id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-14700 | Square of Complex Conjugate is Complex Conjugate of Square | Let $z \in \C$ be a complex number.
Let $\overline z$ denote the complex conjugate of $z$.
Then:
: $\overline {z^2} = \left({\overline z}\right)^2$ | A direct consequence of Product of Complex Conjugates:
: $\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$
for two complex numbers $z_1, z_2 \in \C$.
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Then:
: $\overline {z^2} = \left({\overline z}\right)^2$ | A direct consequence of [[Product of Complex Conjugates]]:
: $\overline {z_1 z_2} = \overline {z_1} \cdot \overline {z_2}$
for two [[Definition:Complex Number|complex numbers]] $z_1, z_2 \in \C$.
{{qed}} | Square of Complex Conjugate is Complex Conjugate of Square | https://proofwiki.org/wiki/Square_of_Complex_Conjugate_is_Complex_Conjugate_of_Square | https://proofwiki.org/wiki/Square_of_Complex_Conjugate_is_Complex_Conjugate_of_Square | [
"Complex Conjugates",
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [
"Product of Complex Conjugates",
"Definition:Complex Number"
] |
proofwiki-14701 | Complex Modulus equals Zero iff Zero | Let $z = a + i b$ be a complex number.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod z = 0 \iff z = 0$ | === Necessary Condition ===
{{begin-eqn}}
{{eqn | l = z
| r = 0
| c =
}}
{{eqn | r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod z
| r = \sqrt {0^2 + 0^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed|lemma}} | Let $z = a + i b$ be a [[Definition:Complex Number|complex number]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod z = 0 \iff z = 0$ | === Necessary Condition ===
{{begin-eqn}}
{{eqn | l = z
| r = 0
| c =
}}
{{eqn | r = 0 + 0 i
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod z
| r = \sqrt {0^2 + 0^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | r = 0
| c =
}}
{{end-eqn}}
{{qed|lemma}} | Complex Modulus equals Zero iff Zero | https://proofwiki.org/wiki/Complex_Modulus_equals_Zero_iff_Zero | https://proofwiki.org/wiki/Complex_Modulus_equals_Zero_iff_Zero | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [] |
proofwiki-14702 | Complex Modulus is Non-Negative | Let $z = a + i b \in \C$ be a complex number.
Let $\cmod z$ be the modulus of $z$.
Then:
:$\cmod z \ge 0$ | {{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {a + b i}
| c = Definition of $z$
}}
{{eqn | r = +\sqrt {a^2 + b^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | o = \ge
| r = 0
| c = {{Defof|Positive Square Root}}
}}
{{end-eqn}}
{{qed}} | Let $z = a + i b \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod z \ge 0$ | {{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {a + b i}
| c = Definition of $z$
}}
{{eqn | r = +\sqrt {a^2 + b^2}
| c = {{Defof|Complex Modulus}}
}}
{{eqn | o = \ge
| r = 0
| c = {{Defof|Positive Square Root}}
}}
{{end-eqn}}
{{qed}} | Complex Modulus is Non-Negative | https://proofwiki.org/wiki/Complex_Modulus_is_Non-Negative | https://proofwiki.org/wiki/Complex_Modulus_is_Non-Negative | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [] |
proofwiki-14703 | Complex Modulus equals Complex Modulus of Conjugate | Let $z \in \C$ be a complex number.
Let $\overline z$ denote the complex conjugate of $z$.
Let $\cmod z$ denote the modulus of $z$.
Then:
:$\cmod z = \cmod {\overline z}$ | Let $z = a + b i$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {a + b i}
| c = Definition of $z$
}}
{{eqn | r = \sqrt {a^2 + b^2}
| c = {{Defof|Complex Modulus}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \cmod {\overline z}
| r = \cmod {\overline {a + b i} }
| c = Definition of... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Let $\cmod z$ denote the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
:$\cmod z = \cmod {\overline z}$ | Let $z = a + b i$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = \cmod {a + b i}
| c = Definition of $z$
}}
{{eqn | r = \sqrt {a^2 + b^2}
| c = {{Defof|Complex Modulus}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \cmod {\overline z}
| r = \cmod {\overline {a + b i} }
| c = Definition... | Complex Modulus equals Complex Modulus of Conjugate | https://proofwiki.org/wiki/Complex_Modulus_equals_Complex_Modulus_of_Conjugate | https://proofwiki.org/wiki/Complex_Modulus_equals_Complex_Modulus_of_Conjugate | [
"Complex Modulus",
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Modulus"
] | [] |
proofwiki-14704 | Square of Complex Modulus equals Complex Modulus of Square | Let $z \in \C$ be a complex number.
Let $\cmod z$ be the modulus of $z$.
Then:
: $\cmod {z^2} = \cmod z^2$ | From Complex Modulus of Product of Complex Numbers:
: $\cmod {z_1 z_2} = \cmod {z_1} \cmod {z_2}$
for $z_1, z_2 \in \C$.
Set $z = z_1 = z_2$ and the result follows.
{{qed}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\cmod z$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Then:
: $\cmod {z^2} = \cmod z^2$ | From [[Complex Modulus of Product of Complex Numbers]]:
: $\cmod {z_1 z_2} = \cmod {z_1} \cmod {z_2}$
for $z_1, z_2 \in \C$.
Set $z = z_1 = z_2$ and the result follows.
{{qed}} | Square of Complex Modulus equals Complex Modulus of Square | https://proofwiki.org/wiki/Square_of_Complex_Modulus_equals_Complex_Modulus_of_Square | https://proofwiki.org/wiki/Square_of_Complex_Modulus_equals_Complex_Modulus_of_Square | [
"Complex Modulus",
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus"
] | [
"Complex Modulus of Product of Complex Numbers"
] |
proofwiki-14705 | Power of Complex Modulus equals Complex Modulus of Power | Let $z \in \C$ be a complex number.
Let $\left\vert{z}\right\vert$ be the modulus of $z$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\left\vert{z^n}\right\vert = \left\vert{z}\right\vert^n$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
:$\left\vert{z^n}\right\vert = \left\vert{z}\right\vert^n$
$P \left({0}\right)$ is the case:
{{begin-eqn}}
{{eqn | l = \left\vert{z^0}\right\vert
| r = \left\vert{1}\right\vert
| c =
}}
{{eqn | r = 1
... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\left\vert{z}\right\vert$ be the [[Definition:Complex Modulus|modulus]] of $z$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\left\vert{z^n}\right\vert = \left\vert{z}\right\vert^n$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the [[Definition:Proposition|proposition]]:
:$\left\vert{z^n}\right\vert = \left\vert{z}\right\vert^n$
$P \left({0}\right)$ is the case:
{{begin-eqn}}
{{eqn | l = \left\vert{z^0}\right\ve... | Power of Complex Modulus equals Complex Modulus of Power | https://proofwiki.org/wiki/Power_of_Complex_Modulus_equals_Complex_Modulus_of_Power | https://proofwiki.org/wiki/Power_of_Complex_Modulus_equals_Complex_Modulus_of_Power | [
"Complex Modulus",
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-14706 | Power of Complex Conjugate is Complex Conjugate of Power | Let $z \in \C$ be a complex number.
Let $\overline z$ denote the complex conjugate of $z$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
:$\overline {z^n} = \left({\overline z}\right)^n$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:
:$\overline {z^n} = \left({\overline z}\right)^n$
$P \left({0}\right)$ is the case:
{{begin-eqn}}
{{eqn | l = \overline {z^0}
| r = \overline 1
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | r = \left(... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of $z$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$\overline {z^n} = \left({\overline z}\right)^n$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the [[Definition:Proposition|proposition]]:
:$\overline {z^n} = \left({\overline z}\right)^n$
$P \left({0}\right)$ is the case:
{{begin-eqn}}
{{eqn | l = \overline {z^0}
| r = \over... | Power of Complex Conjugate is Complex Conjugate of Power | https://proofwiki.org/wiki/Power_of_Complex_Conjugate_is_Complex_Conjugate_of_Power | https://proofwiki.org/wiki/Power_of_Complex_Conjugate_is_Complex_Conjugate_of_Power | [
"Complex Conjugates",
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-14707 | Conjugate of Polynomial is Polynomial of Conjugate | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers.
Let $\alpha \in \C$ be a complex number.
Then:
:$\overline {\map f \alpha} = \map f {\overline \alpha}$
where $\overline \alpha$ denotes the complex conjugate of $\alpha$... | By Power of Complex Conjugate is Complex Conjugate of Power:
:$\overline {\alpha^k} = \paren {\overline \alpha}^k$
for all $k$ between $0$ and $n$.
Then from Product of Complex Conjugates:
:$\overline {a_k \alpha^k} = \overline {a_k} \cdot \overline {\alpha^k}$
But $a_k$ is real.
So by Complex Number equals Conjugate i... | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a [[Definition:Polynomial over Complex Numbers|polynomial over complex numbers]] where $a_0, \ldots, a_n$ are [[Definition:Real Number|real numbers]].
Let $\alpha \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\overline {\ma... | By [[Power of Complex Conjugate is Complex Conjugate of Power]]:
:$\overline {\alpha^k} = \paren {\overline \alpha}^k$
for all $k$ between $0$ and $n$.
Then from [[Product of Complex Conjugates]]:
:$\overline {a_k \alpha^k} = \overline {a_k} \cdot \overline {\alpha^k}$
But $a_k$ is [[Definition:Real Number|real]].... | Conjugate of Polynomial is Polynomial of Conjugate | https://proofwiki.org/wiki/Conjugate_of_Polynomial_is_Polynomial_of_Conjugate | https://proofwiki.org/wiki/Conjugate_of_Polynomial_is_Polynomial_of_Conjugate | [
"Polynomial Theory",
"Complex Conjugates"
] | [
"Definition:Polynomial/Complex Numbers",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [
"Power of Complex Conjugate is Complex Conjugate of Power",
"Product of Complex Conjugates",
"Definition:Real Number",
"Complex Number equals Conjugate iff Wholly Real",
"Sum of Complex Conjugates"
] |
proofwiki-14708 | Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers.
Let $\alpha \in \C$ be a root of $f$.
Then $\overline \alpha$ is also a root of $f$, where $\overline \alpha$ denotes the complex conjugate of $\alpha$.
That is, all comp... | Let $\alpha \in \C$ be a root of $f$.
Then $\map f \alpha = 0$ by definition.
Suppose $\alpha$ is wholly real.
Then by Complex Number equals Conjugate iff Wholly Real:
:$\alpha = \overline \alpha$
and so $\overline \alpha$ is a root of $f$.
Now let $\alpha \in \C$ not be wholly real.
By definition of complex conjugate,... | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a [[Definition:Polynomial over Complex Numbers|polynomial over complex numbers]] where $a_0, \ldots, a_n$ are [[Definition:Real Number|real numbers]].
Let $\alpha \in \C$ be a [[Definition:Root of Polynomial|root]] of $f$.
Then $\overline \alph... | Let $\alpha \in \C$ be a [[Definition:Root of Polynomial|root]] of $f$.
Then $\map f \alpha = 0$ by definition.
Suppose $\alpha$ is [[Definition:Wholly Real|wholly real]].
Then by [[Complex Number equals Conjugate iff Wholly Real]]:
:$\alpha = \overline \alpha$
and so $\overline \alpha$ is a [[Definition:Root of P... | Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs/Proof 1 | https://proofwiki.org/wiki/Complex_Roots_of_Polynomial_with_Real_Coefficients_occur_in_Conjugate_Pairs | https://proofwiki.org/wiki/Complex_Roots_of_Polynomial_with_Real_Coefficients_occur_in_Conjugate_Pairs/Proof_1 | [
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Polynomial Theory",
"Complex Conjugates"
] | [
"Definition:Polynomial/Complex Numbers",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate",
"Definition:Complex Number",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate/Conjugate Pair"
] | [
"Definition:Root of Polynomial",
"Definition:Complex Number/Wholly Real",
"Complex Number equals Conjugate iff Wholly Real",
"Definition:Root of Polynomial",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Conjugate",
"Conjugate of Polynomial is Polynomial of Conjugate",
"Definition:Root ... |
proofwiki-14709 | Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers.
Let $\alpha \in \C$ be a root of $f$.
Then $\overline \alpha$ is also a root of $f$, where $\overline \alpha$ denotes the complex conjugate of $\alpha$.
That is, all comp... | Let $\alpha = p + q i$.
Let $p + q i$ be expressed in exponential form as $\alpha = r e^{i \theta}$.
As $\alpha = r e^{i \theta}$ satisfies $\map f \alpha = 0$, it follows that:
:$a_n r^n e^{n i \theta} + a_{n - 1} r^{n - 1} e^{\paren {n - 1} i \theta} + \dotsb + a_1 r e^{i \theta} + a_0 = 0$
Taking the conjugate of bo... | Let $\map f z = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a [[Definition:Polynomial over Complex Numbers|polynomial over complex numbers]] where $a_0, \ldots, a_n$ are [[Definition:Real Number|real numbers]].
