the focal point of the image is dr. tulp , the doctor who is shown displaying the flexors of the cadaver ’ s left arm . rembrandt notes the doctor ’ s significance by showing him as the only person who wears a hat . seven colleagues surround dr. tulp , and they look in a variety of directions—some gaze at the cadaver , some stare at the lecturer , and some peek directly at the viewer . each face displays a facial expression that is deeply personal and psychological . the cadaver—a recently executed thief named adriaen adriaenszoon—lies nearly parallel to the picture plane . viewing the illuminated body from his head to his feet brings into focus a book—likely andreas vesalius ’ s de humani corporis fabrica ( fabric of the human body , 1543 ) —propped up in the lower right corner . in all , rembrandt shows nine distinct figures , but does so as if they are a unified group . a comparison comparing the anatomy lesson of dr. tulp to a somewhat similar example , the osteology lesson of dr. sebastiaen egbertszoon ( above ) , shows just how different and novel rembrandt ’ s composition was at the time . the sebastiaen erbertszoon painting is a series of six portraits that surround a single human skeleton ; but neither the heads nor the bodies seem to interact with one another in a real or coherent way . in contrast , the figures in rembrandt ’ s tulp seem to truly be a group , one collection of nine rather than nine individuals . if the composition is different from what rembrandt might have seen in amsterdam , the choice of subject is different than what would have been expected in the parts of europe that were catholic . the catholic tenet of resurrection necessitated that dead bodies be interred in a state of wholeness , and this fact explains why leonardo was forced to dissect human bodies in secret . in protestant holland but 113 years after leonardo ’ s death , however , human dissections were not only common practice , they were often public spectacles , complete with food and wine , music and conversation . artistic license if rembrandt was able to create a truly group portrait—one of a single group rather than a collection of individuals—it is important to note that the artist took some understandable artistic license with some parts of the composition . as any anatomy and physiology student today can attest , a dissection of the human body almost always commences with an exploration of the chest and abdominal areas , parts of the human body most likely to decompose first , and only later does the procedure move onwards to the limbs . moreover , it would have been unlikely that a doctor of tulp ’ s importance would have actually dissected the body ; instead , he would have lectured while the menial task of exposing the inner workings of the body would have been left to others . but in paint , a format without sound , rembrandt put tulp in charge not only in costume , but also in action . as the prominent signature in the upper part of the painting indicates , rembrandt was justifiably proud of this large painting . whereas he had previously signed his works with his monogram rhl ( rembrandt harmenszoon of leiden ) , the anatomy lesson of dr. tulp contains rembrant . f [ ecit ] 1632 . this painting and the latin announcement that “ rembrandt made it ” marks the beginning of the painter ’ s mature career . daring , compositionally innovative , and deeply psychological , the anatomy lesson of dr. tulp launched rembrandt to fame and wealth and influenced generations of artists to come . indeed , without tulp , it seems impossible for thomas eakins to have painted the gross clinic , 1876 almost two and a half centuries later . essay by dr. bryan zygmont additional resources : this painting at the mauritshuis dolores mitchell . `` rembrandt 's `` the anatomy lesson of dr. tulp '' : a sinner among the righteous , '' artibus et historiae , vol . 15 , no . 30 ( nov 1994 ) , p. 145 - 156 aloïs riegl and benjamin binstock . `` excertps from `` the dutch group portrait '' , '' october , vol . 74 , ( autumn 1995 ) , p. 3 - 35
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whereas another artist may have simply recreated a previous group image—inserting new heads in place of old ones—rembrandt created something new , and in doing so , completed one of the most recognizable images in the history of painting . dr. nicolaes tulp dr. nicolaes tulp was appointed praelector ( like a professor or lecturer ) of the amsterdam anatomy guild in 1628 . one of the responsibilities of this position was to deliver a yearly public lecture on some aspect of human anatomy .
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does doctor 's like dr.tulp still now do anatomy on dead bodies ?
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a prolific career it would be difficult to overestimate the importance of rembrandt harmenszoon van rijn within the history of western art . indeed , rembrandt is considered one of the foremost artists of the dutch baroque period , and even if he had never picked up a paintbrush , he would have been famous both in his day and ours as a printmaker of particular brilliance and as a prolific teacher . in a career that lasted nearly forty years , rembrandt completed approximately 400 paintings , more than 1,000 drawings , and nearly 300 engravings . although he spent his entire life north of the alps , had he been italian and lived a century or so earlier , he likely would have joined his italian brethren—donatello , leonardo , michelangelo , and raphael , as a member of the famed cartoon teenage mutant ninja turtles . patrons : a wealthy , protestant , and expanding middle class but as time and place would have it , rembrandt was neither italian nor a part of the renaissance . instead , rembrandt was born in leiden in 1606 . this place and time—holland during the height of the expansion of the wealthy mercantile class during the middle half of the seventeenth century—served rembrandt well through his long career . the catholic church often commissioned italian artists at this time to undertake large-scale projects to promote religious ideology in support of the counter-reformation . without the catholic church in holland to commission art , rembrandt and his fellow dutch artists were lavishly supported by a wealthy , protestant , and expanding middle class . this group of patrons enthusiastically commissioned works of art with their increasing discretionary income . new subjects ( including group portraits ) many different types of art became popular during the dutch baroque period . genre paintings—small paintings of everyday life—were exceptionally popular with a middle-class clientele , as were still lifes , landscapes , and prints . the majority of these kinds of art were both affordable and small enough to be easily displayed within an average home . larger and more compositionally complicated , group portraiture also became popular in holland during the seventeenth century . this was a mode of painting that was often placed in a public space so that the image could promote a particular organization . relocating to amsterdam although several of rembrandt ’ s most well-known paintings are group portraits—the anatomy lesson of dr. tulp among others—his early education in leiden , first at a latin school and then later at the university , suggest that he was destined for a vocation other than art . however , by the time he was sixteen he decided he wanted to be a painter and a draughtsman . after finding quick success in leiden during the 1620s , rembrandt relocated to amsterdam in 1631 , a wise professional decision , as this was then one of the wealthiest and largest cities in europe . a group portrait for the amsterdam surgeon 's guild just a year after his arrival , rembrandt was offered the commission to complete a group portrait of the amsterdam surgeon ’ s guild , an image that in time has come to be known rather simply as the anatomy lesson of dr. tulp . it is remarkable that rembrandt received this commission as a newcomer to amsterdam when there were other native-born artists available . thomas de keyser ( above ) and nicolaes pickenoy ( below ) , for example , were older and more experienced in the realm of group portraiture . whereas another artist may have simply recreated a previous group image—inserting new heads in place of old ones—rembrandt created something new , and in doing so , completed one of the most recognizable images in the history of painting . dr. nicolaes tulp dr. nicolaes tulp was appointed praelector ( like a professor or lecturer ) of the amsterdam anatomy guild in 1628 . one of the responsibilities of this position was to deliver a yearly public lecture on some aspect of human anatomy . the lecture in 1632 occurred on 16 january , and this is the scene that rembrandt depicts in paint in the anatomy of lesson of dr. tulp . this is a more complicated composition than it at first appears . understandably , the focal point of the image is dr. tulp , the doctor who is shown displaying the flexors of the cadaver ’ s left arm . rembrandt notes the doctor ’ s significance by showing him as the only person who wears a hat . seven colleagues surround dr. tulp , and they look in a variety of directions—some gaze at the cadaver , some stare at the lecturer , and some peek directly at the viewer . each face displays a facial expression that is deeply personal and psychological . the cadaver—a recently executed thief named adriaen adriaenszoon—lies nearly parallel to the picture plane . viewing the illuminated body from his head to his feet brings into focus a book—likely andreas vesalius ’ s de humani corporis fabrica ( fabric of the human body , 1543 ) —propped up in the lower right corner . in all , rembrandt shows nine distinct figures , but does so as if they are a unified group . a comparison comparing the anatomy lesson of dr. tulp to a somewhat similar example , the osteology lesson of dr. sebastiaen egbertszoon ( above ) , shows just how different and novel rembrandt ’ s composition was at the time . the sebastiaen erbertszoon painting is a series of six portraits that surround a single human skeleton ; but neither the heads nor the bodies seem to interact with one another in a real or coherent way . in contrast , the figures in rembrandt ’ s tulp seem to truly be a group , one collection of nine rather than nine individuals . if the composition is different from what rembrandt might have seen in amsterdam , the choice of subject is different than what would have been expected in the parts of europe that were catholic . the catholic tenet of resurrection necessitated that dead bodies be interred in a state of wholeness , and this fact explains why leonardo was forced to dissect human bodies in secret . in protestant holland but 113 years after leonardo ’ s death , however , human dissections were not only common practice , they were often public spectacles , complete with food and wine , music and conversation . artistic license if rembrandt was able to create a truly group portrait—one of a single group rather than a collection of individuals—it is important to note that the artist took some understandable artistic license with some parts of the composition . as any anatomy and physiology student today can attest , a dissection of the human body almost always commences with an exploration of the chest and abdominal areas , parts of the human body most likely to decompose first , and only later does the procedure move onwards to the limbs . moreover , it would have been unlikely that a doctor of tulp ’ s importance would have actually dissected the body ; instead , he would have lectured while the menial task of exposing the inner workings of the body would have been left to others . but in paint , a format without sound , rembrandt put tulp in charge not only in costume , but also in action . as the prominent signature in the upper part of the painting indicates , rembrandt was justifiably proud of this large painting . whereas he had previously signed his works with his monogram rhl ( rembrandt harmenszoon of leiden ) , the anatomy lesson of dr. tulp contains rembrant . f [ ecit ] 1632 . this painting and the latin announcement that “ rembrandt made it ” marks the beginning of the painter ’ s mature career . daring , compositionally innovative , and deeply psychological , the anatomy lesson of dr. tulp launched rembrandt to fame and wealth and influenced generations of artists to come . indeed , without tulp , it seems impossible for thomas eakins to have painted the gross clinic , 1876 almost two and a half centuries later . essay by dr. bryan zygmont additional resources : this painting at the mauritshuis dolores mitchell . `` rembrandt 's `` the anatomy lesson of dr. tulp '' : a sinner among the righteous , '' artibus et historiae , vol . 15 , no . 30 ( nov 1994 ) , p. 145 - 156 aloïs riegl and benjamin binstock . `` excertps from `` the dutch group portrait '' , '' october , vol . 74 , ( autumn 1995 ) , p. 3 - 35
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understandably , the focal point of the image is dr. tulp , the doctor who is shown displaying the flexors of the cadaver ’ s left arm . rembrandt notes the doctor ’ s significance by showing him as the only person who wears a hat . seven colleagues surround dr. tulp , and they look in a variety of directions—some gaze at the cadaver , some stare at the lecturer , and some peek directly at the viewer .