Let $\alpha \in \C$ be a [[Definition:Root of Polynomial|root]] of $f$.
Then $\overline \alph... | Let $\alpha = p + q i$.
Let $p + q i$ be expressed in [[Definition:Exponential Form of Complex Number|exponential form]] as $\alpha = r e^{i \theta}$.
As $\alpha = r e^{i \theta}$ satisfies $\map f \alpha = 0$, it follows that:
:$a_n r^n e^{n i \theta} + a_{n - 1} r^{n - 1} e^{\paren {n - 1} i \theta} + \dotsb + a_1 ... | Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs/Proof 2 | https://proofwiki.org/wiki/Complex_Roots_of_Polynomial_with_Real_Coefficients_occur_in_Conjugate_Pairs | https://proofwiki.org/wiki/Complex_Roots_of_Polynomial_with_Real_Coefficients_occur_in_Conjugate_Pairs/Proof_2 | [
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Polynomial Theory",
"Complex Conjugates"
] | [
"Definition:Polynomial/Complex Numbers",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate",
"Definition:Complex Number",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate/Conjugate Pair"
] | [
"Definition:Complex Number/Polar Form/Exponential Form",
"Definition:Complex Conjugate",
"Definition:Root of Polynomial",
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Real Number"
] |
proofwiki-14710 | Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers | Let $\alpha, \beta \in \C$ be complex numbers.
Then:
:$\cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2 = 2 \cmod \alpha^2 + 2 \cmod \beta^2$ | Let:
:$\alpha = x_1 + i y_1$
:$\beta = x_2 + i y_2$
Then:
{{begin-eqn}}
{{eqn | o =
| r = \cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2
| c =
}}
{{eqn | r = \cmod {\paren {x_1 + i y_1} + \paren {x_2 + i y_2} }^2 + \cmod {\paren {x_1 + i y_1} - \paren {x_2 + i y_2} }^2
| c = Definition of $\alp... | Let $\alpha, \beta \in \C$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2 = 2 \cmod \alpha^2 + 2 \cmod \beta^2$ | Let:
:$\alpha = x_1 + i y_1$
:$\beta = x_2 + i y_2$
Then:
{{begin-eqn}}
{{eqn | o =
| r = \cmod {\alpha + \beta}^2 + \cmod {\alpha - \beta}^2
| c =
}}
{{eqn | r = \cmod {\paren {x_1 + i y_1} + \paren {x_2 + i y_2} }^2 + \cmod {\paren {x_1 + i y_1} - \paren {x_2 + i y_2} }^2
| c = Definition of $\a... | Sum of Squares of Complex Moduli of Sum and Differences of Complex Numbers | https://proofwiki.org/wiki/Sum_of_Squares_of_Complex_Moduli_of_Sum_and_Differences_of_Complex_Numbers | https://proofwiki.org/wiki/Sum_of_Squares_of_Complex_Moduli_of_Sum_and_Differences_of_Complex_Numbers | [
"Complex Modulus"
] | [
"Definition:Complex Number"
] | [
"Square of Sum",
"Square of Difference"
] |
proofwiki-14711 | Equation of Circle in Complex Plane/Formulation 1 | Let $\C$ be the complex plane.
Let $C$ be a circle in $\C$ whose radius is $r \in \R_{>0}$ and whose center is $\alpha \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} = r$
where $\cmod {\, \cdot \,}$ denotes complex modulus. | Let $z = x + i y$.
Let $\alpha = a + i b$.
Thus:
{{begin-eqn}}
{{eqn | l = \cmod {z - \alpha}
| r = r
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {x + i y - a + i b}
| r = r
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {\paren {x - a} + i \paren {y - b} }
| r = r
| c = {{Def... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $C$ be a [[Definition:Circle|circle]] in $\C$ whose [[Definition:Radius of Circle|radius]] is $r \in \R_{>0}$ and whose [[Definition:Center of Circle|center]] is $\alpha \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} = r$
where $\cmod {\, \cd... | Let $z = x + i y$.
Let $\alpha = a + i b$.
Thus:
{{begin-eqn}}
{{eqn | l = \cmod {z - \alpha}
| r = r
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {x + i y - a + i b}
| r = r
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {\paren {x - a} + i \paren {y - b} }
| r = r
| c = {{... | Equation of Circle in Complex Plane/Formulation 1 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1 | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1 | [
"Equation of Circle in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Complex Modulus"
] | [
"Equation of Circle"
] |
proofwiki-14712 | Equation of Circle in Complex Plane/Formulation 1/Interior | The points in $\C$ which correspond to the interior of $C$ can be defined by:
:$\cmod {z - \alpha} < r$ | From Equation of Circle in Complex Plane, the circle $C$ itself is given by:
:$\cmod {z - \alpha} = r$
{{ProofWanted|This needs to be put into the rigorous context of Jordan curves, so as to define what is actually meant by "interior". At the moment, the understanding is intuitive.}} | The points in $\C$ which correspond to the [[Definition:Interior (Complex Analysis)|interior]] of $C$ can be defined by:
:$\cmod {z - \alpha} < r$ | From [[Equation of Circle in Complex Plane/Formulation 1|Equation of Circle in Complex Plane]], the [[Definition:Circle|circle]] $C$ itself is given by:
:$\cmod {z - \alpha} = r$
{{ProofWanted|This needs to be put into the rigorous context of Jordan curves, so as to define what is actually meant by "interior". At th... | Equation of Circle in Complex Plane/Formulation 1/Interior | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1/Interior | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1/Interior | [
"Equation of Circle in Complex Plane"
] | [
"Definition:Interior (Complex Analysis)"
] | [
"Equation of Circle in Complex Plane/Formulation 1",
"Definition:Circle"
] |
proofwiki-14713 | Equation of Circle in Complex Plane/Formulation 1/Exterior | The points in $\C$ which correspond to the exterior of $C$ can be defined by:
:$\left\lvert{z - \alpha}\right\rvert > r$ | From Equation of Circle in Complex Plane, the circle $C$ itself is given by:
:$\left\lvert{z - \alpha}\right\rvert = r$
{{ProofWanted|This needs to be put into the rigorous context of Jordan curves, so as to define what is actually meant by "exterior". At the moment, the understanding is intuitive.}} | The points in $\C$ which correspond to the [[Definition:Exterior (Complex Analysis)|exterior]] of $C$ can be defined by:
:$\left\lvert{z - \alpha}\right\rvert > r$ | From [[Equation of Circle in Complex Plane/Formulation 1|Equation of Circle in Complex Plane]], the [[Definition:Circle|circle]] $C$ itself is given by:
:$\left\lvert{z - \alpha}\right\rvert = r$
{{ProofWanted|This needs to be put into the rigorous context of Jordan curves, so as to define what is actually meant by "... | Equation of Circle in Complex Plane/Formulation 1/Exterior | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1/Exterior | https://proofwiki.org/wiki/Equation_of_Circle_in_Complex_Plane/Formulation_1/Exterior | [
"Equation of Circle in Complex Plane"
] | [
"Definition:Exterior (Complex Analysis)"
] | [
"Equation of Circle in Complex Plane/Formulation 1",
"Definition:Circle"
] |
proofwiki-14714 | Equation of Imaginary Axis in Complex Plane | Let $\C$ be the complex plane.
Let $z \in \C$ be subject to the condition:
:$\cmod {z - 1} = \cmod {z + 1}$
where $\cmod {\, \cdot \,}$ denotes complex modulus.
Then the locus of $z$ is the imaginary axis. | {{begin-eqn}}
{{eqn | l = \cmod {z - 1}
| r = \cmod {z + 1}
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {z - 1}^2
| r = \cmod {z + 1}^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {z - 1} \paren {\overline {z - 1} }
| r = \paren {z + 1} \paren {\overline {z + 1} }
| c = Mo... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $z \in \C$ be subject to the condition:
:$\cmod {z - 1} = \cmod {z + 1}$
where $\cmod {\, \cdot \,}$ denotes [[Definition:Complex Modulus|complex modulus]].
Then the [[Definition:Locus|locus]] of $z$ is the [[Definition:Imaginary Axis|imaginary axis]]. | {{begin-eqn}}
{{eqn | l = \cmod {z - 1}
| r = \cmod {z + 1}
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {z - 1}^2
| r = \cmod {z + 1}^2
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {z - 1} \paren {\overline {z - 1} }
| r = \paren {z + 1} \paren {\overline {z + 1} }
| c = [[... | Equation of Imaginary Axis in Complex Plane | https://proofwiki.org/wiki/Equation_of_Imaginary_Axis_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Imaginary_Axis_in_Complex_Plane | [
"Geometry of Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Complex Modulus",
"Definition:Locus",
"Definition:Complex Number/Complex Plane/Imaginary Axis"
] | [
"Modulus in Terms of Conjugate",
"Sum of Complex Number with Conjugate",
"Definition:Complex Number/Complex Plane/Imaginary Axis"
] |
proofwiki-14715 | Equation of Line in Complex Plane/Formulation 2 | Let $\C$ be the complex plane.
Let $L$ be the infinite straight line in $\C$ which is the locus of the equation:
:$l x + m y = 1$
Then $L$ may be written as:
:$\map \Re {a z} = 1$
where $a$ is the point in $\C$ defined as:
:$a = l - i m$ | Let $z = x + i y$.
Let $a = l - i m$.
Then:
{{begin-eqn}}
{{eqn | l = \map \Re {a z}
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {a z + \overline {a z} } } 2
| r = 1
| c = Sum of Complex Number with Conjugate
}}
{{eqn | ll= \leadsto
| l = a z + \overline a \cdot \over... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $L$ be the [[Definition:Infinite Straight Line|infinite straight line]] in $\C$ which is the [[Definition:Locus|locus]] of the equation:
:$l x + m y = 1$
Then $L$ may be written as:
:$\map \Re {a z} = 1$
where $a$ is the [[Definition:Point|point]] in $\... | Let $z = x + i y$.
Let $a = l - i m$.
Then:
{{begin-eqn}}
{{eqn | l = \map \Re {a z}
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {a z + \overline {a z} } } 2
| r = 1
| c = [[Sum of Complex Number with Conjugate]]
}}
{{eqn | ll= \leadsto
| l = a z + \overline a \cdo... | Equation of Line in Complex Plane/Formulation 2 | https://proofwiki.org/wiki/Equation_of_Line_in_Complex_Plane/Formulation_2 | https://proofwiki.org/wiki/Equation_of_Line_in_Complex_Plane/Formulation_2 | [
"Equation of Line in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Line/Infinite Straight Line",
"Definition:Locus",
"Definition:Point"
] | [
"Sum of Complex Number with Conjugate",
"Complex Modulus of Product of Complex Numbers"
] |
proofwiki-14716 | Conversion between Cartesian and Polar Coordinates in Plane | Let $S$ be the plane.
Let a Cartesian plane $\CC$ be applied to $S$.
Let a polar coordinate plane $\PP$ be superimposed upon $\CC$ such that:
:$(1): \quad$ The origin of $\CC$ coincides with the pole of $\PP$.
:$(2): \quad$ The $x$-axis of $\CC$ coincides with the polar axis of $\PP$.
Let $p$ be a point in $S$.
Let $p$... | Let $P$ be a point in the plane expressed:
:in Cartesian coordinates as $\tuple {x, y}$
:in polar coordinates as $\polar {r, \theta}$.
:330px
As specified, we identify:
:the origins of both coordinate systems with a distinguished point $O$
:the $x$-axis of $C$ with the polar axis of $P$.
Let a perpendicular $PM$ be dro... | Let $S$ be [[Definition:The Plane|the plane]].
Let a [[Definition:Cartesian Plane|Cartesian plane]] $\CC$ be applied to $S$.
Let a [[Definition:Polar Coordinate Plane|polar coordinate plane]] $\PP$ be superimposed upon $\CC$ such that:
:$(1): \quad$ The [[Definition:Origin|origin]] of $\CC$ coincides with the [[Defi... | Let $P$ be a [[Definition:Point|point]] in [[Definition:The Plane|the plane]] expressed:
:in [[Definition:Cartesian Coordinates|Cartesian coordinates]] as $\tuple {x, y}$
:in [[Definition:Polar Coordinates|polar coordinates]] as $\polar {r, \theta}$.
:[[File:Cartesian-polar-conversion.png|330px]]
As specified, we ... | Conversion between Cartesian and Polar Coordinates in Plane | https://proofwiki.org/wiki/Conversion_between_Cartesian_and_Polar_Coordinates_in_Plane | https://proofwiki.org/wiki/Conversion_between_Cartesian_and_Polar_Coordinates_in_Plane | [
"Conversion between Cartesian and Polar Coordinates in Plane",
"Cartesian Coordinate Systems",
"Polar Coordinates"
] | [
"Definition:Plane Surface/The Plane",
"Definition:Cartesian Plane",
"Definition:Polar Coordinates/Polar Plane",
"Definition:Coordinate System/Origin",
"Definition:Polar Coordinates/Pole",
"Definition:Axis/X-Axis",
"Definition:Polar Coordinates/Polar Axis",
"Definition:Point",
"Definition:Polar Coord... | [
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Cartesian Coordinate System",
"Definition:Polar Coordinates",
"File:Cartesian-polar-conversion.png",
"Definition:Coordinate System/Origin",
"Definition:Coordinate System",
"Definition:Distinct/Singular",
"Definition:Point",
"Def... |
proofwiki-14717 | Polar Form of Reciprocal of Complex Number | Let $z := r \paren {\cos \theta + i \sin \theta} \in \C$ be a complex number expressed in polar form.
Then:
:$\dfrac 1 z = \dfrac {\cos \theta - i \sin \theta} r$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 z
| r = \dfrac {\overline z} {z \overline z}
| c = Inverse for Complex Multiplication
}}
{{eqn | r = \dfrac {r \paren {\cos \theta - i \sin \theta} } {r \paren {\cos \theta + i \sin \theta} r \paren {\cos \theta - i \sin \theta} }
| c = Polar Form of Complex Conjugat... | Let $z := r \paren {\cos \theta + i \sin \theta} \in \C$ be a [[Definition:Polar Form of Complex Number|complex number expressed in polar form]].
Then:
:$\dfrac 1 z = \dfrac {\cos \theta - i \sin \theta} r$ | {{begin-eqn}}
{{eqn | l = \dfrac 1 z
| r = \dfrac {\overline z} {z \overline z}
| c = [[Inverse for Complex Multiplication]]
}}
{{eqn | r = \dfrac {r \paren {\cos \theta - i \sin \theta} } {r \paren {\cos \theta + i \sin \theta} r \paren {\cos \theta - i \sin \theta} }
| c = [[Polar Form of Complex Co... | Polar Form of Reciprocal of Complex Number | https://proofwiki.org/wiki/Polar_Form_of_Reciprocal_of_Complex_Number | https://proofwiki.org/wiki/Polar_Form_of_Reciprocal_of_Complex_Number | [
"Polar Form of Complex Number"
] | [
"Definition:Complex Number/Polar Form"
] | [
"Inverse for Complex Multiplication",
"Polar Form of Complex Conjugate",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14718 | Argument of Complex Conjugate equals Argument of Reciprocal | Let $z \in \C$ be a complex number.
Then:
:$\arg {\overline z} = \arg \dfrac 1 z$
where:
:$\arg$ denotes the argument of a complex number
:$\overline z$ denotes the complex conjugate of $z$. | Let $z$ be expressed in polar form:
:$z := r \paren {\cos \theta + i \sin \theta}$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 z
| r = \dfrac 1 r \paren {\cos \theta - i \sin \theta}
| c = Polar Form of Reciprocal of Complex Number
}}
{{eqn | r = \dfrac 1 r \paren {\map \cos {-\theta} + i \map \sin {-\theta} }... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\arg {\overline z} = \arg \dfrac 1 z$
where:
:$\arg$ denotes the [[Definition:Argument of Complex Number|argument]] of a [[Definition:Complex Number|complex number]]
:$\overline z$ denotes the [[Definition:Complex Conjugate|complex conjugate]] ... | Let $z$ be expressed in [[Definition:Polar Form of Complex Number|polar form]]:
:$z := r \paren {\cos \theta + i \sin \theta}$
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 z
| r = \dfrac 1 r \paren {\cos \theta - i \sin \theta}
| c = [[Polar Form of Reciprocal of Complex Number]]
}}
{{eqn | r = \dfrac 1 r \... | Argument of Complex Conjugate equals Argument of Reciprocal | https://proofwiki.org/wiki/Argument_of_Complex_Conjugate_equals_Argument_of_Reciprocal | https://proofwiki.org/wiki/Argument_of_Complex_Conjugate_equals_Argument_of_Reciprocal | [
"Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Argument of Complex Number",
"Definition:Complex Number",
"Definition:Complex Conjugate"
] | [
"Definition:Complex Number/Polar Form",
"Polar Form of Reciprocal of Complex Number",
"Cosine Function is Even",
"Sine Function is Odd",
"Definition:Argument of Complex Number",
"Definition:Complex Number"
] |
proofwiki-14719 | Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle | Let $z \in \C$ be a complex number.
Let $z$ be interpreted as a vector in the complex plane.
Let $w \in \C$ be the complex number defined as $z$ multiplied by the imaginary unit $i$:
:$w = i z$
Then $w$ can be interpreted as the vector $z$ after being rotated through a right angle in an anticlockwise direction. | :600px
Let $z$ be expressed in polar form as:
:$z = r \left({\cos \theta + i \sin \theta}\right)$
From Polar Form of Complex Number: $i$:
:$i = \cos \dfrac \pi 2 + i \sin \dfrac \pi 2$
and so:
: the modulus of $i$ is $1$
: the argument of $i$ is $\dfrac \pi 2$.
By Product of Complex Numbers in Polar Form:
: the modulus... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $z$ be interpreted as a [[Definition:Complex Number as Vector|vector]] in the [[Definition:Complex Plane|complex plane]].
Let $w \in \C$ be the [[Definition:Complex Number|complex number]] defined as $z$ [[Definition:Complex Multiplication|multipli... | :[[File:Rotation-by-i.png|600px]]
Let $z$ be expressed in [[Definition:Polar Form of Complex Number|polar form]] as:
:$z = r \left({\cos \theta + i \sin \theta}\right)$
From [[Polar Form of Complex Number/Examples/i|Polar Form of Complex Number: $i$]]:
:$i = \cos \dfrac \pi 2 + i \sin \dfrac \pi 2$
and so:
: the [[... | Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle | https://proofwiki.org/wiki/Multiplication_by_Imaginary_Unit_is_Equivalent_to_Rotation_through_Right_Angle | https://proofwiki.org/wiki/Multiplication_by_Imaginary_Unit_is_Equivalent_to_Rotation_through_Right_Angle | [
"Complex Multiplication",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number as Vector",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number",
"Definition:Multiplication/Complex Numbers",
"Definition:Complex Number/Imaginary Unit",
"Definition:Complex Number as Vector",
"Definition:Rotation (Geometry)/Pl... | [
"File:Rotation-by-i.png",
"Definition:Complex Number/Polar Form",
"Polar Form of Complex Number/Examples/i",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"Product of Complex Numbers in Polar Form",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"Defi... |
proofwiki-14720 | Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles | Let $z \in \C$ be a complex number.
Let $z$ be interpreted as a vector in the complex plane.
Let $w \in \C$ be the complex number defined as $z$ multiplied by $-1$:
:$w = \left({-1}\right) z$
Then $w$ can be interpreted as the vector $z$ after being rotated through two right angles.
The direction of rotation is usually... | :600px
By definition of the imaginary unit:
:$-1 = i^2$
and so:
:$-1 \times z = i \paren {i z}$
From Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle, multiplication by $i$ is equivalent to rotation through a right angle, in an anticlockwise direction.
So multiplying by $i^2$ is equivalent... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $z$ be interpreted as a [[Definition:Complex Number as Vector|vector]] in the [[Definition:Complex Plane|complex plane]].
Let $w \in \C$ be the [[Definition:Complex Number|complex number]] defined as $z$ [[Definition:Complex Multiplication|multipli... | :[[File:Rotation-by-minus-1.png|600px]]
By definition of the [[Definition:Imaginary Unit|imaginary unit]]:
:$-1 = i^2$
and so:
:$-1 \times z = i \paren {i z}$
From [[Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle]], [[Definition:Complex Multiplication|multiplication]] by $i$ is equ... | Multiplication of Complex Number by -1 is Equivalent to Rotation through Two Right Angles | https://proofwiki.org/wiki/Multiplication_of_Complex_Number_by_-1_is_Equivalent_to_Rotation_through_Two_Right_Angles | https://proofwiki.org/wiki/Multiplication_of_Complex_Number_by_-1_is_Equivalent_to_Rotation_through_Two_Right_Angles | [
"Complex Multiplication",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number as Vector",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number",
"Definition:Multiplication/Complex Numbers",
"Definition:Complex Number as Vector",
"Definition:Rotation (Geometry)/Plane",
"Definition:Right Angle",
"Definitio... | [
"File:Rotation-by-minus-1.png",
"Definition:Complex Number/Imaginary Unit",
"Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle",
"Definition:Multiplication/Complex Numbers",
"Definition:Rotation (Geometry)/Plane",
"Definition:Right Angle",
"Definition:Anticlockwise",
"Defi... |
proofwiki-14721 | Diagonals of Rhombus Intersect at Right Angles | Let $ABCD$ be a rhombus.
The diagonals $AC$ and $BD$ of $ABCD$ intersect each other at right angles. | :400px
{{WLOG}}, let $ABCD$ be embedded in the complex plane so that vertex $A$ coincides with the origin $0 + 0 i$.
Let $AB$ and $AD$ be represented by the complex numbers $a$ and $b$ respectively, expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.
By Geometrical Interpretation of Complex Addition, the dia... | Let $ABCD$ be a [[Definition:Rhombus|rhombus]].
The [[Definition:Diagonal of Parallelogram|diagonals]] $AC$ and $BD$ of $ABCD$ [[Definition:Intersection (Geometry)|intersect]] each other at [[Definition:Right Angle|right angles]]. | :[[File:Rhombus-diagonals-complex-plane.png|400px]]
{{WLOG}}, let $ABCD$ be embedded in the [[Definition:Complex Plane|complex plane]] so that [[Definition:Vertex of Polygon|vertex]] $A$ coincides with the [[Definition:Origin|origin]] $0 + 0 i$.
Let $AB$ and $AD$ be represented by the [[Definition:Complex Number|com... | Diagonals of Rhombus Intersect at Right Angles | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Intersect_at_Right_Angles | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Intersect_at_Right_Angles | [
"Parallelograms"
] | [
"Definition:Quadrilateral/Rhombus",
"Definition:Diameter of Parallelogram",
"Definition:Intersection (Geometry)",
"Definition:Right Angle"
] | [
"File:Rhombus-diagonals-complex-plane.png",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Coordinate System/Origin",
"Definition:Complex Number",
"Definition:Complex Number as Vector",
"Geometrical Interpretation of Complex Addition",
"Definition:Diameter of Paral... |
proofwiki-14722 | Geometrical Interpretation of Complex Addition | Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.
Then the diagonal $OC$ of $OACB$ corresponds to $\mathbf a + \mathbf b$, ... | :400px
Let $a = a_x + i a_y$ and $b = b_x + i b_y$.
Then by definition of complex addition:
:$a + b = \paren {a_x + b_x} + i \paren {a_y + b_y}$
Thus $\mathbf a + \mathbf b$ is the vector whose components are $a_x + b_x$ and $a_y + b_y$.
Similarly, we have:
:$b + a = \paren {b_x + a_x} + i \paren {b_y + a_y}$
Thus $\ma... | Let $a, b \in \C$ be [[Definition:Complex Number|complex numbers]] expressed as [[Definition:Complex Number as Vector|vectors]] $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two [[Definition:Adjacent Sides|adjacent sides]] of the [[Definition:Parallelogram|parallelogram]] $OACB$ such that $OA$ corresp... | :[[File:Complex-Addition-as-Parallelogram.png|400px]]
Let $a = a_x + i a_y$ and $b = b_x + i b_y$.
Then by definition of [[Definition:Complex Addition|complex addition]]:
:$a + b = \paren {a_x + b_x} + i \paren {a_y + b_y}$
Thus $\mathbf a + \mathbf b$ is the [[Definition:Complex Number as Vector|vector]] whose [[D... | Geometrical Interpretation of Complex Addition | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Addition | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Addition | [
"Complex Addition",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number as Vector",
"Definition:Polygon/Adjacent/Sides",
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Vector Sum",
"Definition:Complex Number as Vector"
] | [
"File:Complex-Addition-as-Parallelogram.png",
"Definition:Addition/Complex Numbers",
"Definition:Complex Number as Vector",
"Definition:Vector Quantity/Component",
"Definition:Complex Number as Vector",
"Definition:Vector Quantity/Component",
"Definition:Diameter of Parallelogram"
] |
proofwiki-14723 | Geometrical Interpretation of Complex Subtraction | Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.
Then the diagonal $BA$ of $OACB$ corresponds to $\mathbf a - \mathbf b$, ... | :400px
By definition of vector addition:
:$OB + BA = OA$
That is:
:$\mathbf b + \vec {BA} = \mathbf a$
which leads directly to:
:$\vec {BA} = \mathbf a - \mathbf b$
{{qed}} | Let $a, b \in \C$ be [[Definition:Complex Number|complex numbers]] expressed as [[Definition:Complex Number as Vector|vectors]] $\mathbf a$ and $\mathbf b$ respectively.