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is n't it possible that a person confident of their skill might make such an attractive offer ?
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the emperor constantine , called constantine the great , was significant for several reasons . these include his political transformation of the roman empire , his support for christianity , and his founding of constantinople ( modern day istanbul ) . constantine ’ s status as an agent of change also extended into the realms of art and architecture . the triumphal arch of constantine in rome is not only a superb example of the ideological and stylistic changes constantine ’ s reign brought to art , but also demonstrates the emperor ’ s careful adherence to traditional forms of roman imperial art and architecture . location and appearance the arch of constantine is located along the via triumphalis in rome , and it is situated between the flavian amphitheater ( better known as the colosseum ) and the temple of venus and roma . this location was significant , as the arch was a highly visible example of connective architecture that linked the area of the forum romanum ( roman forum ) to the major entertainment and public bathing complexes of central rome . the monumental arch stands approximately 20 meters high , 25 meters wide , and 7 meters deep . three portals punctuate the exceptional width of the arch , each flanked by partially engaged corinthian columns . the central opening is approximately 12 meters high , above which are identical inscribed marble panels , one on each side , that read : to the emperor caesar flavius constantinus , the greatest , pious , fortunate , the senate and people of rome , by inspiration of divinity and his own great mind with his righteous arms on both the tyrant and his faction in one instant in rightful battle he avenged the republic , dedicated this arch as a memorial to his military victory the end of the tetrarchy beginning in the late 3rd century , the roman empire was ruled by four co-emperors ( two senior emperors and two junior emperors ) , in an effort to bring political stability after the turbulent 3rd century . but in 312 c.e. , constantine took control over the western roman empire by defeating his co-emperor maxentius at the battle of the milvian bridge ( and soon after became the sole ruler of the empire ) . the inscription on the arch refers to maxentius as the tyrant and portrays constantine as the rightful ruler of the western empire . curiously , the inscription also attributes the victory to constantine ’ s “ great mind ” and the inspiration of a singular divinity . the mention of divine inspiration has been interpreted by some scholars as a coded reference to constantine ’ s developing interest in christian monotheism . sculpture from different eras perhaps the most striking feature of the arch is its eclectic and stylistically varied relief sculptures . some aspects of the sculpture are quite standard , like the victoria ( or nike ) figures that occupy the spandrels above the central archway or the typical architectural moldings found in most imperial roman public and religious architecture ( below ) . other sculpted elements , however , show a multiplicity of styles . in fact , most scholars accept that many of the sculptures of the arch were spolia ( the reuse of building stone or decorative sculpture on a new monument ) taken from older monuments dating to the 2nd century c.e . although there is some scholarly disagreement on the origins of the sculptures , their imperial style corresponds to those of the reigns of trajan ( ruled 98-117 c.e.—the figures surmounting the decorative columns ) , hadrian ( ruled 117-138 c.e.—the middle register roundels ) , and marcus aurelius ( ruled 161-180 c.e.—the large panel reliefs on the top registers ) . most of the reliefs feature the emperors participating in codified activities that demonstrate the ruler ’ s authority and piety by addressing troops , defeating enemies , distributing largesse , and offering sacrifices . some sculptural elements of the structure also date to constantine ’ s reign , most notably the frieze which is located immediately above the portals . these relief sculptures are of a drastically different style and narrative content when compared with the spoliated ( older , borrowed ) sections ( below , left ) ; constantine ’ s relief sculptures ( below , right ) feature squat and blocky figures that are more abstract than they are naturalistic . the constantinian reliefs also depict historical , rather than general events related to constantine , including his rise to power and victory over maxentius at the milvian bridge . there is also a scene of constantine distributing largesse ( funds ) to the public—recalling the scenes of emperors from the earlier sculptures . clarity of form regarding style , the relief figures from constantine ’ s age still seem like outliers . yet in comparison to the idealized naturalism of the earlier sculptural elements ( for example , in the roundels in the image below ) , the thick , bold outlines of the constantinian figures render them remarkably legible to passersby . while the constantinian figures lack natural aesthetics , their clarity of form ensured that they were informative and communicated constantine ’ s official ( and celebrated ) history to viewers of his own time . analysis and meaning until relatively recently , art historians viewed the blocky sculptures and use of spolia in the arch as signs of poor craftsmanship , deficient artistry , and economic decline in the late roman empire ( this reading is now almost wholly rejected by art historians ) . even one of the most prolific and influential art historians of the modern age , bernard berenson , titled his short book on the arch , the arch of constantine : the decline of form . more recently , however , analysis of the arch has focused on the political and ideological goals of constantine and the objectives of the artists , which has highlighted new possibilities for the interpretation of the arch . if , indeed , the spoliated ( older ) material from the arch can be traced to the reigns of trajan , hadrian , and marcus aurelius , then it situates constantine as one worthy of the same level of reverence as those emperors—all of whom earned deserved levels of acclaim . this was vitally important to constantine , who had himself essentially bypassed lawful succession and usurped power from others . moreover , constantine encouraged major social changes in rome , such as decriminalizing christianity . any religious change was a threat to the ruling and political classes of rome . by aligning himself with well-regarded emperors of rome ’ s 2nd-century c.e . golden age , constantine was signaling that he intended to model his rule after earlier , successful leaders . essay by dr. andrew findley additional resources : list of rulers of the roman empire on the metropolitan museum of art 's heilbrunn timeline of art history the roman empire on the metropolitan museum of art 's heilbrunn timeline of art history byzantium ( ca . 330–1453 ) on the metropolitan museum of art 's heilbrunn timeline of art history mary beard , the roman triumph ( cambridge , mass . : belknap , 2009 ) . bernard berenson , the arch of constantine : the decline of form ( london : chapman & amp ; hall , 1954 ) . r. ross holloway “ the spolia on the arch of constantine ” quaderni ticinesi numismatica e antichità classiche 14 ( 1985 ) , pp . 261-273 . ernst kitzinger , byzantine art in the making : main lines of stylistic development in mediterranean art , 3rd-7th century ( cambridge , mass . : harvard university press , 1977 ) . jaś elsner , `` from the culture of spolia to the cult of relics : the arch of constantine and the genesis of late antique forms , ” papers of the british school at rome 68 ( 2000 ) , pp . 149–184 . elizabeth marlowe , “ framing the sun : the arch of constantine and the roman cityscape , ” the art bulletin vol . 88 no . 2 ( june 2006 ) , pp . 223-242 . mark wilson jones , `` genesis and mimesis : the design of the arch of constantine in rome , ” the journal of the society of architectural historians 59 ( march 2000 ) , pp . 50–77 .
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most of the reliefs feature the emperors participating in codified activities that demonstrate the ruler ’ s authority and piety by addressing troops , defeating enemies , distributing largesse , and offering sacrifices . some sculptural elements of the structure also date to constantine ’ s reign , most notably the frieze which is located immediately above the portals . these relief sculptures are of a drastically different style and narrative content when compared with the spoliated ( older , borrowed ) sections ( below , left ) ; constantine ’ s relief sculptures ( below , right ) feature squat and blocky figures that are more abstract than they are naturalistic .
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what was the date of publication ?
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a new city seventh-century north africa was not the easiest place to establish a new city . it required battling byzantines ; convincing berbers , the indigenous people of north africa , to accept centralized muslim rule ; and persuading middle eastern merchants to move to north africa . so , in 670 ce , conquering general sidi okba constructed a friday mosque ( masjid-i jami_ or_jami ) in what was becoming kairouan in modern day tunisia . a friday mosque is used for communal prayers on the muslim holy day , friday . the mosque was a critical addition , communicating that kairouan would become a cosmopolitan metropolis under strong muslim control , an important distinction at this time and place . known as the great mosque of kairouan , it is an early example of a hypostyle mosque that also reflects how pre-islamic and eastern islamic art and motifs were incorporated into the religious architecture of islamic north africa . the aesthetics signified the great mosque and kairouan , and , thus , its patrons , were just as important as the religious structures , cities , and rulers of other empires in this region , and that kairouan was part of the burgeoning islamic empire . the aghlabids during the eighth century , sidi okba ’ s mosque was rebuilt at least twice as kairouan prospered . however , the mosque we see today is essentially ninth century . the aghlabids ( 800-909 c.e . ) were the semi-independent rulers of much of north africa . in 836 , prince ziyadat allah i tore down most of the earlier mudbrick structure and rebuilt it in more permanent stone , brick , and wood . the prayer hall or sanctuary is supported by rows of columns and there is an open courtyard , that are characteristic of a hypostyle plan . in the late ninth century , another aghlabid ruler embellished the courtyard entrance to the prayer space and added a dome over the central arches and portal . the dome emphasizes the placement of the mihrab , or prayer niche ( below ) , which is on the same central axis and also under a cupola to signify its importance . the dome is an architectural element borrowed from roman and byzantine architecture . the small windows in the drum of the dome above the mihrab space let natural light into what was an otherwise dim interior . rays fall around the most significant area of the mosque , the mihrab . the drum rests on squinches , small arches decorated with shell over rosette designs similar to examples in roman , byzantine , and umayyad islamic art . the stone dome is constructed of twenty four ribs that each have a small corbel at their base , so the dome looks like a cut cantaloupe , according to the architectural historian k. a. c. creswell . other architectural elements link the great mosque of kairouan with earlier and contemporary islamic religious structures and pre-islamic buildings . they also show the joint religious and secular importance of the great mosque of kairouan . like other hypostyle mosques , such as the prophet ’ s mosque in medina , the mosque of kairouan is roughly rectangular . wider aisles leading to the mihrab and along the qibla wall give it a t-plan . the sanctuary roof and courtyard porticos are supported by repurposed roman and byzantine columns and capitals . the lower portion of the mihrab is decorated with openwork marble panels in floral and geometric vine designs . though the excessively decorated mihrab is unique , the panels are from the syrian area . around the mihrab are lustre tiles from iraq . they also feature stylized floral patterns like byzantine and eastern islamic examples . since it was used for friday prayer , the mosque has a ninth-century minbar , a narrow wooden pulpit where the weekly sermon was delivered . it is said to be the oldest surviving wooden minbar . like christian pulpits , the minbar made the prayer leader more visible and audible . because a ruler ’ s legitimacy could rest upon the mention of his name during the sermon , the minbar served both religious and secular purposes . the minbar is made from teak imported from asia , an expensive material exemplifying kairouan ’ s commercial reach . the side of the minbar closest to the mihrab is composed of elaborately carved latticework with vegetal , floral , and geometric designs evocative of those used in byzantine and umayyad architecture . the minaret dates from the early ninth century , or at least its lower portion does . perhaps inspired by roman lighthouses , the massive square kairouan minaret is about thirty two meters tall , over one hundred feet , making it one of the highest structures around . so in addition to functioning as a place to call for prayer , the minaret identifies the mosque ’ s presence and location in the city while helping to define the city ’ s religious identity . as it was placed just off the mihrab axis , it also affirmed the mihrab ’ s importance . the mosque continued to be modified after the aghlabids , showing that it remained religiously and socially significant even as kairouan fell into decline . a zirid , al-mu ‘ izz ibn badis ( ruled 1016-62 ce ) , commissioned a wooden maqsura , an enclosed space within a mosque that was reserved for the ruler and his associates . the maqsura is assembled from cutwork wooden screens topped with bands of carved abstracted vegetal motifs set into geometric frames , kufic-style script inscriptions , and merlons , which look like the crenellations a top a fortress wall . maqsuras are said to indicate political instability in a society . they remove a ruler from the rest of the worshippers . so , the enclosure , along with its inscription , protected the lives and affirmed the status of persons allowed inside . in the thirteenth century , the hafsids gave the mosque a more fortified look when they added buttresses to support falling exterior walls , a practice continued in later centuries . in 1294 , caliph al-mustansir restored the courtyard and added monumental portals , such as bab al-ma on the east and the domed bab lalla rejana on the west . additional gates were constructed in later centuries . carved stone panels inside the mosque and on the exterior acted like billboards advertising which patron was responsible for construction and restoration . an intellectual center the great mosque was literally and figuratively at the center of kairouan activity , growth , and prestige . though the mosque is now near the northwest city ramparts established in the eleventh century , when sidi okba founded kairouan , it was probably closer to the center of town , near what was the governor ’ s residence and the main thoroughfare , a symbolically prominent and physical visible part of the city . by the mid-tenth century , kairouan became a thriving settlement with marketplaces , agriculture imported from surrounding towns , cisterns supplying water , and textile and ceramic manufacturing areas . it was a political capital , a pilgrimage city , and intellectual center , particularly for the maliki school of sunni islam and the sciences . the great mosque had fifteen thoroughfares leading from it into a city that may have had a circular layout like baghdad , the capital of the islamic empire during kairouan ’ s heyday . as a friday mosque , it was one of if not the largest buildings in town . the great mosque of kairouan was a public structure , set along roads that served a city with a vibrant commercial , educational , and religious life . as such , it assumed the important function of representing a cosmopolitan and urbane kairouan , one of the first cities organized under muslim rule in north africa . even today , the great mosque of kairouan reflects the time and place in which it was built . text by dr. colette apelian additional resources : sheila s. blair and jonathan m. bloom , the art and architecture of islam 1250-1800 ( new york : yale university press , 1995 ) . k. a. c. creswell and james w. allan , a short account of early muslim architecture ( cairo : the american university of cairo press , 1989 ) . richard ettinghausen , oleg grabar , marilyn jenkins-madina , islamic art and architecture , 650-1250 ( new york : yale university press , 2001 ) . peter harrison , castles of god : fortified religious buildings of the world ( rochester : boydell press , 2004 ) . robert hillenbrand , islamic architecture : form , function , and meaning ( new york : columbia university press , 1994 ) . john d. hoag , islamic architecture , new york : electa/rizzoli , 1975 . ira lapidus , a history of islamic societies , 2nd edition ( new york : cambridge university press , 2002 ) . mourad rammah , “ kairouan , ” museum with no frontiers , islamic art in the mediterranean , ifriqiya . thirteen centuries of art and architecture in tunisia ( vienna : electa , 2002 ) . andrew petersen , dictionary of islamic architecture ( new york : routledge , 1999 ) .
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the lower portion of the mihrab is decorated with openwork marble panels in floral and geometric vine designs . though the excessively decorated mihrab is unique , the panels are from the syrian area . around the mihrab are lustre tiles from iraq .
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clearly just from the written context we can tell that a `` cupola '' is some sort of a chandelier ... is there any difference though other than language ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work .
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can someone check / confirm if the next challenge many waves is still working on the 1st step ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line .
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just wondering ... can you make real waves with waves ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity .
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how can i change the color of every circle that was draw in the function ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next .
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like the first bal is red , the second is blue , etc ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity .
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what is the difference between var draw = function ( ) and draw = function ( ) ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity .
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for exsample noise wave but with smooth curves ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there .
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why the pulse on the string is transverse when it is reflected back ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity .
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does the verticle speed of element on a string depends upon wave speed ?
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there . when we oscillate a single circle up and down according to the sine function , what we are doing is looking at a single point along the x-axis of a wave pattern . with a little panache and a for loop , we can place a whole bunch of these oscillating circles next to each other . this wavy pattern could be used in the design of the body or appendages of a creature , as well as to simulate a soft surface ( such as water ) . here , we ’ re going to encounter the same questions of amplitude ( height of pattern ) and period . instead of period referring to time , however , since we ’ re looking at the full wave , we can talk about period as the width ( in pixels ) of a full wave cycle . and just as with simple oscillation , we have the option of computing the wave pattern according to a precise period or simply following the model of angular velocity . let ’ s go with the simpler case , angular velocity . we know we need to start with an angle , an angular velocity , and an amplitude : var angle = 0 ; var anglevel = 0.2 ; var amplitude = 100 ; then we ’ re going to loop through all of the x values where we want to draw a point of the wave . let ’ s say every 24 pixels for now . in that loop , we ’ re going to want to do three things : calculate y location according to amplitude and sine of the angle . draw a circle at the ( x , y ) location . increment the angle according to angular velocity . for ( var x = 0 ; x & lt ; = width ; x += 24 ) { // calculate y location according to amplitude and sine of angle var y = amplitude * sin ( angle ) ; // draw a circle at the x , y location ellipse ( x , y+height/2 , 48 , 48 ) ; // increment the angle according to angular velocity angle += anglevel ; } let ’ s look at the results with different values for anglevel : notice how , although we ’ re not precisely computing the period of the wave , the higher the angular velocity , the shorter the period . it ’ s also worth noting that as the period becomes shorter , it becomes more and more difficult to make out the wave itself as the distance between the individual points increases . one option we have is to use beginshape ( ) and endshape ( ) to connect the points with a line . the above example is static . the wave never changes , never undulates , and that 's what we 've been building up to . this additional step of animating the wave is a bit tricky . your first instinct might be to say : “ hey , no problem , we ’ ll just let angle be a global variable and let it increment from one cycle through draw ( ) to another. ” while it ’ s a nice thought , it wo n't work . if you look at the statically drawn wave , the righthand edge doesn ’ t match the lefthand ; where it ends in one cycle of draw ( ) can ’ t be where it starts in the next . instead , what we need to do is have a variable dedicated entirely to tracking what value of angle the wave should start with . this angle ( which we ’ ll call startangle ) increments with its own angular velocity . here it is , with the start angle incorporated . try changing the different numbers to see what happens to the oscillating wave . this `` natural simulations '' course is a derivative of `` the nature of code '' by daniel shiffman , used under a creative commons attribution-noncommercial 3.0 unported license .
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if you ’ re saying to yourself , “ um , this is all great and everything , but what i really want is to draw a wave onscreen , ” well , then , the time has come . the thing is , we ’ re about 90 % there .
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what is anglemode mean in khanacademy ?