Let $OA$ and $OB$ be two [[Definition:Adjacent Sides|adjacent sides]] of the [[Definition:Parallelogram|parallelogram]] $OACB$ such that $OA$ corresp... | :[[File:Complex-Subtraction-as-Parallelogram.png|400px]]
By definition of [[Definition:Vector Sum|vector addition]]:
:$OB + BA = OA$
That is:
:$\mathbf b + \vec {BA} = \mathbf a$
which leads directly to:
:$\vec {BA} = \mathbf a - \mathbf b$
{{qed}} | Geometrical Interpretation of Complex Subtraction | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Subtraction | https://proofwiki.org/wiki/Geometrical_Interpretation_of_Complex_Subtraction | [
"Complex Addition",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number as Vector",
"Definition:Polygon/Adjacent/Sides",
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Subtraction/Complex Numbers",
"Definition:Complex Number as Vector"
] | [
"File:Complex-Subtraction-as-Parallelogram.png",
"Definition:Vector Sum"
] |
proofwiki-14724 | Condition for Collinearity of Points in Complex Plane/Formulation 1 | Let $z_1$, $z_2$ and $z_3$ be points in the complex plane.
Then $z_1$, $z_2$ and $z_3$ are collinear {{iff}}:
:$\dfrac {z_1 - z_3} {z_3 - z_2} = \lambda$
where $\lambda \in \R$ is a real number.
If this is the case, then $z_3$ divides the line segment in the ratio $\lambda$.
If $\lambda > 0$ then $z_3$ is between $z_1$... | By Geometrical Interpretation of Complex Subtraction:
:$z_1 - z_3$ can be represented as the line segment from $z_3$ to $z_1$
:$z_3 - z_2$ can be represented as the line segment from $z_2$ to $z_3$.
Thus we have that $z_1$, $z_2$ and $z_3$ are collinear {{iff}} $z_1 - z_3$ is parallel to $z_3 - z_2$, when expressed as ... | Let $z_1$, $z_2$ and $z_3$ be [[Definition:Point|points]] in the [[Definition:Complex Plane|complex plane]].
Then $z_1$, $z_2$ and $z_3$ are [[Definition:Collinear Points|collinear]] {{iff}}:
:$\dfrac {z_1 - z_3} {z_3 - z_2} = \lambda$
where $\lambda \in \R$ is a [[Definition:Real Number|real number]].
If this is t... | By [[Geometrical Interpretation of Complex Subtraction]]:
:$z_1 - z_3$ can be represented as the [[Definition:Line Segment|line segment]] from $z_3$ to $z_1$
:$z_3 - z_2$ can be represented as the [[Definition:Line Segment|line segment]] from $z_2$ to $z_3$.
Thus we have that $z_1$, $z_2$ and $z_3$ are [[Definition:... | Condition for Collinearity of Points in Complex Plane/Formulation 1 | https://proofwiki.org/wiki/Condition_for_Collinearity_of_Points_in_Complex_Plane/Formulation_1 | https://proofwiki.org/wiki/Condition_for_Collinearity_of_Points_in_Complex_Plane/Formulation_1 | [
"Condition for Collinearity of Points in Complex Plane"
] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Collinear/Points",
"Definition:Real Number",
"Definition:Line/Segment",
"Definition:Line/Segment"
] | [
"Geometrical Interpretation of Complex Subtraction",
"Definition:Line/Segment",
"Definition:Line/Segment",
"Definition:Collinear/Points",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Segment",
"Complex Multiplication as Geometrical Transformation",
"Definition:Argument of Complex Number",
... |
proofwiki-14725 | Equation of Circular Arc in Complex Plane | Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane.
Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane.
Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$.
Then the equation:
:$\arg \dfrac {z - b} {z - a} =... | :420px
By Geometrical Interpretation of Complex Subtraction:
:$z - a$ represents the line from $A$ to $Z$
:$z - b$ represents the line from $B$ to $Z$
{{begin-eqn}}
{{eqn | l = \arg \dfrac {z - b} {z - a}
| r = \lambda
| c =
}}
{{eqn | ll= \leadsto
| l = \map \arg {z - b} - \map \arg {z - a}
| ... | Let $a, b \in \C$ be [[Definition:Complex Number|complex]] [[Definition:Constant|constants]] representing the [[Definition:Point|points]] $A$ and $B$ respectively in the [[Definition:Complex Plane|complex plane]].
Let $z \in \C$ be a [[Definition:Complex Variable|complex variable]] representing the [[Definition:Point|... | :[[File:Circular-Arc-in-Complex-Plane.png|420px]]
By [[Geometrical Interpretation of Complex Subtraction]]:
:$z - a$ represents the [[Definition:Line Segment|line]] from $A$ to $Z$
:$z - b$ represents the [[Definition:Line Segment|line]] from $B$ to $Z$
{{begin-eqn}}
{{eqn | l = \arg \dfrac {z - b} {z - a}
| ... | Equation of Circular Arc in Complex Plane | https://proofwiki.org/wiki/Equation_of_Circular_Arc_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Circular_Arc_in_Complex_Plane | [
"Geometry of Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Constant",
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Variable/Complex",
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Real Number",
"Definition:Constant ",
"Definition:Circle/Arc",
"Definition:... | [
"File:Circular-Arc-in-Complex-Plane.png",
"Geometrical Interpretation of Complex Subtraction",
"Definition:Line/Segment",
"Definition:Line/Segment",
"Argument of Quotient equals Difference of Arguments",
"Definition:Angle",
"Definition:Constant",
"Definition:Angle",
"Definition:Subtend",
"Inscribe... |
proofwiki-14726 | Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm implies Limit of Sequence | The following definition of the concept of the real exponential function: | Let $\exp x$ be the real function defined as the inverse of the natural logarithm:
:$y = \exp x \iff x = \ln y$
Let $\left \langle {x_n} \right \rangle$ be the sequence in $\R$ defined as:
:$x_n = \paren {1 + \dfrac x n}^n$
First it needs to be noted that $\sequence {x_n}$ does indeed converge to a limit.
From Equivale... | The following definition of the concept of the [[Definition:Real Exponential Function|real exponential function]]: | Let $\exp x$ be the [[Definition:Real Function|real function]] defined as the [[Definition:Exponential Function/Real/Inverse of Natural Logarithm|inverse of the natural logarithm]]:
:$y = \exp x \iff x = \ln y$
Let $\left \langle {x_n} \right \rangle$ be the [[Definition:Sequence|sequence in $\R$]] defined as:
:$x_n ... | Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm implies Limit of Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Inverse_of_Natural_Logarithm_implies_Limit_of_Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Inverse_of_Natural_Logarithm_implies_Limit_of_Sequence | [
"Equivalence of Definitions of Exponential Function"
] | [
"Definition:Exponential Function/Real"
] | [
"Definition:Real Function",
"Definition:Exponential Function/Real/Inverse of Natural Logarithm",
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion",
"Series of Power over Factorial... |
proofwiki-14727 | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential | The following definition of the concept of the real exponential function: | Let the restriction of the exponential function to the rationals be defined as:
:$\ds \exp \restriction_\Q: x \mapsto \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
Thus, let $e$ be Euler's Number defined as:
:$e = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac 1 n}^n$
For $x = 0$:
{{begin-eqn}}
{{eqn | l =... | The following definition of the concept of the [[Definition:Real Exponential Function|real exponential function]]: | Let the [[Definition:Restriction of Mapping|restriction]] of the [[Definition:Real Exponential Function|exponential function]] to the [[Definition:Rational Number|rationals]] be defined as:
:$\ds \exp \restriction_\Q: x \mapsto \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
Thus, let $e$ be [[Definition:Eule... | Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Extension of Rational Exponential | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Extension_of_Rational_Exponential | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Limit_of_Sequence_implies_Extension_of_Rational_Exponential | [
"Equivalence of Definitions of Exponential Function"
] | [
"Definition:Exponential Function/Real"
] | [
"Definition:Restriction/Mapping",
"Definition:Exponential Function/Real",
"Definition:Rational Number",
"Definition:Euler's Number/Limit of Sequence",
"Exponent Combination Laws",
"Definition:Continuous Real Function",
"Power Function to Rational Power permits Unique Continuous Extension",
"Power Func... |
proofwiki-14728 | Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation | The following definition of the concept of the real exponential function: | Let $\exp x$ be the real function defined as the extension of rational exponential.
Then we have:
{{begin-eqn}}
{{eqn | l = \map {D_x} {\exp x}
| r = \lim_{h \mathop \to 0} \frac {\map \exp {x + h} - \exp x} h
| c = {{Defof|Derivative}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\exp x \cdot \exp h - \exp... | The following definition of the concept of the [[Definition:Real Exponential Function|real exponential function]]: | Let $\exp x$ be the [[Definition:Real Function|real function]] defined as the [[Definition:Exponential Function/Real/Extension of Rational Exponential|extension of rational exponential]].
Then we have:
{{begin-eqn}}
{{eqn | l = \map {D_x} {\exp x}
| r = \lim_{h \mathop \to 0} \frac {\map \exp {x + h} - \exp x}... | Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Extension_of_Rational_Exponential_implies_Differential_Equation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Real_Exponential_Function/Extension_of_Rational_Exponential_implies_Differential_Equation | [
"Equivalence of Definitions of Exponential Function"
] | [
"Definition:Exponential Function/Real"
] | [
"Definition:Real Function",
"Definition:Exponential Function/Real/Extension of Rational Exponential",
"Exponential of Sum",
"Combination Theorem for Limits of Functions/Real/Multiple Rule",
"Derivative of Exponential at Zero/Proof 2",
"Derivative of Exponential at Zero/Proof 2",
"Definition:Circular Pro... |
proofwiki-14729 | Euler's Number as Limit of 1 + Reciprocal of n to nth Power | :$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = e$
where $e$ denotes Euler's number. | By definition of the real exponential function as the limit of a sequence:
:$(1): \quad \exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
By definition of Euler's number:
:$e = e^1 = \exp 1$
The result follows by setting $x = 1$ in $(1)$.
{{qed}} | :$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = e$
where $e$ denotes [[Definition:Euler's Number|Euler's number]]. | By definition of the [[Definition:Exponential Function/Real/Limit of Sequence|real exponential function as the limit of a sequence]]:
:$(1): \quad \exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
By definition of [[Definition:Euler's Number/Exponential Function|Euler's number]]:
:$e = e^1 = \exp 1... | Euler's Number as Limit of 1 + Reciprocal of n to nth Power/Proof 1 | https://proofwiki.org/wiki/Euler's_Number_as_Limit_of_1_+_Reciprocal_of_n_to_nth_Power | https://proofwiki.org/wiki/Euler's_Number_as_Limit_of_1_+_Reciprocal_of_n_to_nth_Power/Proof_1 | [
"Euler's Number as Limit of 1 + Reciprocal of n to nth Power",
"Euler's Number"
] | [
"Definition:Euler's Number"
] | [
"Definition:Exponential Function/Real/Limit of Sequence",
"Definition:Euler's Number/Exponential Function"
] |
proofwiki-14730 | Exponential of Sum/Complex Numbers/General Result | Let $m \in \N_{>0}$ be a natural number.
Let $z_1, z_2, \ldots, z_m \in \C$ be complex numbers.
Let $\exp z$ be the exponential of $z$.
Then:
:$\ds \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$ | The proof proceeds by induction.
For all $m \in \N_{>0}$, let $\map P m$ be the proposition:
:$\ds \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \map \exp {\sum_{j \mathop = 1}^1 z_j}
| r = \exp z_j
| c =
}}
{{eqn | r =... | Let $m \in \N_{>0}$ be a [[Definition:Natural Number|natural number]].
Let $z_1, z_2, \ldots, z_m \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\exp z$ be the [[Definition:Complex Exponential Function|exponential of $z$]].
Then:
:$\ds \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $m \in \N_{>0}$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\ds \map \exp {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \map \exp {\sum_... | Exponential of Sum/Complex Numbers/General Result | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers/General_Result | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers/General_Result | [
"Exponential of Sum"
] | [
"Definition:Natural Numbers",
"Definition:Complex Number",
"Definition:Exponential Function/Complex"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-14731 | Exponential of Sum/Complex Numbers/General Result/Corollary | Let $m \in \Z_{>0}$ be a positive integer.
Let $z \in \C$ be a complex number.
Let $\exp z$ be the exponential of $z$.
Then:
: $\ds \exp \paren {m z} = \paren {\exp z}^m$ | From Exponential of Sum: Complex Numbers: General Result:
:$\ds \exp \paren {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$
for complex numberst $z_1, z_2, \ldots, z_m \in \C$.
The result follows by setting $z = z_1 = z_2 = \cdots = z_m$.
{{qed}} | Let $m \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let $z \in \C$ be a [[Definition:Complex|complex number]].
Let $\exp z$ be the [[Definition:Complex Exponential Function|exponential of $z$]].
Then:
: $\ds \exp \paren {m z} = \paren {\exp z}^m$ | From [[Exponential of Sum/Complex Numbers/General Result|Exponential of Sum: Complex Numbers: General Result]]:
:$\ds \exp \paren {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$
for [[Definition:Complex Number|complex numbers]]t $z_1, z_2, \ldots, z_m \in \C$.