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key points habituation is a simple learned behavior in which an animal gradually stops responding to a repeated stimulus . imprinting is a specialized form of learning that occurs during a brief period in young animals—e.g. , ducks imprinting on their mother . in classical conditioning , a new stimulus is associated with a pre-existing response through repeated pairing of new and previously known stimuli . in operant conditioning , an animal learns to perform a behavior more or less frequently through a reward or punishment that follows the behavior . some animals , especially primates , are capable of more complex forms of learning , such as problem-solving and the construction of mental maps . introduction if you own a dog—or have a friend who owns a dog—you probably know that dogs can be trained to do things like sit , beg , roll over , and play dead . these are examples of learned behaviors , and dogs can be capable of significant learning . by some estimates , a very clever dog has cognitive abilities on par with a two-and-a-half-year-old human ! $ ^1 $ in general , a learned behavior is one that an organism develops as a result of experience . learned behaviors contrast with innate behaviors , which are genetically hardwired and can be performed without any prior experience or training . of course , some behaviors have both learned and innate elements . for instance , zebra finches are genetically preprogrammed to learn a song , but the song they sing depends on what they hear from their fathers . in this article , we 'll take a look at some examples of learned behaviors in animals . we 'll start with simple ones like habituation and imprinting , then work our way up to complex cases like operant conditioning and cognitive learning . simple learned behaviors learned behaviors , even though they may have innate components or underpinnings , allow an individual organism to adapt to changes in the environment . learned behaviors are modified by previous experiences ; examples of simple learned behaviors include habituation and imprinting . habituation habituation is a simple form of learning in which an animal stops responding to a stimulus , or cue , after a period of repeated exposure . this is a form of non-associative learning , meaning that the stimulus is not linked with any punishment or reward . for example , prairie dogs typically sound an alarm call when threatened by a predator . at first , they will give this alarm call in response to hearing human steps , which indicate the presence of a large and potentially hungry animal . however , the prairie dogs gradually become habituated to the sound of human footsteps , as they repeatedly experience the sound without anything bad happening . eventually , they stop giving the alarm call in response to footsteps . in this example , habituation is specific to the sound of human footsteps , as the animals still respond to the sounds of potential predators . imprinting imprinting is a simple and highly specific type of learning that occurs at a particular age or life stage during the development of certain animals , such as ducks and geese . when ducklings hatch , they imprint on the first adult animal they see , typically their mother . once a duckling has imprinted on its mother , the sight of the mother acts as a cue to trigger a suite of survival-promoting behaviors , such as following the mother around and imitating her . how do we know this is not an innate behavior , in which the duckling is hardwired to follow around a female duck ? that is , how do we know imprinting is a learning process conditioned by experience ? if newborn ducks or geese see a human before they see their mother , they will imprint on the human and follow it around just as they would follow their real mother . an interesting case of imprinting being used for good comes from efforts to rehabilitate the endangered whooping crane by raising chicks in captivity . biologists dress up in full whooping crane costume while caring for the young birds , ensuring that they do n't imprint on humans but rather on the bird dummies that are part of the costume . eventually , they teach the birds to migrate using an ultralight aircraft , preparing them for release into the wild. $ ^ { 2,3 } $ conditioned behaviors conditioned behaviors are the result of associative learning , which takes two forms : classical conditioning and operant conditioning . classical conditioning in classical conditioning , a response already associated with one stimulus is associated with a second stimulus to which it had no previous connection . the most famous example of classical conditioning comes from ivan pavlov ’ s experiments in which dogs were conditioned to drool—a response previously associated with food—upon hearing the sound of a bell . as pavlov observed , and as you may have noticed too , dogs salivate , or drool , in response to the sight or smell of food . this is something dogs do innately , without any need for learning . in the language of classical conditioning , this existing stimulus-response pair can be broken into an unconditioned stimulus , the sight or smell of food , and an unconditioned response , drooling . in pavlov 's experiments , every time a dog was given food , another stimulus was provided alongside the unconditioned stimulus . specifically , a bell was rung at the same time the dog received food . this ringing of the bell , paired with food , is an example of a conditioning stimulus—a new stimulus delivered in parallel with the unconditioned stimulus . over time , the dogs learned to associate the ringing of the bell with food and to respond by drooling . eventually , they would respond with drool when the bell was rung , even when the unconditioned stimulus , the food , was absent . this new , artificially formed stimulus-response pair consists of a conditioned stimulus , the bell ringing , and a conditioned response , drooling . is the unconditioned response , drooling in response to food , exactly identical to the conditioned response , drooling in response to the bell ? not necessarily . pavlov discovered that the saliva in the conditioned dogs was actually different in composition than the saliva of unconditioned dogs . operant conditioning operant conditioning is a bit different than classical conditioning in that it does not rely on an existing stimulus-response pair . instead , whenever an organism performs a behavior—or an intermediate step on the way to the complete behavior—it is given a reward or a punishment . at first , the organism may perform the behavior—e.g. , pressing a lever—purely by chance . through reinforcement , the organism is induced to perform the behavior more or less frequently . one prominent early investigator of operant conditioning was the psychologist b. f. skinner , the inventor of the skinner box , see image below . skinner put rats in boxes containing a lever that would dispense food when pushed by the rat . the rat would initially push the lever a few times by accident , and would then begin to associate pushing the lever with getting the food . over time , the rat would push the lever more and more frequently in order to obtain the food . not all of skinner 's experiments involved pleasant treats . the bottom of the box consisted of a metal grid that could deliver an electric shock to rats as a punishment . when the rat got an electric shock each time it performed a certain behavior , it quickly learned to stop performing the behavior . as these examples show , both positive and negative reinforcement can be used to shape an organism 's behavior in operant conditioning . ouch ! poor rats ! operant conditioning is the basis of most animal training . for instance , you might give your dog a biscuit or a `` good dog ! '' every time it sits , rolls over , or refrains from barking . on the other hand , cows in a field surrounded by an electrified fence will quickly learn to avoid brushing up against the fence. $ ^4 $ as these examples illustrate , operant conditioning through reinforcement can cause animals to engage in behaviors they would not have naturally performed or to avoid behaviors that are normally part of their repertoire . learning and cognition humans , other primates , and some non-primate animals are capable of sophisticated learning that does not fit under the heading of classical or operant conditioning . let 's look at some examples of problem-solving and complex spatial learning in nonhuman animals . problem-solving in chimpanzees the german scientist wolfgang köhler did some of the earliest studies on problem-solving in chimpanzees . he found that the chimps were capable of abstract thought and could think their way through possible solutions to a puzzle , envisioning the result of a solution even before they carried it out . for example , in one experiment , köhler hung a banana in the chimpanzees ' cage , too high for them to reach . several boxes were also placed randomly on the floor . faced with this dilemma , some of the chimps—after a few false starts and some frustration—stacked the boxes one on top of the other , climbed on top of them , and got the banana . this behavior suggests they could visualize the result of stacking the boxes before they actually carried out the action. $ ^5 $ spatial learning in rats learning that extends beyond simple association is not limited to primates . for instance , maze-running experiments done in the 1920s—maze shown below—demonstrated that rats were capable of complex spatial learning. $ ^ { 6,7 } $ in these experiments , rats were divided into three groups : group i : rats got food at the end of the maze from day one . group ii : rats were placed in the maze on six consecutive days before receiving food at the end of the maze on day seven . group iii : rats were placed in the maze for two consecutive days before receiving food at the end of the maze on day three . not surprisingly , rats given a food reward from day one appeared to learn faster—had a more rapid drop in their number of errors while running the maze—than rats not given an initial reward . what was most striking , however , was what happened after the group ii and iii rats were given food . in both groups , the day after the food had been provided , the rats showed a sharp drop in number of errors , almost catching up to the group i rats . this pattern suggested that the group ii and iii rats had , in fact , been learning efficiently , building a mental map , in the previous days . they just did n't have much reason to demonstrate their learning until the food showed up ! these results show that rats are capable of complex spatial learning , even in the absence of a direct reward , in other words , without reinforcement . later experiments confirmed that the rats make a representation of the maze in their minds—a cognitive map—rather than simply learning a conditioned series of turns .
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learned behaviors contrast with innate behaviors , which are genetically hardwired and can be performed without any prior experience or training . of course , some behaviors have both learned and innate elements . for instance , zebra finches are genetically preprogrammed to learn a song , but the song they sing depends on what they hear from their fathers .
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what is the difference between innate and learned behaviours ?
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sex , booze and 18th-century britain if you ever needed proof that the sex , booze and a rock ’ n ’ roll lifestyle was not a twentieth century invention , you need look no further than the satirical prints of william hogarth . he held up a moralising mirror to eighteenth-century britain ; the harlots , the womanisers—even the clergy could not escape . hogarth ’ s prints play out the sins of eighteenth-century london in a kind of visual theatre that was entirely new and novel in their day . the first example of these prints , which hogarth himself termed ‘ modern moral subjects ’ , was a harlot ’ s progress . in this series , we meet the fresh faced moll hackabout as she arrives for the first time in london . moll is soon preyed upon by a brothel keeper and she descends into prostitution . the tale ends with her premature death from a sexually transmitted disease , aged just twenty-three . despite its dark subject matter a harlot ’ s progress was a huge success . these prints reference characters types who were well known to their contemporary london audience . for example moll ’ s madam was the real-life elizabeth needham , keeper of an exclusive london brothel and her first patron , the renowned love-rat and convicted rapist , colonel francis charteris . a rake ’ s progress these sly nods to the bad guys of the day not only made the prints hugely relevant and enjoyable to their target audience but it also made them incredibly popular . a rake ’ s progress ( 1735 ) was hogarth ’ s second series and proved to be just as well loved . the main character is tom rakewell—a rake being a old fashioned term for a man of loose morals or a womaniser . tom ’ s name is intentionally general and in a modern equivalent , he might be called ‘ mr . immoral. ’ tom is not unique , he could be any number of people in eighteenth-century britain . the series opens with a chaotic scene : tom ’ s father , who was a rich merchant , has died and tom has returned from oxford university to collect and spend his late father ’ s wealth . he also wastes no time in rejecting his pregnant fiance , sarah young , by attempting to pay her off . hogarth laces all his prints with clues to help us decode the scene . here we can see sarah sobbing into a hankie whilst holding her engagement ring in her hand . her mother stands behind her angrily clutching the love letters tom once wrote to her daughter and he holds out a handful of coins in an attempt to get rid of them . sarah pops up throughout these prints representing a more wholesome life that he could have had . a fashionable life by the next scene ( `` surrounded by artists and professors , '' above ) tom has already moved from his cosy , if slightly shabby family home into his new bachelor pad surrounded by a dance master , a music teacher , a poet , a tailor , a landscape gardener , a body guard and a jockey all offering their services to help tom complete his fashionable lifestyle . he is dressed in his nightclothes indicating that he has just woken : those who wish to exploit his new found wealth are wasting no time . tom ’ s fashionable life also comes with fashionable vices and soon we see him in the rose tavern with a group of prostitutes ( see `` the tavern scene , '' below ) . he sits on the lap of one who caresses him with one hand whilst robbing him of his watch with the other . portraits of roman emperors hang on the wall behind them but the only one that has not been defaced is nero ’ s . this is perhaps hard for a modern audience to identify but there would have been a significant number of hogarth ’ s classically educated audience who got the gag : nero was a corrupt womaniser who fiercely persecuted christians . to the very classically aware georgians ( george ii was then king ) the message was clear , christian morals are not to be found here . a decadant decline tom ’ s decadent lifestyle does not last for long and by the third scene his sedan chair is intercepted by bailiffs as he is en route to the queen ’ s birthday party . it is at this point that our heroine sarah young comes to the rescue . she is now working as a milliner and kindly pays tom ’ s bail . although sarah has saved tom from the bailiffs , she can not save him from himself . by the next scene he is marrying a wealthy old hag . the old woman ’ s eyes lust eagerly towards the ring and tom ’ s towards her maid . in the background sarah young and her mother struggle to voice their objections but are held back by some of the guests . tom is wealthy again but he is no better with his money now than he was last time and soon he is on his knees in a gambling den having just squandered the lot ( see `` the gaming house , '' below ) . excessive gambling was a real problem in the eighteenth century and whole family fortunes could be lost in one evening . later in the century , george iii ’ s son , who later became george iv , had to ask parliament for money to help him pay off his gambling debts . it was given to him but it was not long before he needed even more . debtors ' prison like so many others in eighteenth-century britain , tom finds himself in debtors ’ prison , quickly coming to the end of his tether . on the one side his wife derides him for squandering their fortune , on the other the beer-boy and the jailer harass him to settle his weekly bill . sarah young , who has come to visit tom with their child , has fainted on seeing him in this hopeless situation . this must have been very personally relevant for hogarth as his father spent much of his childhood in a debtors ' prison . my favourite part of the series is played out in the background of this scene . tom ’ s fellow inmates are trying various schemes to get enough money to buy their freedom . however , their choice of projects cleverly illustrate just how impossible it was to get out of debt in georgian britain . one man is attempting to turn lead into gold while the other is working on solving the national debt crisis . even tom has written a play , though we can clearly see a rejection letter for it lying on the table . bedlam the stresses of the previous scene have proven to be more than tom can bear and in the final scene he is found languishing in bedlam—london ’ s notorious mental hospital ( see `` the madhouse , '' above ) . the mark on his chest suggests that he has stabbed himself in a failed suicide attempt . fashionable young women , the likes of which tom would have socialised with just a short while ago , observe the scene in amusement . all that he has left is the company and care of the faithful sarah young . why so popular ? one answer is that it appealed to the contemporary concern about people from the middle classes who tried to live like aristocrats . this was a popular issue at this time as the merchant trade was creating social mobility on a scale never seen before . however if we scratch the surface of a rake ’ s progress we can see that a number of different types of people implicated . for example , tom was on his way to the queen ’ s birthday party when he was stopped by bailiffs and therefore his lifestyle is actually being encouraged and supported by aristocrats . in a rake ’ s progress , everyone from the queen to the priest that performs his marriage of convenience , to common prostitutes , are part of the problem . but it is not just hogarth ’ s ‘ take no prisoners ’ approach to social commentary that made him so popular . printed satire was actually already very common place and central london was full of bookshops and print sellers that displayed this kind of work . what hogarth did do that was so completely novel was to tell a story through pictures , a rake ’ s progress is like a story board for a play . in fact , hogarth ’ s series were adapted into plays and pantomimes during his lifetime . his visual drama offered his audience a new way to enjoy satire . it is for this reason that to find comparisons and inspirations we should be looking at authors such as hogarth ’ s friend and fellow moraliser , henry fielding or jonathan swift—author of gulliver ’ s travels , rather than contemporary artists . the title a rake ’ s progress was referencing john bunyan ’ s the pilgrims progress . we can be quite sure that most people would have gotten this reference as it is thought that , at this time , this was the most read book in britain after the bible . hogarth successfully borrows from popular culture in order to express complicated ideas through an enjoyable and totally accessible story . of course hogarth wasn ’ t the first to do this , but he did it so well , he is celebrated to this day . essay by sophie harland additional resources the original painted series from 1733 at sir john soane 's museum hogarth exhibition at tate britain a short biography of the artist from the national gallery
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sex , booze and 18th-century britain if you ever needed proof that the sex , booze and a rock ’ n ’ roll lifestyle was not a twentieth century invention , you need look no further than the satirical prints of william hogarth . he held up a moralising mirror to eighteenth-century britain ; the harlots , the womanisers—even the clergy could not escape .
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what era of art is this from ?
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan . when it was rebuilt in the 12th century , it ushered in a new era of shoguns and helped to found japan ’ s most celebrated school of sculpture . it was built to impress . twice . buddhism , emperor shomu and the creation of todai-ji the roots of todai-ji are found in the arrival of buddhism in japan in the 6th century . buddhism made its way from india along the silk route through central asia , china and korea . mahayana buddhism was officially introduced to the japanese imperial court around 552 by an emissary from a korean king who offered the japanese emperor kimmei a gilded bronze statue of the buddha , a copy of the buddhist sutras ( sacred writings ) and a letter stating : “ this doctrine can create religious merit and retribution without measure and bounds and so lead on to a full appreciation of the highest wisdom. ” buddhism quickly became associated with the imperial court whose members became the patrons of early buddhist art and architecture . this connection between sacred and secular power would define japan ’ s ruling elite for centuries to come . these early buddhist projects also reveal the receptivity of japan to foreign ideas and goods—as buddhist monks and craftspeople came to japan . buddhism ’ s influence grew in the nara era ( 710-794 ) during the reign of emperor shomu and his consort , empress komyo who fused buddhist doctrine and political policy—promoting buddhism as the protector of the state . in 741 , reportedly following the empress ’ wishes , shomu ordered temples , monasteries and convents to be built throughout japan ’ s 66 provinces . this national system of monasteries , known as the kokubun-ji , would be under the jurisdiction of the new imperial todai-ji ( “ great eastern temple ” ) to be built in the capital of nara . building todai-ji why build such on such an unprecedented scale ? emperor shomu ’ s motives seem to have been a mix of the spiritual and the pragmatic : in his bid to unite various japanese clans under his centralized rule , shomu also promoted spiritual unity . todai-ji would be the chief temple of the kokubin-ji system and be the center of national ritual . its construction brought together the best craftspeople in japan with the latest building technology . it was architecture to impress—displaying the power , prestige and piety of the imperial house of japan . however the project was not without its critics . every person in japan was required to contribute through a special tax to its construction and the court chronicle , the shoku nihon-gi , notes that , `` ... the people are made to suffer by the construction of todai-ji and the clans worry over their suffering. ” bronze buddha todai-ji included the usual components of a buddhist complex . at its symbolic heart was the massive hondō ( main hall ) , also called the daibutsuden ( great buddha hall ) , which when completed in 752 , measured 50 meters by 86 meters and was supported by 84 massive cypress pillars . it held a huge bronze buddha figure ( the daibutsu ) created between 743 to 752 . subsequently , two nine-story pagodas , a lecture hall and quarters for the monks were added to the complex . the statue was inspired by similar statues of the buddha in china and was commissioned by emperor shomu in 743 . this colossal buddha required all the available copper in japan and workers used an estimated 163,000 cubic feet of charcoal to produce the metal alloy and form the bronze figure . it was completed in 749 , though the snail-curl hair ( one of the 32 signs of the buddha ’ s divinity ) took an additional two years . when completed , the entire japanese court , government officials and buddhist dignitaries from china and india attended the buddha ’ s “ eye-opening ” ceremony . overseen by the empress koken and attended by the retired emperor shomu and empress komyo , an indian monk named bodhisena is recorded as painting in the buddha ’ s eyes , symbolically imbuing it with life . the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) . the petal surfaces ( image left ) are etched with fleshy figures with swelling chests , full faces and swirling drapery in a style typical of the elegant naturalism of nara era imagery . the petals are the only reminders of the original statue , which was destroyed by fire in the 12th century . today ’ s statue is a 17th century replacement but remains a revered figure with an annual ritual cleaning ceremony each august . chogen and the rebuilding of todai-ji in the kamakura era ( 1185-1333 ) the genpei civil war ( 1180-85 ) saw countless temples destroyed as buddhist clergy took sides in clan warfare . japan ’ s principal temple todai-ji sided with the eventually victorious minamoto clan but was burned by the soon-to-be defeated taira clan in 1180 . the destruction of this revered temple shocked japan . at the war ’ s end , the reconstruction of todai-ji was one of the first projects undertaken by minamoto yoritomo who , as the new ruling shogun , was eager to present the minamoto as national saviors . the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 . his experience of song dynasty buddhist architecture inspired the rebuilding of the temples of nara , in what became know as the “ great buddha ” or the “ indian ” style . the key-surviving example of this style is todai-ji ’ s great south gate—nandaimon—which dates to 1199 . an elaborate bracketing system supports the broad-eaved , two-tiered roof . the nandaimon holds the 2 massive wooden sculptures of guardian kings ( kongō rikishi ) by masters of the kei school of sculpture . kei school of sculpture the large scale rebuilding after the genpei civil war created a multitude of commissions for builders , carpenters and sculptors . this concentration of talent led to the emergence of the kei school of sculpture—considered by many to be the peak of japanese sculpture . noted for its austere realism and the dynamic , muscularity of its figures , the kei school reflects the buddhism and warrior-centered culture of the kamakura era ( 1185–1333 ) . unkei is considered the leading figure of the kei school , with a career spanning over 30 years . his distinctive style emerged in his work on the refurbishment of the many nara temples/shrines , most particularly todai-ji . unkei ’ s fierce guardian figure ungyō in the nandaimon is typical of unkei ’ s powerful , dynamic bodies . it stands in dramatic contrapposto opposite the other muscular guardian king , agyō , created with kaikei and other kei sculptors . both figures are fashioned of cypress wood and stand over eight meters tall . they were made using the joint block technique ( yosegi zukuri ) , that used eight or nine large wood blocks over which another layer of wooden planks were attached . the outer wood was then carved and painted . only a few traces of color remain . ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground . this technology ushered in a new , larger scale in japanese architecture . monumental timber framed architecture requires enormous amounts of wood . the wood of choice was cypress , which grows up to 40 meters tall and has a straight tight grain that easily splits into long beams and is resistant to rot . the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji . 8th century todai-ji had two 9-storey pagodas and a 50 x 86 meter great hall supported by 84 massive cypress pillars that used at least 2200 acres of local forest . after todai-ji ’ s destruction in 1180 , it was rebuilt under the supervision of the monk chogen , who solicited aid from all over western japan . builders had to travel hundreds of kilometers from kinai to find suitable wood . whole forests were cleared to find tall cypresses for pillars , which were then transported at great cost : 118 dams were built to raise river levels in order to transport the massive pillars . and that was only the pillars—wood for the rest of the structure came from at least ten provinces . todai-ji ’ s reconstructed main hall was only half the size of the original and its pagodas several stories shorter . the availability or scarcity of quality local wood was a major factor in the design and evolution of architecture in japan . for example , the growing scarcity of cypress of structural dimensions led to innovations that allowed carpenters to work with less straight-grained woods , like red pine and zelkova . essay by dr. deanna macdonald additional resources : historic monuments of ancient nara ( unesco ) video from unesco on the great buddha sculpture japan , 500-100 a.d. on the metropolitan museum of art 's heilbrunn timeline of art history japan restores old temple gods
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the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji .
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is there any original `` old growth '' forest left in japan ?