The result follows by setting ... | Exponential of Sum/Complex Numbers/General Result/Corollary | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers/General_Result/Corollary | https://proofwiki.org/wiki/Exponential_of_Sum/Complex_Numbers/General_Result/Corollary | [
"Exponential of Sum"
] | [
"Definition:Positive/Integer",
"Definition:Complex",
"Definition:Exponential Function/Complex"
] | [
"Exponential of Sum/Complex Numbers/General Result",
"Definition:Complex Number"
] |
proofwiki-14732 | Exponential of Complex Number plus 2 pi i | :$\map \exp {z + 2 \pi i} = \map \exp z$ | {{begin-eqn}}
{{eqn | l = \map \exp {z + 2 \pi i}
| r = \map \exp z \, \map \exp {2 \pi i}
| c = Exponential of Sum: Complex Numbers
}}
{{eqn | r = \map \exp z \times 1
| c = Euler's Formula Example: $e^{2 i \pi}$
}}
{{eqn | r = \map \exp z
}}
{{end-eqn}}
{{qed}} | :$\map \exp {z + 2 \pi i} = \map \exp z$ | {{begin-eqn}}
{{eqn | l = \map \exp {z + 2 \pi i}
| r = \map \exp z \, \map \exp {2 \pi i}
| c = [[Exponential of Sum/Complex Numbers|Exponential of Sum: Complex Numbers]]
}}
{{eqn | r = \map \exp z \times 1
| c = [[Euler's Formula/Examples/e^2 i pi|Euler's Formula Example: $e^{2 i \pi}$]]
}}
{{eqn | ... | Exponential of Complex Number plus 2 pi i | https://proofwiki.org/wiki/Exponential_of_Complex_Number_plus_2_pi_i | https://proofwiki.org/wiki/Exponential_of_Complex_Number_plus_2_pi_i | [
"Exponential Function"
] | [] | [
"Exponential of Sum/Complex Numbers",
"Euler's Formula/Examples/e^2 i pi"
] |
proofwiki-14733 | Sum over k from 1 to n of n Choose k by Sine of n Theta | :$\ds \sum_{k \mathop = 1}^n \dbinom n k \sin k \theta = \paren {2 \cos \dfrac \theta 2}^n \sin \dfrac {n \theta} 2$ | {{begin-eqn}}
{{eqn | l = \paren {1 + e^{i \theta} }^n
| r = \sum_{k \mathop = 0}^n \dbinom n k e^{i k \theta}
| c = Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^n \dbinom n k \paren {\cos k \theta + i \sin k \theta}
| c = Euler's Formula
}}
{{eqn | ll= \leadsto
| l = \map \Im {\paren {1... | :$\ds \sum_{k \mathop = 1}^n \dbinom n k \sin k \theta = \paren {2 \cos \dfrac \theta 2}^n \sin \dfrac {n \theta} 2$ | {{begin-eqn}}
{{eqn | l = \paren {1 + e^{i \theta} }^n
| r = \sum_{k \mathop = 0}^n \dbinom n k e^{i k \theta}
| c = [[Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^n \dbinom n k \paren {\cos k \theta + i \sin k \theta}
| c = [[Euler's Formula]]
}}
{{eqn | ll= \leadsto
| l = \map \Im {\... | Sum over k from 1 to n of n Choose k by Sine of n Theta | https://proofwiki.org/wiki/Sum_over_k_from_1_to_n_of_n_Choose_k_by_Sine_of_n_Theta | https://proofwiki.org/wiki/Sum_over_k_from_1_to_n_of_n_Choose_k_by_Sine_of_n_Theta | [
"Sine Function",
"Binomial Coefficients"
] | [] | [
"Binomial Theorem",
"Euler's Formula",
"Definition:Complex Number/Imaginary Part",
"Definition:Zeroth",
"Euler's Cosine Identity",
"Euler's Formula"
] |
proofwiki-14734 | Point of Perpendicular Intersection on Real Line from Points in Complex Plane | Let $a, b \in \C$ be complex numbers represented by the points $A$ and $B$ respectively in the complex plane.
Let $x \in \R$ be a real number represented by the point $X$ on the real axis such that $AXB$ is a right triangle with $X$ as the right angle.
Then:
:$x = \dfrac {a_x - b_x \pm \sqrt {a_x^2 + b_x^2 + 2 a_x b_x ... | From Geometrical Interpretation of Complex Subtraction, the lines $XA$ and $XB$ can be represented by the complex numbers $a - x$ and $b - x$.
:400px
From Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle $a - x$ and $b - x$ are perpendicular {{iff}} either:
:$a - x = r i \paren {b - x}$
fo... | Let $a, b \in \C$ be [[Definition:Complex Number|complex numbers]] represented by the [[Definition:Point|points]] $A$ and $B$ respectively in the [[Definition:Complex Plane|complex plane]].
Let $x \in \R$ be a [[Definition:Real Number|real number]] represented by the [[Definition:Point|point]] $X$ on the [[Definition:... | From [[Geometrical Interpretation of Complex Subtraction]], the [[Definition:Line Segment|lines]] $XA$ and $XB$ can be represented by the [[Definition:Complex Number|complex numbers]] $a - x$ and $b - x$.
:[[File:Perpendicular-intersection-on-real-axis.png|400px]]
From [[Multiplication by Imaginary Unit is Equivale... | Point of Perpendicular Intersection on Real Line from Points in Complex Plane | https://proofwiki.org/wiki/Point_of_Perpendicular_Intersection_on_Real_Line_from_Points_in_Complex_Plane | https://proofwiki.org/wiki/Point_of_Perpendicular_Intersection_on_Real_Line_from_Points_in_Complex_Plane | [
"Geometry of Complex Plane",
"Point of Perpendicular Intersection on Real Line from Points in Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Real Number",
"Definition:Point",
"Definition:Complex Number/Complex Plane/Real Axis",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Right Angle"
] | [
"Geometrical Interpretation of Complex Subtraction",
"Definition:Line/Segment",
"Definition:Complex Number",
"File:Perpendicular-intersection-on-real-axis.png",
"Multiplication by Imaginary Unit is Equivalent to Rotation through Right Angle",
"Definition:Right Angle/Perpendicular",
"Definition:Real Numb... |
proofwiki-14735 | Condition for Points in Complex Plane to form Parallelogram/Examples/2+i, 3+2i, 2+3i, 1+2i | The points in the complex plane represented by the complex numbers:
:$2 + i, 3 + 2 i, 2 + 3 i, 1 + 2 i$
are the vertices of a square. | Let us label the points:
{{begin-eqn}}
{{eqn | l = A
| o = :=
| r = 2 + i
| c =
}}
{{eqn | l = B
| o = :=
| r = 3 + 2 i
| c =
}}
{{eqn | l = C
| o = :=
| r = 2 + 3 i
| c =
}}
{{eqn | l = D
| o = :=
| r = 1 + 2 i
| c =
}}
{{end-eqn}}
From Geomet... | The [[Definition:Point|points]] in the [[Definition:Complex Plane|complex plane]] represented by the [[Definition:Complex Number|complex numbers]]:
:$2 + i, 3 + 2 i, 2 + 3 i, 1 + 2 i$
are the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Square (Geometry)|square]]. | Let us label the [[Definition:Point|points]]:
{{begin-eqn}}
{{eqn | l = A
| o = :=
| r = 2 + i
| c =
}}
{{eqn | l = B
| o = :=
| r = 3 + 2 i
| c =
}}
{{eqn | l = C
| o = :=
| r = 2 + 3 i
| c =
}}
{{eqn | l = D
| o = :=
| r = 1 + 2 i
| c =
}}
{... | Condition for Points in Complex Plane to form Parallelogram/Examples/2+i, 3+2i, 2+3i, 1+2i | https://proofwiki.org/wiki/Condition_for_Points_in_Complex_Plane_to_form_Parallelogram/Examples/2+i,_3+2i,_2+3i,_1+2i | https://proofwiki.org/wiki/Condition_for_Points_in_Complex_Plane_to_form_Parallelogram/Examples/2+i,_3+2i,_2+3i,_1+2i | [
"Condition for Points in Complex Plane to form Parallelogram"
] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Number",
"Definition:Polygon/Vertex",
"Definition:Quadrilateral/Square"
] | [
"Definition:Point",
"Geometrical Interpretation of Complex Subtraction",
"Definition:Subtraction/Complex Numbers",
"Definition:Complex Number",
"Definition:Vector Form of Complex Number",
"Definition:Subtraction/Complex Numbers",
"Definition:Complex Number",
"Definition:Quadrilateral/Parallelogram",
... |
proofwiki-14736 | Equation of Ellipse in Complex Plane | Let $\C$ be the complex plane.
Let $E$ be an ellipse in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} + \cmod {z - \beta} = d$
where $\cmod {\, \cdot \,}$ denotes complex modulus. | By definition of complex modulus:
:$\cmod {z - \alpha}$ is the distance from $z$ to $\alpha$
:$\cmod {z - \beta}$ is the distance from $z$ to $\beta$.
Thus $\cmod {z - \alpha} + \cmod {z - \beta}$ is the sum of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.
This is precisely the equidistance property of the... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $E$ be an [[Definition:Ellipse|ellipse]] in $\C$ whose [[Definition:Major Axis of Ellipse|major axis]] is $d \in \R_{>0}$ and whose [[Definition:Focus of Ellipse|foci]] are at $\alpha, \beta \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} + \c... | By definition of [[Definition:Complex Modulus|complex modulus]]:
:$\cmod {z - \alpha}$ is the [[Definition:Distance between Points|distance]] from $z$ to $\alpha$
:$\cmod {z - \beta}$ is the [[Definition:Distance between Points|distance]] from $z$ to $\beta$.
Thus $\cmod {z - \alpha} + \cmod {z - \beta}$ is the [[Defi... | Equation of Ellipse in Complex Plane | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Ellipse_in_Complex_Plane | [
"Equation of Ellipse in Complex Plane",
"Ellipses",
"Geometry of Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Ellipse",
"Definition:Ellipse/Major Axis",
"Definition:Ellipse/Focus",
"Definition:Complex Modulus"
] | [
"Definition:Complex Modulus",
"Definition:Distance between Points",
"Definition:Distance between Points",
"Definition:Addition/Real Numbers",
"Definition:Distance between Points",
"Definition:Ellipse/Equidistance",
"Equidistance of Ellipse equals Major Axis",
"Definition:Constant",
"Definition:Dista... |
proofwiki-14737 | Condition for Collinearity of Points in Complex Plane/Formulation 2 | Let $z_1, z_2, z_3$ be distinct complex numbers.
Then:
:$z_1, z_2, z_3$ are collinear in the complex plane
{{iff}}:
::$\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$
:where:
::$\alpha + \beta + \gamma = 0$
::not all of $\alpha, \beta, \gamma$ are zero. | === Sufficient Condition ===
Let $z_1, z_2, z_3$ be collinear.
Then by Condition for Collinearity of Points in Complex Plane: Formulation 1 there exists a real number $b$ such that:
:$z_2 - z_1 = b \paren {z_3 - z_1}$
Then:
{{begin-eqn}}
{{eqn | l = z_2 - z_1
| r = b \paren {z_3 - z_1}
| c =
}}
{{eqn | ll=... | Let $z_1, z_2, z_3$ be [[Definition:Distinct Elements|distinct]] [[Definition:Complex Number|complex numbers]].
Then:
:$z_1, z_2, z_3$ are [[Definition:Collinear Points|collinear]] in the [[Definition:Complex Plane|complex plane]]
{{iff}}:
::$\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 =... | === Sufficient Condition ===
Let $z_1, z_2, z_3$ be [[Definition:Collinear Points|collinear]].
Then by [[Condition for Collinearity of Points in Complex Plane/Formulation 1|Condition for Collinearity of Points in Complex Plane: Formulation 1]] there exists a [[Definition:Real Number|real number]] $b$ such that:
:$z_... | Condition for Collinearity of Points in Complex Plane/Formulation 2 | https://proofwiki.org/wiki/Condition_for_Collinearity_of_Points_in_Complex_Plane/Formulation_2 | https://proofwiki.org/wiki/Condition_for_Collinearity_of_Points_in_Complex_Plane/Formulation_2 | [
"Condition for Collinearity of Points in Complex Plane"
] | [
"Definition:Distinct/Plural",
"Definition:Complex Number",
"Definition:Collinear/Points",
"Definition:Complex Number/Complex Plane"
] | [
"Definition:Collinear/Points",
"Condition for Collinearity of Points in Complex Plane/Formulation 1",
"Definition:Real Number",
"Condition for Collinearity of Points in Complex Plane/Formulation 1",
"Definition:Collinear/Points"
] |
proofwiki-14738 | Quadrilateral in Complex Plane is Cyclic iff Cross Ratio of Vertices is Real | Let $z_1, z_2, z_3, z_4$ be distinct complex numbers.