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan . when it was rebuilt in the 12th century , it ushered in a new era of shoguns and helped to found japan ’ s most celebrated school of sculpture . it was built to impress . twice . buddhism , emperor shomu and the creation of todai-ji the roots of todai-ji are found in the arrival of buddhism in japan in the 6th century . buddhism made its way from india along the silk route through central asia , china and korea . mahayana buddhism was officially introduced to the japanese imperial court around 552 by an emissary from a korean king who offered the japanese emperor kimmei a gilded bronze statue of the buddha , a copy of the buddhist sutras ( sacred writings ) and a letter stating : “ this doctrine can create religious merit and retribution without measure and bounds and so lead on to a full appreciation of the highest wisdom. ” buddhism quickly became associated with the imperial court whose members became the patrons of early buddhist art and architecture . this connection between sacred and secular power would define japan ’ s ruling elite for centuries to come . these early buddhist projects also reveal the receptivity of japan to foreign ideas and goods—as buddhist monks and craftspeople came to japan . buddhism ’ s influence grew in the nara era ( 710-794 ) during the reign of emperor shomu and his consort , empress komyo who fused buddhist doctrine and political policy—promoting buddhism as the protector of the state . in 741 , reportedly following the empress ’ wishes , shomu ordered temples , monasteries and convents to be built throughout japan ’ s 66 provinces . this national system of monasteries , known as the kokubun-ji , would be under the jurisdiction of the new imperial todai-ji ( “ great eastern temple ” ) to be built in the capital of nara . building todai-ji why build such on such an unprecedented scale ? emperor shomu ’ s motives seem to have been a mix of the spiritual and the pragmatic : in his bid to unite various japanese clans under his centralized rule , shomu also promoted spiritual unity . todai-ji would be the chief temple of the kokubin-ji system and be the center of national ritual . its construction brought together the best craftspeople in japan with the latest building technology . it was architecture to impress—displaying the power , prestige and piety of the imperial house of japan . however the project was not without its critics . every person in japan was required to contribute through a special tax to its construction and the court chronicle , the shoku nihon-gi , notes that , `` ... the people are made to suffer by the construction of todai-ji and the clans worry over their suffering. ” bronze buddha todai-ji included the usual components of a buddhist complex . at its symbolic heart was the massive hondō ( main hall ) , also called the daibutsuden ( great buddha hall ) , which when completed in 752 , measured 50 meters by 86 meters and was supported by 84 massive cypress pillars . it held a huge bronze buddha figure ( the daibutsu ) created between 743 to 752 . subsequently , two nine-story pagodas , a lecture hall and quarters for the monks were added to the complex . the statue was inspired by similar statues of the buddha in china and was commissioned by emperor shomu in 743 . this colossal buddha required all the available copper in japan and workers used an estimated 163,000 cubic feet of charcoal to produce the metal alloy and form the bronze figure . it was completed in 749 , though the snail-curl hair ( one of the 32 signs of the buddha ’ s divinity ) took an additional two years . when completed , the entire japanese court , government officials and buddhist dignitaries from china and india attended the buddha ’ s “ eye-opening ” ceremony . overseen by the empress koken and attended by the retired emperor shomu and empress komyo , an indian monk named bodhisena is recorded as painting in the buddha ’ s eyes , symbolically imbuing it with life . the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) . the petal surfaces ( image left ) are etched with fleshy figures with swelling chests , full faces and swirling drapery in a style typical of the elegant naturalism of nara era imagery . the petals are the only reminders of the original statue , which was destroyed by fire in the 12th century . today ’ s statue is a 17th century replacement but remains a revered figure with an annual ritual cleaning ceremony each august . chogen and the rebuilding of todai-ji in the kamakura era ( 1185-1333 ) the genpei civil war ( 1180-85 ) saw countless temples destroyed as buddhist clergy took sides in clan warfare . japan ’ s principal temple todai-ji sided with the eventually victorious minamoto clan but was burned by the soon-to-be defeated taira clan in 1180 . the destruction of this revered temple shocked japan . at the war ’ s end , the reconstruction of todai-ji was one of the first projects undertaken by minamoto yoritomo who , as the new ruling shogun , was eager to present the minamoto as national saviors . the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 . his experience of song dynasty buddhist architecture inspired the rebuilding of the temples of nara , in what became know as the “ great buddha ” or the “ indian ” style . the key-surviving example of this style is todai-ji ’ s great south gate—nandaimon—which dates to 1199 . an elaborate bracketing system supports the broad-eaved , two-tiered roof . the nandaimon holds the 2 massive wooden sculptures of guardian kings ( kongō rikishi ) by masters of the kei school of sculpture . kei school of sculpture the large scale rebuilding after the genpei civil war created a multitude of commissions for builders , carpenters and sculptors . this concentration of talent led to the emergence of the kei school of sculpture—considered by many to be the peak of japanese sculpture . noted for its austere realism and the dynamic , muscularity of its figures , the kei school reflects the buddhism and warrior-centered culture of the kamakura era ( 1185–1333 ) . unkei is considered the leading figure of the kei school , with a career spanning over 30 years . his distinctive style emerged in his work on the refurbishment of the many nara temples/shrines , most particularly todai-ji . unkei ’ s fierce guardian figure ungyō in the nandaimon is typical of unkei ’ s powerful , dynamic bodies . it stands in dramatic contrapposto opposite the other muscular guardian king , agyō , created with kaikei and other kei sculptors . both figures are fashioned of cypress wood and stand over eight meters tall . they were made using the joint block technique ( yosegi zukuri ) , that used eight or nine large wood blocks over which another layer of wooden planks were attached . the outer wood was then carved and painted . only a few traces of color remain . ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground . this technology ushered in a new , larger scale in japanese architecture . monumental timber framed architecture requires enormous amounts of wood . the wood of choice was cypress , which grows up to 40 meters tall and has a straight tight grain that easily splits into long beams and is resistant to rot . the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji . 8th century todai-ji had two 9-storey pagodas and a 50 x 86 meter great hall supported by 84 massive cypress pillars that used at least 2200 acres of local forest . after todai-ji ’ s destruction in 1180 , it was rebuilt under the supervision of the monk chogen , who solicited aid from all over western japan . builders had to travel hundreds of kilometers from kinai to find suitable wood . whole forests were cleared to find tall cypresses for pillars , which were then transported at great cost : 118 dams were built to raise river levels in order to transport the massive pillars . and that was only the pillars—wood for the rest of the structure came from at least ten provinces . todai-ji ’ s reconstructed main hall was only half the size of the original and its pagodas several stories shorter . the availability or scarcity of quality local wood was a major factor in the design and evolution of architecture in japan . for example , the growing scarcity of cypress of structural dimensions led to innovations that allowed carpenters to work with less straight-grained woods , like red pine and zelkova . essay by dr. deanna macdonald additional resources : historic monuments of ancient nara ( unesco ) video from unesco on the great buddha sculpture japan , 500-100 a.d. on the metropolitan museum of art 's heilbrunn timeline of art history japan restores old temple gods
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan .
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in the photo of ungyo and agyo , are the labels reversed ?
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan . when it was rebuilt in the 12th century , it ushered in a new era of shoguns and helped to found japan ’ s most celebrated school of sculpture . it was built to impress . twice . buddhism , emperor shomu and the creation of todai-ji the roots of todai-ji are found in the arrival of buddhism in japan in the 6th century . buddhism made its way from india along the silk route through central asia , china and korea . mahayana buddhism was officially introduced to the japanese imperial court around 552 by an emissary from a korean king who offered the japanese emperor kimmei a gilded bronze statue of the buddha , a copy of the buddhist sutras ( sacred writings ) and a letter stating : “ this doctrine can create religious merit and retribution without measure and bounds and so lead on to a full appreciation of the highest wisdom. ” buddhism quickly became associated with the imperial court whose members became the patrons of early buddhist art and architecture . this connection between sacred and secular power would define japan ’ s ruling elite for centuries to come . these early buddhist projects also reveal the receptivity of japan to foreign ideas and goods—as buddhist monks and craftspeople came to japan . buddhism ’ s influence grew in the nara era ( 710-794 ) during the reign of emperor shomu and his consort , empress komyo who fused buddhist doctrine and political policy—promoting buddhism as the protector of the state . in 741 , reportedly following the empress ’ wishes , shomu ordered temples , monasteries and convents to be built throughout japan ’ s 66 provinces . this national system of monasteries , known as the kokubun-ji , would be under the jurisdiction of the new imperial todai-ji ( “ great eastern temple ” ) to be built in the capital of nara . building todai-ji why build such on such an unprecedented scale ? emperor shomu ’ s motives seem to have been a mix of the spiritual and the pragmatic : in his bid to unite various japanese clans under his centralized rule , shomu also promoted spiritual unity . todai-ji would be the chief temple of the kokubin-ji system and be the center of national ritual . its construction brought together the best craftspeople in japan with the latest building technology . it was architecture to impress—displaying the power , prestige and piety of the imperial house of japan . however the project was not without its critics . every person in japan was required to contribute through a special tax to its construction and the court chronicle , the shoku nihon-gi , notes that , `` ... the people are made to suffer by the construction of todai-ji and the clans worry over their suffering. ” bronze buddha todai-ji included the usual components of a buddhist complex . at its symbolic heart was the massive hondō ( main hall ) , also called the daibutsuden ( great buddha hall ) , which when completed in 752 , measured 50 meters by 86 meters and was supported by 84 massive cypress pillars . it held a huge bronze buddha figure ( the daibutsu ) created between 743 to 752 . subsequently , two nine-story pagodas , a lecture hall and quarters for the monks were added to the complex . the statue was inspired by similar statues of the buddha in china and was commissioned by emperor shomu in 743 . this colossal buddha required all the available copper in japan and workers used an estimated 163,000 cubic feet of charcoal to produce the metal alloy and form the bronze figure . it was completed in 749 , though the snail-curl hair ( one of the 32 signs of the buddha ’ s divinity ) took an additional two years . when completed , the entire japanese court , government officials and buddhist dignitaries from china and india attended the buddha ’ s “ eye-opening ” ceremony . overseen by the empress koken and attended by the retired emperor shomu and empress komyo , an indian monk named bodhisena is recorded as painting in the buddha ’ s eyes , symbolically imbuing it with life . the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) . the petal surfaces ( image left ) are etched with fleshy figures with swelling chests , full faces and swirling drapery in a style typical of the elegant naturalism of nara era imagery . the petals are the only reminders of the original statue , which was destroyed by fire in the 12th century . today ’ s statue is a 17th century replacement but remains a revered figure with an annual ritual cleaning ceremony each august . chogen and the rebuilding of todai-ji in the kamakura era ( 1185-1333 ) the genpei civil war ( 1180-85 ) saw countless temples destroyed as buddhist clergy took sides in clan warfare . japan ’ s principal temple todai-ji sided with the eventually victorious minamoto clan but was burned by the soon-to-be defeated taira clan in 1180 . the destruction of this revered temple shocked japan . at the war ’ s end , the reconstruction of todai-ji was one of the first projects undertaken by minamoto yoritomo who , as the new ruling shogun , was eager to present the minamoto as national saviors . the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 . his experience of song dynasty buddhist architecture inspired the rebuilding of the temples of nara , in what became know as the “ great buddha ” or the “ indian ” style . the key-surviving example of this style is todai-ji ’ s great south gate—nandaimon—which dates to 1199 . an elaborate bracketing system supports the broad-eaved , two-tiered roof . the nandaimon holds the 2 massive wooden sculptures of guardian kings ( kongō rikishi ) by masters of the kei school of sculpture . kei school of sculpture the large scale rebuilding after the genpei civil war created a multitude of commissions for builders , carpenters and sculptors . this concentration of talent led to the emergence of the kei school of sculpture—considered by many to be the peak of japanese sculpture . noted for its austere realism and the dynamic , muscularity of its figures , the kei school reflects the buddhism and warrior-centered culture of the kamakura era ( 1185–1333 ) . unkei is considered the leading figure of the kei school , with a career spanning over 30 years . his distinctive style emerged in his work on the refurbishment of the many nara temples/shrines , most particularly todai-ji . unkei ’ s fierce guardian figure ungyō in the nandaimon is typical of unkei ’ s powerful , dynamic bodies . it stands in dramatic contrapposto opposite the other muscular guardian king , agyō , created with kaikei and other kei sculptors . both figures are fashioned of cypress wood and stand over eight meters tall . they were made using the joint block technique ( yosegi zukuri ) , that used eight or nine large wood blocks over which another layer of wooden planks were attached . the outer wood was then carved and painted . only a few traces of color remain . ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground . this technology ushered in a new , larger scale in japanese architecture . monumental timber framed architecture requires enormous amounts of wood . the wood of choice was cypress , which grows up to 40 meters tall and has a straight tight grain that easily splits into long beams and is resistant to rot . the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji . 8th century todai-ji had two 9-storey pagodas and a 50 x 86 meter great hall supported by 84 massive cypress pillars that used at least 2200 acres of local forest . after todai-ji ’ s destruction in 1180 , it was rebuilt under the supervision of the monk chogen , who solicited aid from all over western japan . builders had to travel hundreds of kilometers from kinai to find suitable wood . whole forests were cleared to find tall cypresses for pillars , which were then transported at great cost : 118 dams were built to raise river levels in order to transport the massive pillars . and that was only the pillars—wood for the rest of the structure came from at least ten provinces . todai-ji ’ s reconstructed main hall was only half the size of the original and its pagodas several stories shorter . the availability or scarcity of quality local wood was a major factor in the design and evolution of architecture in japan . for example , the growing scarcity of cypress of structural dimensions led to innovations that allowed carpenters to work with less straight-grained woods , like red pine and zelkova . essay by dr. deanna macdonald additional resources : historic monuments of ancient nara ( unesco ) video from unesco on the great buddha sculpture japan , 500-100 a.d. on the metropolitan museum of art 's heilbrunn timeline of art history japan restores old temple gods
|
ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground .
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why is there a deer running up to those people in the 6th picture ?
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan . when it was rebuilt in the 12th century , it ushered in a new era of shoguns and helped to found japan ’ s most celebrated school of sculpture . it was built to impress . twice . buddhism , emperor shomu and the creation of todai-ji the roots of todai-ji are found in the arrival of buddhism in japan in the 6th century . buddhism made its way from india along the silk route through central asia , china and korea . mahayana buddhism was officially introduced to the japanese imperial court around 552 by an emissary from a korean king who offered the japanese emperor kimmei a gilded bronze statue of the buddha , a copy of the buddhist sutras ( sacred writings ) and a letter stating : “ this doctrine can create religious merit and retribution without measure and bounds and so lead on to a full appreciation of the highest wisdom. ” buddhism quickly became associated with the imperial court whose members became the patrons of early buddhist art and architecture . this connection between sacred and secular power would define japan ’ s ruling elite for centuries to come . these early buddhist projects also reveal the receptivity of japan to foreign ideas and goods—as buddhist monks and craftspeople came to japan . buddhism ’ s influence grew in the nara era ( 710-794 ) during the reign of emperor shomu and his consort , empress komyo who fused buddhist doctrine and political policy—promoting buddhism as the protector of the state . in 741 , reportedly following the empress ’ wishes , shomu ordered temples , monasteries and convents to be built throughout japan ’ s 66 provinces . this national system of monasteries , known as the kokubun-ji , would be under the jurisdiction of the new imperial todai-ji ( “ great eastern temple ” ) to be built in the capital of nara . building todai-ji why build such on such an unprecedented scale ? emperor shomu ’ s motives seem to have been a mix of the spiritual and the pragmatic : in his bid to unite various japanese clans under his centralized rule , shomu also promoted spiritual unity . todai-ji would be the chief temple of the kokubin-ji system and be the center of national ritual . its construction brought together the best craftspeople in japan with the latest building technology . it was architecture to impress—displaying the power , prestige and piety of the imperial house of japan . however the project was not without its critics . every person in japan was required to contribute through a special tax to its construction and the court chronicle , the shoku nihon-gi , notes that , `` ... the people are made to suffer by the construction of todai-ji and the clans worry over their suffering. ” bronze buddha todai-ji included the usual components of a buddhist complex . at its symbolic heart was the massive hondō ( main hall ) , also called the daibutsuden ( great buddha hall ) , which when completed in 752 , measured 50 meters by 86 meters and was supported by 84 massive cypress pillars . it held a huge bronze buddha figure ( the daibutsu ) created between 743 to 752 . subsequently , two nine-story pagodas , a lecture hall and quarters for the monks were added to the complex . the statue was inspired by similar statues of the buddha in china and was commissioned by emperor shomu in 743 . this colossal buddha required all the available copper in japan and workers used an estimated 163,000 cubic feet of charcoal to produce the metal alloy and form the bronze figure . it was completed in 749 , though the snail-curl hair ( one of the 32 signs of the buddha ’ s divinity ) took an additional two years . when completed , the entire japanese court , government officials and buddhist dignitaries from china and india attended the buddha ’ s “ eye-opening ” ceremony . overseen by the empress koken and attended by the retired emperor shomu and empress komyo , an indian monk named bodhisena is recorded as painting in the buddha ’ s eyes , symbolically imbuing it with life . the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) . the petal surfaces ( image left ) are etched with fleshy figures with swelling chests , full faces and swirling drapery in a style typical of the elegant naturalism of nara era imagery . the petals are the only reminders of the original statue , which was destroyed by fire in the 12th century . today ’ s statue is a 17th century replacement but remains a revered figure with an annual ritual cleaning ceremony each august . chogen and the rebuilding of todai-ji in the kamakura era ( 1185-1333 ) the genpei civil war ( 1180-85 ) saw countless temples destroyed as buddhist clergy took sides in clan warfare . japan ’ s principal temple todai-ji sided with the eventually victorious minamoto clan but was burned by the soon-to-be defeated taira clan in 1180 . the destruction of this revered temple shocked japan . at the war ’ s end , the reconstruction of todai-ji was one of the first projects undertaken by minamoto yoritomo who , as the new ruling shogun , was eager to present the minamoto as national saviors . the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 . his experience of song dynasty buddhist architecture inspired the rebuilding of the temples of nara , in what became know as the “ great buddha ” or the “ indian ” style . the key-surviving example of this style is todai-ji ’ s great south gate—nandaimon—which dates to 1199 . an elaborate bracketing system supports the broad-eaved , two-tiered roof . the nandaimon holds the 2 massive wooden sculptures of guardian kings ( kongō rikishi ) by masters of the kei school of sculpture . kei school of sculpture the large scale rebuilding after the genpei civil war created a multitude of commissions for builders , carpenters and sculptors . this concentration of talent led to the emergence of the kei school of sculpture—considered by many to be the peak of japanese sculpture . noted for its austere realism and the dynamic , muscularity of its figures , the kei school reflects the buddhism and warrior-centered culture of the kamakura era ( 1185–1333 ) . unkei is considered the leading figure of the kei school , with a career spanning over 30 years . his distinctive style emerged in his work on the refurbishment of the many nara temples/shrines , most particularly todai-ji . unkei ’ s fierce guardian figure ungyō in the nandaimon is typical of unkei ’ s powerful , dynamic bodies . it stands in dramatic contrapposto opposite the other muscular guardian king , agyō , created with kaikei and other kei sculptors . both figures are fashioned of cypress wood and stand over eight meters tall . they were made using the joint block technique ( yosegi zukuri ) , that used eight or nine large wood blocks over which another layer of wooden planks were attached . the outer wood was then carved and painted . only a few traces of color remain . ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground . this technology ushered in a new , larger scale in japanese architecture . monumental timber framed architecture requires enormous amounts of wood . the wood of choice was cypress , which grows up to 40 meters tall and has a straight tight grain that easily splits into long beams and is resistant to rot . the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji . 8th century todai-ji had two 9-storey pagodas and a 50 x 86 meter great hall supported by 84 massive cypress pillars that used at least 2200 acres of local forest . after todai-ji ’ s destruction in 1180 , it was rebuilt under the supervision of the monk chogen , who solicited aid from all over western japan . builders had to travel hundreds of kilometers from kinai to find suitable wood . whole forests were cleared to find tall cypresses for pillars , which were then transported at great cost : 118 dams were built to raise river levels in order to transport the massive pillars . and that was only the pillars—wood for the rest of the structure came from at least ten provinces . todai-ji ’ s reconstructed main hall was only half the size of the original and its pagodas several stories shorter . the availability or scarcity of quality local wood was a major factor in the design and evolution of architecture in japan . for example , the growing scarcity of cypress of structural dimensions led to innovations that allowed carpenters to work with less straight-grained woods , like red pine and zelkova . essay by dr. deanna macdonald additional resources : historic monuments of ancient nara ( unesco ) video from unesco on the great buddha sculpture japan , 500-100 a.d. on the metropolitan museum of art 's heilbrunn timeline of art history japan restores old temple gods
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the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 .
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how long did it take to construct todai-ji ?