Then:
:$z_1, z_2, z_3, z_4$ define the vertices of a cyclic quadrilateral
{{iff}} their cross-ratio:
:$\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$
is wholly real. | Let $z_1 z_2 z_3 z_4$ be a cyclic quadrilateral.
By Geometrical Interpretation of Complex Subtraction, the four sides of $z_1 z_2 z_3 z_4$ can be defined as $z_1 - z_2$, $z_3 - z_2$, $z_3 - z_4$ and $z_1 - z_4$.
:400px
From Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles, the opposite angles of $z_1 z_2... | Let $z_1, z_2, z_3, z_4$ be [[Definition:Distinct Elements|distinct]] [[Definition:Complex Number|complex numbers]].
Then:
:$z_1, z_2, z_3, z_4$ define the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]]
{{iff}} their [[Definition:Complex Cross-Ratio|cross-ratio]... | Let $z_1 z_2 z_3 z_4$ be a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]].
By [[Geometrical Interpretation of Complex Subtraction]], the four [[Definition:Side of Polygon|sides]] of $z_1 z_2 z_3 z_4$ can be defined as $z_1 - z_2$, $z_3 - z_2$, $z_3 - z_4$ and $z_1 - z_4$.
:[[File:Cyclic-Quadrilateral-in-Co... | Quadrilateral in Complex Plane is Cyclic iff Cross Ratio of Vertices is Real | https://proofwiki.org/wiki/Quadrilateral_in_Complex_Plane_is_Cyclic_iff_Cross_Ratio_of_Vertices_is_Real | https://proofwiki.org/wiki/Quadrilateral_in_Complex_Plane_is_Cyclic_iff_Cross_Ratio_of_Vertices_is_Real | [
"Geometry of Complex Plane",
"Cyclic Quadrilaterals"
] | [
"Definition:Distinct/Plural",
"Definition:Complex Number",
"Definition:Polygon/Vertex",
"Definition:Cyclic Quadrilateral",
"Definition:Cross-Ratio/Complex Analysis",
"Definition:Complex Number/Wholly Real"
] | [
"Definition:Cyclic Quadrilateral",
"Geometrical Interpretation of Complex Subtraction",
"Definition:Polygon/Side",
"File:Cyclic-Quadrilateral-in-Complex-Plane.png",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Definition:Polygon/Opposite",
"Definition:Angular Measure/Radian",
"C... |
proofwiki-14739 | Equation relating Points of Parallelogram in Complex Plane | Let $ABVU$ be a parallelogram in the complex plane whose vertices correspond to the complex numbers $a, b, v, u$ respectively.
Let $\angle BAU = \alpha$.
Let $\cmod {UA} = \lambda \cmod {AB}$.
:510px
Then:
:$u = \paren {1 - q} a + q b$
:$v = -q a + \paren {1 + q} b$
where:
:$q = \lambda e^{i \alpha}$ | From Geometrical Interpretation of Complex Subtraction, the four sides of $UABC$ can be defined as:
{{begin-eqn}}
{{eqn | l = UA
| r = a - u
| c =
}}
{{eqn | ll= \leadsto
| l = AU
| r = u - a
| c = (easier in this form)
}}
{{eqn | l = AB
| r = b - a
| c =
}}
{{eqn | l = UV
... | Let $ABVU$ be a [[Definition:Parallelogram|parallelogram]] in the [[Definition:Complex Plane|complex plane]] whose [[Definition:Vertex of Polygon|vertices]] correspond to the [[Definition:Complex Number|complex numbers]] $a, b, v, u$ respectively.
Let $\angle BAU = \alpha$.
Let $\cmod {UA} = \lambda \cmod {AB}$.
:[[... | From [[Geometrical Interpretation of Complex Subtraction]], the four [[Definition:Side of Polygon|sides]] of $UABC$ can be defined as:
{{begin-eqn}}
{{eqn | l = UA
| r = a - u
| c =
}}
{{eqn | ll= \leadsto
| l = AU
| r = u - a
| c = (easier in this form)
}}
{{eqn | l = AB
| r = b -... | Equation relating Points of Parallelogram in Complex Plane | https://proofwiki.org/wiki/Equation_relating_Points_of_Parallelogram_in_Complex_Plane | https://proofwiki.org/wiki/Equation_relating_Points_of_Parallelogram_in_Complex_Plane | [
"Parallelograms",
"Geometry of Complex Plane"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Complex Number",
"File:Parallelogram-in-Complex-Plane.png"
] | [
"Geometrical Interpretation of Complex Subtraction",
"Definition:Polygon/Side",
"Product of Complex Numbers in Exponential Form",
"Opposite Sides and Angles of Parallelogram are Equal"
] |
proofwiki-14740 | Circle of Apollonius in Complex Plane | Let $\C$ be the complex plane.
Let $\lambda \in \R$ be a real number such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a circle of Apollonius $C$ in $\C$ such that:
:if $\lambda < 0$, then $a$ is inside $C$ and $b$ i... | By the geometry, the locus described by this equation is a circle of Apollonius.
{{finish|etc.}} | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $\lambda \in \R$ be a [[Definition:Real Number|real number]] such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a [[Definition:Circle of Apollonius|... | By the geometry, the [[Definition:Locus|locus]] described by this equation is a [[Definition:Circle of Apollonius|circle of Apollonius]].
{{finish|etc.}} | Circle of Apollonius in Complex Plane | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane | [
"Equation of Circle in Complex Plane",
"Circle of Apollonius",
"Circle of Apollonius in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Real Number",
"Definition:Circle of Apollonius",
"Definition:Perpendicular Bisector",
"Definition:Line/Segment"
] | [
"Definition:Locus",
"Definition:Circle of Apollonius"
] |
proofwiki-14741 | Circle of Apollonius in Complex Plane | Let $\C$ be the complex plane.
Let $\lambda \in \R$ be a real number such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a circle of Apollonius $C$ in $\C$ such that:
:if $\lambda < 0$, then $a$ is inside $C$ and $b$ i... | A point $P$ on this circle is $2$ times the distance from $z = 3$ as it is from $z = -3$.
:thumb400px
Let $z = x + i y$.
{{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z - 3} {z + 3} }
| r = 2
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \cmod {z - 3}
| r = 2 \cmod {z + 3}
| c =
}}
{{eqn | ll= \... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $\lambda \in \R$ be a [[Definition:Real Number|real number]] such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a [[Definition:Circle of Apollonius|... | A point $P$ on this [[Definition:Circle|circle]] is $2$ times the [[Definition:Distance between Points|distance]] from $z = 3$ as it is from $z = -3$.
:[[File:Circle-of-Apollonius-(z-3)-(z+3).png|thumb|400px]]
Let $z = x + i y$.
{{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z - 3} {z + 3} }
| r = 2
| c =
}}
... | Circle of Apollonius in Complex Plane/Examples/mod z-3 over z+3 = 2/Proof 1 | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane/Examples/mod_z-3_over_z+3_=_2/Proof_1 | [
"Equation of Circle in Complex Plane",
"Circle of Apollonius",
"Circle of Apollonius in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Real Number",
"Definition:Circle of Apollonius",
"Definition:Perpendicular Bisector",
"Definition:Line/Segment"
] | [
"Definition:Circle",
"Definition:Distance between Points",
"File:Circle-of-Apollonius-(z-3)-(z+3).png",
"Equation of Circle in Complex Plane"
] |
proofwiki-14742 | Circle of Apollonius in Complex Plane | Let $\C$ be the complex plane.
Let $\lambda \in \R$ be a real number such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a circle of Apollonius $C$ in $\C$ such that:
:if $\lambda < 0$, then $a$ is inside $C$ and $b$ i... | {{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z - 3} {z + 3} }
| r = 2
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {\dfrac {z - 3} {z + 3} } \paren {\dfrac {\overline z - 3} {\overline z + 3} }
| r = 4
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = z \overline z + 5 \overline z + 5 z ... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $\lambda \in \R$ be a [[Definition:Real Number|real number]] such that $\lambda \ne 0$ and $\lambda \ne 1$.
Let $a, b \in \C$ such that $a \ne b$.
The equation:
:$\cmod {\dfrac {z - a} {z - b} } = \lambda$
decribes a [[Definition:Circle of Apollonius|... | {{begin-eqn}}
{{eqn | l = \cmod {\dfrac {z - 3} {z + 3} }
| r = 2
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {\dfrac {z - 3} {z + 3} } \paren {\dfrac {\overline z - 3} {\overline z + 3} }
| r = 4
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = z \overline z + 5 \overline z + 5 z ... | Circle of Apollonius in Complex Plane/Examples/mod z-3 over z+3 = 2/Proof 2 | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane | https://proofwiki.org/wiki/Circle_of_Apollonius_in_Complex_Plane/Examples/mod_z-3_over_z+3_=_2/Proof_2 | [
"Equation of Circle in Complex Plane",
"Circle of Apollonius",
"Circle of Apollonius in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Real Number",
"Definition:Circle of Apollonius",
"Definition:Perpendicular Bisector",
"Definition:Line/Segment"
] | [
"Equation of Circle in Complex Plane"
] |
proofwiki-14743 | Circle of Apollonius is Circle | Let $A, B$ be distinct points in the plane.
Let $\lambda \in \R_{>0}$ be a strictly positive real number.
Let $X$ be the locus of points in the plane such that:
:$XA = \lambda \paren {XB}$
Then $X$ is in the form of a circle, known as a circle of Apollonius.
:400px
If $\lambda < 1$, then $A$ is inside the circle, and $... | :340px
Let $P$ be an arbitrary point such that $\dfrac {AP} {PB} = \lambda$.
Let $\angle APB$ be bisected internally and externally to intersect $AB$ at $X$ and $Y$ respectively.
Then by Angle Bisector Theorem:
:$\dfrac {AX} {XB} = \dfrac {AP} {PB} = \lambda$
and by Angle Bisector Theorem (Exterior Angle):
:$\dfrac {AY... | Let $A, B$ be [[Definition:Distinct Elements|distinct]] [[Definition:Point|points]] in [[Definition:The Plane|the plane]].
Let $\lambda \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $X$ be the [[Definition:Locus|locus]] of [[Definition:Point|points]] in [[Definitio... | :[[File:Circle-of-Apollonius-Construction.png|340px]]
Let $P$ be an arbitrary [[Definition:Point|point]] such that $\dfrac {AP} {PB} = \lambda$.
Let $\angle APB$ be [[Definition:Angle Bisector|bisected]] [[Definition:Internal Angle Bisector|internally]] and [[Definition:External Angle Bisector|externally]] to [[Defin... | Circle of Apollonius is Circle/Proof 1 | https://proofwiki.org/wiki/Circle_of_Apollonius_is_Circle | https://proofwiki.org/wiki/Circle_of_Apollonius_is_Circle/Proof_1 | [
"Circle of Apollonius is Circle",
"Circle of Apollonius"
] | [
"Definition:Distinct/Plural",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Strictly Positive/Real Number",
"Definition:Locus",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Circle",
"Definition:Circle of Apollonius",
"File:Circle-of-Apollonius.png",... | [
"File:Circle-of-Apollonius-Construction.png",
"Definition:Point",
"Definition:Angle Bisector",
"Definition:Angle Bisector/Internal",
"Definition:Angle Bisector/External",
"Definition:Intersection (Geometry)",
"Angle Bisector Theorem",
"Angle Bisector Theorem/Exterior Angle",
"Definition:Point",
"B... |
proofwiki-14744 | Sextuple Angle Formulas/Cosine | :$\cos 6 \theta = 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta - 1$ | {{begin-eqn}}
{{eqn | l = \cos 6 \theta + i \sin 6 \theta
| r = \paren {\cos \theta + i \sin \theta}^6
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^6 + \binom 6 1 \paren {\cos \theta}^5 \paren {i \sin \theta} + \binom 6 2 \paren {\cos \theta}^4 \paren {i \sin \theta}^2
| c = Binomial ... | :$\cos 6 \theta = 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta - 1$ | {{begin-eqn}}
{{eqn | l = \cos 6 \theta + i \sin 6 \theta
| r = \paren {\cos \theta + i \sin \theta}^6
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^6 + \binom 6 1 \paren {\cos \theta}^5 \paren {i \sin \theta} + \binom 6 2 \paren {\cos \theta}^4 \paren {i \sin \theta}^2
| c = [[Bin... | Sextuple Angle Formulas/Cosine | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Cosine | [
"Cosine Function",
"Sextuple Angle Formulas",
"Sextuple Angle Formula for Cosine"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Real Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14745 | Sextuple Angle Formulas/Sine | :$\dfrac {\sin 6 \theta} {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 6 \theta + i \sin 6 \theta
| r = \paren {\cos \theta + i \sin \theta}^6
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^6 + \binom 6 1 \paren {\cos \theta}^5 \paren {i \sin \theta} + \binom 6 2 \paren {\cos \theta}^4 \paren {i \sin \theta}^2
| c = Binomial ... | :$\dfrac {\sin 6 \theta} {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 6 \theta + i \sin 6 \theta
| r = \paren {\cos \theta + i \sin \theta}^6
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^6 + \binom 6 1 \paren {\cos \theta}^5 \paren {i \sin \theta} + \binom 6 2 \paren {\cos \theta}^4 \paren {i \sin \theta}^2
| c = [[Bin... | Sextuple Angle Formulas/Sine | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Sine | https://proofwiki.org/wiki/Sextuple_Angle_Formulas/Sine | [
"Sine Function",
"Sextuple Angle Formula for Sine",
"Sextuple Angle Formulas"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Imaginary Part",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14746 | Cosine of Angle plus Integer Multiple of Pi | :$\map \cos {\theta + n \pi} = \paren {-1}^n \cos \theta$ | {{begin-eqn}}
{{eqn | l = \map \cos {\theta + n \pi}
| r = \cos \theta \cos n \pi - \sin \theta \sin n \pi
| c = Cosine of Sum
}}
{{eqn | r = \cos \theta \cos n \pi
| c = Sine of Integer Multiple of Pi
}}
{{eqn | r = \paren {-1}^n \cos \theta
| c = Cosine of Integer Multiple of Pi
}}
{{end-eqn}}... | :$\map \cos {\theta + n \pi} = \paren {-1}^n \cos \theta$ | {{begin-eqn}}
{{eqn | l = \map \cos {\theta + n \pi}
| r = \cos \theta \cos n \pi - \sin \theta \sin n \pi
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos \theta \cos n \pi
| c = [[Sine of Integer Multiple of Pi]]
}}
{{eqn | r = \paren {-1}^n \cos \theta
| c = [[Cosine of Integer Multiple of Pi]]
}}... | Cosine of Angle plus Integer Multiple of Pi | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Cosine_of_Angle_plus_Integer_Multiple_of_Pi | [
"Cosine Function"
] | [] | [
"Cosine of Sum",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-14747 | Sine of Angle plus Integer Multiple of Pi | :$\map \sin {\theta + n \pi} = \paren {-1}^n \sin \theta$ | {{begin-eqn}}
{{eqn | l = \map \sin {\theta + n \pi}
| r = \sin \theta \cos n \pi + \cos \theta \sin n \pi
| c = Sine of Sum
}}
{{eqn | r = \sin \theta \cos n \pi
| c = Sine of Integer Multiple of Pi
}}
{{eqn | r = \paren {-1}^n \sin \theta
| c = Cosine of Integer Multiple of Pi
}}
{{end-eqn}}
{... | :$\map \sin {\theta + n \pi} = \paren {-1}^n \sin \theta$ | {{begin-eqn}}
{{eqn | l = \map \sin {\theta + n \pi}
| r = \sin \theta \cos n \pi + \cos \theta \sin n \pi
| c = [[Sine of Sum]]
}}
{{eqn | r = \sin \theta \cos n \pi
| c = [[Sine of Integer Multiple of Pi]]
}}
{{eqn | r = \paren {-1}^n \sin \theta
| c = [[Cosine of Integer Multiple of Pi]]
}}
{... | Sine of Angle plus Integer Multiple of Pi | https://proofwiki.org/wiki/Sine_of_Angle_plus_Integer_Multiple_of_Pi | https://proofwiki.org/wiki/Sine_of_Angle_plus_Integer_Multiple_of_Pi | [
"Sine Function"
] | [] | [
"Sine of Sum",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-14748 | Complex Division/Examples/(1 + sin theta + i cos theta) (1 + sin theta - i cos theta)^-1 | :$\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta} = \sin \theta + i \cos \theta$ | {{begin-eqn}}
{{eqn | l = \frac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta}
| r = \frac {\paren {1 + \sin \theta + i \cos \theta}^2} {\paren {1 + \sin \theta - i \cos \theta} \paren {1 + \sin \theta + i \cos \theta} }
| c =
}}
{{eqn | r = \frac {\paren {1 + \sin \theta}^2 + 2 i \cos... | :$\dfrac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta} = \sin \theta + i \cos \theta$ | {{begin-eqn}}
{{eqn | l = \frac {1 + \sin \theta + i \cos \theta} {1 + \sin \theta - i \cos \theta}
| r = \frac {\paren {1 + \sin \theta + i \cos \theta}^2} {\paren {1 + \sin \theta - i \cos \theta} \paren {1 + \sin \theta + i \cos \theta} }
| c =
}}
{{eqn | r = \frac {\paren {1 + \sin \theta}^2 + 2 i \cos... | Complex Division/Examples/(1 + sin theta + i cos theta) (1 + sin theta - i cos theta)^-1 | https://proofwiki.org/wiki/Complex_Division/Examples/(1_+_sin_theta_+_i_cos_theta)_(1_+_sin_theta_-_i_cos_theta)^-1 | https://proofwiki.org/wiki/Complex_Division/Examples/(1_+_sin_theta_+_i_cos_theta)_(1_+_sin_theta_-_i_cos_theta)^-1 | [
"Examples of Complex Division"
] | [] | [
"Sum of Squares of Sine and Cosine"
] |
proofwiki-14749 | Sum of 1 + sin pi by 5 plus i cos pi by 5 to Fifth Power plus i times its Conjugate | :$\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 = 0$ | {{begin-eqn}}
{{eqn | r = \paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5
| o =
| c =
}}
{{eqn | r = \frac {\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5} {\paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5} \paren {1 + \sin \d... | :$\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5 = 0$ | {{begin-eqn}}
{{eqn | r = \paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5 + i \paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5
| o =
| c =
}}
{{eqn | r = \frac {\paren {1 + \sin \dfrac \pi 5 + i \cos \dfrac \pi 5}^5} {\paren {1 + \sin \dfrac \pi 5 - i \cos \dfrac \pi 5}^5} \paren {1 + \sin \d... | Sum of 1 + sin pi by 5 plus i cos pi by 5 to Fifth Power plus i times its Conjugate | https://proofwiki.org/wiki/Sum_of_1_+_sin_pi_by_5_plus_i_cos_pi_by_5_to_Fifth_Power_plus_i_times_its_Conjugate | https://proofwiki.org/wiki/Sum_of_1_+_sin_pi_by_5_plus_i_cos_pi_by_5_to_Fifth_Power_plus_i_times_its_Conjugate | [
"Examples of Complex Powers"
] | [] | [
"Complex Division/Examples/(1 + sin theta + i cos theta) (1 + sin theta - i cos theta)^-1",
"De Moivre's Formula",
"Sine of Straight Angle",
"Cosine of Straight Angle"
] |
proofwiki-14750 | Complex Roots of Unity include 1 | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.
Then $1 \in U_n$.
That is, $1$ is always one of the complex $n$th roots of unity of any $n$. | By definition of integer power:
:$1^n = 1$
for all $n$.
Hence the result.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Set|set]] of [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
Then $1 \in U_n$.
That is, $1$ is always one of the [[Definition:Complex Roots of ... | By definition of [[Definition:Integer Power|integer power]]:
:$1^n = 1$
for all $n$.
Hence the result.
{{qed}} | Complex Roots of Unity include 1 | https://proofwiki.org/wiki/Complex_Roots_of_Unity_include_1 | https://proofwiki.org/wiki/Complex_Roots_of_Unity_include_1 | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex"
] | [
"Definition:Power (Algebra)/Integer"
] |
proofwiki-14751 | Positive Real Complex Root of Unity | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.
The only $x \in U_n$ such that $x \in \R_{>0}$ is:
:$x = 1$
That is, $1$ is the only complex $n$th root of unity which is a positive real number. | We have that $1$ is a positive real number.
The result follows from Existence and Uniqueness of Positive Root of Positive Real Number.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Set|set]] of [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
The only $x \in U_n$ such that $x \in \R_{>0}$ is:
:$x = 1$
That is, $1$ is the [[... | We have that $1$ is a [[Definition:Positive Real Number|positive real number]].
The result follows from [[Existence and Uniqueness of Positive Root of Positive Real Number]].
{{qed}} | Positive Real Complex Root of Unity | https://proofwiki.org/wiki/Positive_Real_Complex_Root_of_Unity | https://proofwiki.org/wiki/Positive_Real_Complex_Root_of_Unity | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Definition:Unique",
"Definition:Root of Unity/Complex",
"Definition:Positive/Real Number"
] | [
"Definition:Positive/Real Number",
"Existence and Uniqueness of Positive Root of Positive Real Number"
] |
proofwiki-14752 | Real Complex Roots of Unity for Odd Index | Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is odd.
Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.
The only $x \in U_n$ such that $x \in \R$ is:
:$x = 1$
That is, $1$ is the only complex $n$th root of unity which is a real number. | From Positive Real Complex Root of Unity, we have that $1$ is the only positive real number in $U_n$.
{{AimForCont}} $z \in \R$ such that $z \in U_n$ and $z < 0$.
From Odd Power of Negative Real Number is Negative, $z^n < 0$.
But this contradicts the fact that $z_n = 1 > 0$.
Hence by Proof by Contradiction it follows t... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] such that $n$ is [[Definition:Odd Integer|odd]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Set|set]] of [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
The only $x \in U_n$ such that $x \i... | From [[Positive Real Complex Root of Unity]], we have that $1$ is the only [[Definition:Positive Real Number|positive real number]] in $U_n$.
{{AimForCont}} $z \in \R$ such that $z \in U_n$ and $z < 0$.
From [[Odd Power of Negative Real Number is Negative]], $z^n < 0$.
But this [[Definition:Contradiction|contradicts... | Real Complex Roots of Unity for Odd Index | https://proofwiki.org/wiki/Real_Complex_Roots_of_Unity_for_Odd_Index | https://proofwiki.org/wiki/Real_Complex_Roots_of_Unity_for_Odd_Index | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Odd Integer",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Definition:Unique",
"Definition:Root of Unity/Complex",
"Definition:Real Number"
] | [
"Positive Real Complex Root of Unity",
"Definition:Positive/Real Number",
"Odd Power of Negative Real Number is Negative",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Negative/Real Number",
"Definition:Real Number"
] |
proofwiki-14753 | Real Complex Roots of Unity for Even Index | Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is even.
Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.
The only $x \in U_n$ such that $x \in \R$ are:
:$x = 1$ or $x \in -1$
That is, $1$ and $-1$ are the only complex $n$th roots of unity which are real number. | From Positive Real Complex Root of Unity, we have that $1$ is the only element of $U_n$ which is a positive real number.
We note that $\paren {-1}^n = 1$ as $n$ is even.
Thus $-1$ is also an element of $U_n$.
Now let $z \in U_n$ such that $\cmod z \ne 1$.
Let $z > 0$.
From Positive Power Function on Non-negative Reals ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] such that $n$ is [[Definition:Even Integer|even]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Set|set]] of [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
The only $x \in U_n$ such that $x ... | From [[Positive Real Complex Root of Unity]], we have that $1$ is the only element of $U_n$ which is a [[Definition:Positive Real Number|positive real number]].
We note that $\paren {-1}^n = 1$ as $n$ is [[Definition:Even Integer|even]].
Thus $-1$ is also an element of $U_n$.
Now let $z \in U_n$ such that $\cmod z \... | Real Complex Roots of Unity for Even Index | https://proofwiki.org/wiki/Real_Complex_Roots_of_Unity_for_Even_Index | https://proofwiki.org/wiki/Real_Complex_Roots_of_Unity_for_Even_Index | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Even Integer",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex",
"Definition:Real Number"
] | [
"Positive Real Complex Root of Unity",
"Definition:Positive/Real Number",
"Definition:Even Integer",
"Positive Power Function on Non-negative Reals is Strictly Increasing",
"Positive Power Function on Negative Reals is Strictly Decreasing"
] |
proofwiki-14754 | Modulus of Complex Root of Unity equals 1 | Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is even.
Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity.
Let $z \in U_n$.
Then:
:$\cmod z = 1$
where $\cmod z$ denotes the modulus of $z$. | {{begin-eqn}}
{{eqn | l = z^n
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {z^n}
| r = \cmod 1
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod z^n
| r = 1
| c = Power of Complex Modulus equals Complex Modulus of Power
}}
{{eqn | ll= \leadsto... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] such that $n$ is [[Definition:Even Integer|even]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Set|set]] of [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
Let $z \in U_n$.
Then:
:$\cmod z ... | {{begin-eqn}}
{{eqn | l = z^n
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod {z^n}
| r = \cmod 1
| c =
}}
{{eqn | r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = \cmod z^n
| r = 1
| c = [[Power of Complex Modulus equals Complex Modulus of Power]]
}}
{{eqn | ll= \lea... | Modulus of Complex Root of Unity equals 1 | https://proofwiki.org/wiki/Modulus_of_Complex_Root_of_Unity_equals_1 | https://proofwiki.org/wiki/Modulus_of_Complex_Root_of_Unity_equals_1 | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Even Integer",
"Definition:Set",
"Definition:Root of Unity/Complex",
"Definition:Complex Modulus"
] | [
"Power of Complex Modulus equals Complex Modulus of Power",
"Positive Real Complex Root of Unity"
] |
proofwiki-14755 | First Complex Root of Unity is Primitive | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n$ denote the complex $n$th roots of unity:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha_1 = \exp \paren {\dfrac {2 \pi i} n}$ denote the first complex root of unity.