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built to impress , twice when completed in the 740s , todai-ji ( or “ great eastern temple ” ) was the largest building project ever on japanese soil . its creation reflects the complex intermingling of buddhism and politics in early japan . when it was rebuilt in the 12th century , it ushered in a new era of shoguns and helped to found japan ’ s most celebrated school of sculpture . it was built to impress . twice . buddhism , emperor shomu and the creation of todai-ji the roots of todai-ji are found in the arrival of buddhism in japan in the 6th century . buddhism made its way from india along the silk route through central asia , china and korea . mahayana buddhism was officially introduced to the japanese imperial court around 552 by an emissary from a korean king who offered the japanese emperor kimmei a gilded bronze statue of the buddha , a copy of the buddhist sutras ( sacred writings ) and a letter stating : “ this doctrine can create religious merit and retribution without measure and bounds and so lead on to a full appreciation of the highest wisdom. ” buddhism quickly became associated with the imperial court whose members became the patrons of early buddhist art and architecture . this connection between sacred and secular power would define japan ’ s ruling elite for centuries to come . these early buddhist projects also reveal the receptivity of japan to foreign ideas and goods—as buddhist monks and craftspeople came to japan . buddhism ’ s influence grew in the nara era ( 710-794 ) during the reign of emperor shomu and his consort , empress komyo who fused buddhist doctrine and political policy—promoting buddhism as the protector of the state . in 741 , reportedly following the empress ’ wishes , shomu ordered temples , monasteries and convents to be built throughout japan ’ s 66 provinces . this national system of monasteries , known as the kokubun-ji , would be under the jurisdiction of the new imperial todai-ji ( “ great eastern temple ” ) to be built in the capital of nara . building todai-ji why build such on such an unprecedented scale ? emperor shomu ’ s motives seem to have been a mix of the spiritual and the pragmatic : in his bid to unite various japanese clans under his centralized rule , shomu also promoted spiritual unity . todai-ji would be the chief temple of the kokubin-ji system and be the center of national ritual . its construction brought together the best craftspeople in japan with the latest building technology . it was architecture to impress—displaying the power , prestige and piety of the imperial house of japan . however the project was not without its critics . every person in japan was required to contribute through a special tax to its construction and the court chronicle , the shoku nihon-gi , notes that , `` ... the people are made to suffer by the construction of todai-ji and the clans worry over their suffering. ” bronze buddha todai-ji included the usual components of a buddhist complex . at its symbolic heart was the massive hondō ( main hall ) , also called the daibutsuden ( great buddha hall ) , which when completed in 752 , measured 50 meters by 86 meters and was supported by 84 massive cypress pillars . it held a huge bronze buddha figure ( the daibutsu ) created between 743 to 752 . subsequently , two nine-story pagodas , a lecture hall and quarters for the monks were added to the complex . the statue was inspired by similar statues of the buddha in china and was commissioned by emperor shomu in 743 . this colossal buddha required all the available copper in japan and workers used an estimated 163,000 cubic feet of charcoal to produce the metal alloy and form the bronze figure . it was completed in 749 , though the snail-curl hair ( one of the 32 signs of the buddha ’ s divinity ) took an additional two years . when completed , the entire japanese court , government officials and buddhist dignitaries from china and india attended the buddha ’ s “ eye-opening ” ceremony . overseen by the empress koken and attended by the retired emperor shomu and empress komyo , an indian monk named bodhisena is recorded as painting in the buddha ’ s eyes , symbolically imbuing it with life . the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) . the petal surfaces ( image left ) are etched with fleshy figures with swelling chests , full faces and swirling drapery in a style typical of the elegant naturalism of nara era imagery . the petals are the only reminders of the original statue , which was destroyed by fire in the 12th century . today ’ s statue is a 17th century replacement but remains a revered figure with an annual ritual cleaning ceremony each august . chogen and the rebuilding of todai-ji in the kamakura era ( 1185-1333 ) the genpei civil war ( 1180-85 ) saw countless temples destroyed as buddhist clergy took sides in clan warfare . japan ’ s principal temple todai-ji sided with the eventually victorious minamoto clan but was burned by the soon-to-be defeated taira clan in 1180 . the destruction of this revered temple shocked japan . at the war ’ s end , the reconstruction of todai-ji was one of the first projects undertaken by minamoto yoritomo who , as the new ruling shogun , was eager to present the minamoto as national saviors . the aristocracy and the warrior elite contributed funds and the buddhist priest shunjobo chogen was placed in charge of reconstruction . todai-ji again became the largest building project in japan . chōgen was unique in his generation in that he made three trips to china between 1167-1176 . his experience of song dynasty buddhist architecture inspired the rebuilding of the temples of nara , in what became know as the “ great buddha ” or the “ indian ” style . the key-surviving example of this style is todai-ji ’ s great south gate—nandaimon—which dates to 1199 . an elaborate bracketing system supports the broad-eaved , two-tiered roof . the nandaimon holds the 2 massive wooden sculptures of guardian kings ( kongō rikishi ) by masters of the kei school of sculpture . kei school of sculpture the large scale rebuilding after the genpei civil war created a multitude of commissions for builders , carpenters and sculptors . this concentration of talent led to the emergence of the kei school of sculpture—considered by many to be the peak of japanese sculpture . noted for its austere realism and the dynamic , muscularity of its figures , the kei school reflects the buddhism and warrior-centered culture of the kamakura era ( 1185–1333 ) . unkei is considered the leading figure of the kei school , with a career spanning over 30 years . his distinctive style emerged in his work on the refurbishment of the many nara temples/shrines , most particularly todai-ji . unkei ’ s fierce guardian figure ungyō in the nandaimon is typical of unkei ’ s powerful , dynamic bodies . it stands in dramatic contrapposto opposite the other muscular guardian king , agyō , created with kaikei and other kei sculptors . both figures are fashioned of cypress wood and stand over eight meters tall . they were made using the joint block technique ( yosegi zukuri ) , that used eight or nine large wood blocks over which another layer of wooden planks were attached . the outer wood was then carved and painted . only a few traces of color remain . ecology , craftsmanship & amp ; early buddhist art in japan the grand buddhist architectural and sculptural projects of early japan share a common material—wood–and are thus closely linked to the natural environment and to the long history of wood craftsmanship in japan . when korean craftsmen brought buddhist temple architecture to japan in the 6th century , japanese carpenters were already using complex wooden joints ( instead of nails ) to hold buildings together . the korean 's technology allowed for the support of larger , tile-roof structures that used brackets and sturdy foundation pillars to funnel weight to the ground . this technology ushered in a new , larger scale in japanese architecture . monumental timber framed architecture requires enormous amounts of wood . the wood of choice was cypress , which grows up to 40 meters tall and has a straight tight grain that easily splits into long beams and is resistant to rot . the 8th century campaign to construct buddhist temples in every japanese province under imperial control ( mostly in the kinai area , today home to osaka and kyoto ) is estimated to have resulted in the construction 600-850 temples using 3 million cubic meters of wood . as the years progressed kinai ’ s old growth forests were exhausted and builders had to travel farther for wood . by far the most prestigious and wood-demanding project was the imperial monastery of todai-ji . 8th century todai-ji had two 9-storey pagodas and a 50 x 86 meter great hall supported by 84 massive cypress pillars that used at least 2200 acres of local forest . after todai-ji ’ s destruction in 1180 , it was rebuilt under the supervision of the monk chogen , who solicited aid from all over western japan . builders had to travel hundreds of kilometers from kinai to find suitable wood . whole forests were cleared to find tall cypresses for pillars , which were then transported at great cost : 118 dams were built to raise river levels in order to transport the massive pillars . and that was only the pillars—wood for the rest of the structure came from at least ten provinces . todai-ji ’ s reconstructed main hall was only half the size of the original and its pagodas several stories shorter . the availability or scarcity of quality local wood was a major factor in the design and evolution of architecture in japan . for example , the growing scarcity of cypress of structural dimensions led to innovations that allowed carpenters to work with less straight-grained woods , like red pine and zelkova . essay by dr. deanna macdonald additional resources : historic monuments of ancient nara ( unesco ) video from unesco on the great buddha sculpture japan , 500-100 a.d. on the metropolitan museum of art 's heilbrunn timeline of art history japan restores old temple gods
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the emperor shōmu himself is said to have sat in front of great buddha and vowed himself to be a servant of the three treasures of buddhism : the buddha , buddhist law , and buddhist monastic community . no images of the ceremony survive but a nara period scroll painting depicts a sole , humbly small figure at the daibutsu 's base suggesting its awe-inspiring presence . the daibutsu sits upon a bronze lotus petal pedestal that is engraved with images of the shaka ( the historical buddha , known also as shakyamuni ) buddha and varied bodhisattvas ( sacred beings ) .
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wait , what happened between the end of the kofun period ( 533 ) and the start of the nara period ( 710 ) ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship .
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that four-faced linga in the photo is meant to be a phallic representation of shiva ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley .
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perhaps , were the indigenous cultures the dravidian peoples ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities .
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what is the gift of edward nagel ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior .
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from what period was shiva first mentioned in the hinduism ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship .
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what 's the relation of hindu with hong kong ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities .
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so no one knows who founded hinduism ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley .
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perhaps , were the indigenous cultures the dravidian peoples ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities .
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are there any religions that are older than hinduism ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia .
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what is the gangetic region ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder .
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according with the last paragraph , does vedas show a time period in hindu culture ?
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hinduism is one of the world ’ s oldest religions . it has complex roots , and involves a vast array of practices and a host of deities . its plethora of forms and beliefs reflects the tremendous diversity of india , where most of its one billion followers reside . hinduism is more than a religion . it is a culture , a way of life , and a code of behavior . this is reflected in a term indians use to describe the hindu religion : sanatana dharma , which means eternal faith , or the eternal way things are ( truth ) . the word hinduism derives from a persian term denoting the inhabitants of the land beyond the indus , a river in present-day pakistan . by the early nineteenth century the term had entered popular english usage to describe the predominant religious traditions of south asia , and it is now used by hindus themselves . hindu beliefs and practices are enormously diverse , varying over time and among individuals , communities , and regional areas . unlike buddhism , jainism , or sikhism , hinduism has no historical founder . its authority rests instead upon a large body of sacred texts that provide hindus with rules governing rituals , worship , pilgrimage , and daily activities , among many other things . although the oldest of these texts may date back four thousand years , the earliest surviving hindu images and temples were created some two thousand years later . what are the roots of hinduism ? hinduism developed over many centuries from a variety of sources : cultural practices , sacred texts , and philosophical movements , as well as local popular beliefs . the combination of these factors is what accounts for the varied and diverse nature of hindu practices and beliefs . hinduism developed from several sources : prehistoric and neolithic culture , which left material evidence including abundant rock and cave paintings of bulls and cows , indicating an early interest in the sacred nature of these animals . the indus valley civilization , located in what is now pakistan and northwestern india , which flourished between approximately 2500 and 1700 b.c.e. , and persisted with some regional presence as late as 800 b.c.e . the civilization reached its high point in the cities of harrapa and mohenjo-daro . although the physical remains of these large urban complexes have not produced a great deal of explicit religious imagery , archaeologists have recovered some intriguing items , including an abundance of seals depicting bulls , among these a few exceptional examples illustrating figures seated in yogic positions ; terracotta female figures that suggest fertility ; and small anthropomorphic sculptures made of stone and bronze . material evidence found at these sites also includes prototypes of stone linga ( phallic emblems of the hindu god shiva ) . later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia . these peoples brought with them ritual life including fire sacrifices presided over by priests , and a set of hymns and poems collectively known as the vedas . the indigenous beliefs of the pre-vedic peoples of the subcontinent of india encompassed a variety of local practices based on agrarian fertility cults and local nature spirits . vedic writings refer to the worship of images , tutelary divinities , and the phallus .
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later textual sources assert that indigenous peoples of this area engaged in linga worship . according to recent theories , indus valley peoples migrated to the gangetic region of india and blended with indigenous cultures , after the decline of civilization in the indus valley . a separate group of indo-european speaking people migrated to the subcontinent from west asia .
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why did the indus valley people choose to migrate ?
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