Then $\alpha_1$ is a primitive complex root of unity. | From Condition for Complex Root of Unity to be Primitive:
:$\gcd \set {n, k} = 1$
where:
:$\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$
Here we have:
:$k = 1$
and:
:$\gcd \set {n, 1} = 1$
Hence the result.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n$ denote the [[Definition:Complex Roots of Unity|complex $n$th roots of unity]]:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha_1 = \exp \paren {\dfrac {2 \pi i} n}$ denote the [[Definition:First Complex Root of Unit... | From [[Condition for Complex Root of Unity to be Primitive]]:
:$\gcd \set {n, k} = 1$
where:
:$\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$
Here we have:
:$k = 1$
and:
:$\gcd \set {n, 1} = 1$
Hence the result.
{{qed}} | First Complex Root of Unity is Primitive | https://proofwiki.org/wiki/First_Complex_Root_of_Unity_is_Primitive | https://proofwiki.org/wiki/First_Complex_Root_of_Unity_is_Primitive | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex/First",
"Definition:Root of Unity/Complex/Primitive"
] | [
"Condition for Complex Root of Unity to be Primitive"
] |
proofwiki-14756 | Powers of Primitive Complex Root of Unity form Complete Set | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n$ denote the complex $n$th roots of unity:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$ denote the $k$th complex root of unity.
Let $\alpha_k$ be a primitive complex root of unity.
Let $V_k = \set { {\alpha_k}^r: r \i... | From Roots of Unity under Multiplication form Cyclic Group, $\struct {U_n, \times}$ is a group.
The result follows from Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n$ denote the [[Definition:Complex Roots of Unity|complex $n$th roots of unity]]:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha_k = \exp \paren {\dfrac {2 k \pi i} n}$ denote the [[Definition:First Complex Root of Un... | From [[Roots of Unity under Multiplication form Cyclic Group]], $\struct {U_n, \times}$ is a [[Definition:Group|group]].
The result follows from [[Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order]].
{{qed}} | Powers of Primitive Complex Root of Unity form Complete Set | https://proofwiki.org/wiki/Powers_of_Primitive_Complex_Root_of_Unity_form_Complete_Set | https://proofwiki.org/wiki/Powers_of_Primitive_Complex_Root_of_Unity_form_Complete_Set | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex/First",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Set",
"Definition:Root of Unity/Complex"
] | [
"Roots of Unity under Multiplication form Cyclic Group",
"Definition:Group",
"Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order"
] |
proofwiki-14757 | Sum of Powers of Primitive Complex Roots of Unity | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n$ denote the complex $n$th roots of unity:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha = \exp \paren {\dfrac {2 k \pi i} n}$ denote a primitive complex $n$th root of unity.
Let $s \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l... | First we address the case where $n \divides s$.
Then:
{{begin-eqn}}
{{eqn | l = s
| r = q n
| c = for some $q \in \Z_{>0}$
}}
{{eqn | ll= \leadsto
| l = \alpha^{j s}
| r = \alpha^{j q n}
| c =
}}
{{eqn | r = \paren {\alpha^n}^{j q}
| c =
}}
{{eqn | r = 1^{j q}
| c =
}}
{{eqn... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n$ denote the [[Definition:Complex Roots of Unity|complex $n$th roots of unity]]:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha = \exp \paren {\dfrac {2 k \pi i} n}$ denote a [[Definition:Primitive Complex Root of Un... | First we address the case where $n \divides s$.
Then:
{{begin-eqn}}
{{eqn | l = s
| r = q n
| c = for some $q \in \Z_{>0}$
}}
{{eqn | ll= \leadsto
| l = \alpha^{j s}
| r = \alpha^{j q n}
| c =
}}
{{eqn | r = \paren {\alpha^n}^{j q}
| c =
}}
{{eqn | r = 1^{j q}
| c =
}}
{{e... | Sum of Powers of Primitive Complex Roots of Unity | https://proofwiki.org/wiki/Sum_of_Powers_of_Primitive_Complex_Roots_of_Unity | https://proofwiki.org/wiki/Sum_of_Powers_of_Primitive_Complex_Roots_of_Unity | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Division Theorem",
"Sum of Geometric Sequence"
] |
proofwiki-14758 | Difference of Two Powers/Examples/Difference of Two Cubes/Corollary | :$x^3 - 1 = \paren {x - 1} \paren {x^2 + x + 1}$ | From Difference of Two Cubes:
:$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$
The result follows directly by setting $y = 1$.
{{qed}} | :$x^3 - 1 = \paren {x - 1} \paren {x^2 + x + 1}$ | From [[Difference of Two Cubes]]:
:$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$
The result follows directly by setting $y = 1$.
{{qed}} | Difference of Two Powers/Examples/Difference of Two Cubes/Corollary | https://proofwiki.org/wiki/Difference_of_Two_Powers/Examples/Difference_of_Two_Cubes/Corollary | https://proofwiki.org/wiki/Difference_of_Two_Powers/Examples/Difference_of_Two_Cubes/Corollary | [
"Third Powers",
"Difference of Two Cubes"
] | [] | [
"Difference of Two Powers/Examples/Difference of Two Cubes"
] |
proofwiki-14759 | Sum of Cube Roots of Unity | Let $U_3 = \set {1, \omega, \omega^2}$ denote the Cube Roots of Unity.
Then:
:$1 + \omega + \omega^2 = 0$ | {{begin-eqn}}
{{eqn | l = 1 + \omega + \omega^2
| r = 1 + \paren {-\dfrac 1 2 + \dfrac {\sqrt 3} 2} + \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2}
| c = Cube Roots of Unity
}}
{{eqn | r = 1 - \frac 1 2 - \frac 1 2 + \dfrac {\sqrt 3} 2 - \dfrac {\sqrt 3} 2
| c =
}}
{{eqn | r = 0
| c =
}}
{{end-eqn... | Let $U_3 = \set {1, \omega, \omega^2}$ denote the [[Cube Roots of Unity]].
Then:
:$1 + \omega + \omega^2 = 0$ | {{begin-eqn}}
{{eqn | l = 1 + \omega + \omega^2
| r = 1 + \paren {-\dfrac 1 2 + \dfrac {\sqrt 3} 2} + \paren {-\dfrac 1 2 - \dfrac {\sqrt 3} 2}
| c = [[Cube Roots of Unity]]
}}
{{eqn | r = 1 - \frac 1 2 - \frac 1 2 + \dfrac {\sqrt 3} 2 - \dfrac {\sqrt 3} 2
| c =
}}
{{eqn | r = 0
| c =
}}
{{end... | Sum of Cube Roots of Unity/Proof 1 | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity/Proof_1 | [
"Sum of Cube Roots of Unity",
"Cube Roots of Unity"
] | [
"Complex Roots of Unity/Examples/Cube Roots"
] | [
"Complex Roots of Unity/Examples/Cube Roots"
] |
proofwiki-14760 | Sum of Cube Roots of Unity | Let $U_3 = \set {1, \omega, \omega^2}$ denote the Cube Roots of Unity.
Then:
:$1 + \omega + \omega^2 = 0$ | From Sum of Powers of Primitive Complex Roots of Unity:
:$\ds \sum_{j \mathop = 0}^{n - 1} \alpha^{j s} = \begin {cases} n & : n \divides s \\ 0 & : n \nmid s \end {cases}$
Here we have that $n = 3$ and $s = 1$.
Thus $n$ is not a divisor of $s$.
Hence the result.
{{qed}} | Let $U_3 = \set {1, \omega, \omega^2}$ denote the [[Cube Roots of Unity]].
Then:
:$1 + \omega + \omega^2 = 0$ | From [[Sum of Powers of Primitive Complex Roots of Unity]]:
:$\ds \sum_{j \mathop = 0}^{n - 1} \alpha^{j s} = \begin {cases} n & : n \divides s \\ 0 & : n \nmid s \end {cases}$
Here we have that $n = 3$ and $s = 1$.
Thus $n$ is not a [[Definition:Divisor of Integer|divisor]] of $s$.
Hence the result.
{{qed}} | Sum of Cube Roots of Unity/Proof 2 | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity/Proof_2 | [
"Sum of Cube Roots of Unity",
"Cube Roots of Unity"
] | [
"Complex Roots of Unity/Examples/Cube Roots"
] | [
"Sum of Powers of Primitive Complex Roots of Unity",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-14761 | Sum of Cube Roots of Unity | Let $U_3 = \set {1, \omega, \omega^2}$ denote the Cube Roots of Unity.
Then:
:$1 + \omega + \omega^2 = 0$ | Observe:
{{begin-eqn}}
{{eqn | l = \paren {1 - \omega} \paren {1 + \omega + \omega^2}
| r = 1 - \omega^3
| c = Difference of Two Cubes
}}
{{eqn | r = 1 - 1
| c = Cube Roots of Unity
}}
{{eqn | r = 0
}}
{{end-eqn}}
As $\omega \ne 1$, it follows:
:$1 + \omega + \omega^2 = 0$
{{qed}} | Let $U_3 = \set {1, \omega, \omega^2}$ denote the [[Cube Roots of Unity]].
Then:
:$1 + \omega + \omega^2 = 0$ | Observe:
{{begin-eqn}}
{{eqn | l = \paren {1 - \omega} \paren {1 + \omega + \omega^2}
| r = 1 - \omega^3
| c = [[Difference of Two Cubes]]
}}
{{eqn | r = 1 - 1
| c = [[Cube Roots of Unity]]
}}
{{eqn | r = 0
}}
{{end-eqn}}
As $\omega \ne 1$, it follows:
:$1 + \omega + \omega^2 = 0$
{{qed}} | Sum of Cube Roots of Unity/Proof 3 | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity | https://proofwiki.org/wiki/Sum_of_Cube_Roots_of_Unity/Proof_3 | [
"Sum of Cube Roots of Unity",
"Cube Roots of Unity"
] | [
"Complex Roots of Unity/Examples/Cube Roots"
] | [
"Difference of Two Powers/Examples/Difference of Two Cubes",
"Complex Roots of Unity/Examples/Cube Roots"
] |
proofwiki-14762 | Sum of Cubes of Three Indeterminates Minus 3 Times their Product | For indeterminates $x, y, z$:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
where $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | {{begin-eqn}}
{{eqn | r = \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}
| o =
| c =
}}
{{eqn | r = x^2 + \omega^2 x y + \omega x z + \omega x y + \omega^3 y^2 + \omega^2 y z + x \omega^2 z + \omega^4 y z + \omega^3 z^2
| c =
}}
{{eqn | r = x^2 + y^2 + z^2 + \paren {\omega + ... | For [[Definition:Indeterminate|indeterminates]] $x, y, z$:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
where $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | {{begin-eqn}}
{{eqn | r = \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}
| o =
| c =
}}
{{eqn | r = x^2 + \omega^2 x y + \omega x z + \omega x y + \omega^3 y^2 + \omega^2 y z + x \omega^2 z + \omega^4 y z + \omega^3 z^2
| c =
}}
{{eqn | r = x^2 + y^2 + z^2 + \paren {\omega + ... | Sum of Cubes of Three Indeterminates Minus 3 Times their Product/Proof 1 | https://proofwiki.org/wiki/Sum_of_Cubes_of_Three_Indeterminates_Minus_3_Times_their_Product | https://proofwiki.org/wiki/Sum_of_Cubes_of_Three_Indeterminates_Minus_3_Times_their_Product/Proof_1 | [
"Algebra",
"Cube Roots of Unity",
"Sum of Cubes of Three Indeterminates Minus 3 Times their Product"
] | [
"Definition:Indeterminate"
] | [
"Sum of Cube Roots of Unity"
] |
proofwiki-14763 | Sum of Cubes of Three Indeterminates Minus 3 Times their Product | For indeterminates $x, y, z$:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
where $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | Consider the determinant:
:$\Delta = \begin {vmatrix} x & z & y \\ y & x & z \\ z & y & x \end {vmatrix}$
We have:
{{begin-eqn}}
{{eqn | l = \Delta
| r = x \paren {x^2 - y z} - z \paren {y x - z^2} + y \paren {y^2 - x z}
| c = Determinant of Order 3
}}
{{eqn | r = x^3 + y^3 + z^3 - 3 x y z
| c =
}}
{... | For [[Definition:Indeterminate|indeterminates]] $x, y, z$:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
where $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | Consider the [[Definition:Determinant|determinant]]:
:$\Delta = \begin {vmatrix} x & z & y \\ y & x & z \\ z & y & x \end {vmatrix}$
We have:
{{begin-eqn}}
{{eqn | l = \Delta
| r = x \paren {x^2 - y z} - z \paren {y x - z^2} + y \paren {y^2 - x z}
| c = [[Determinant of Order 3]]
}}
{{eqn | r = x^3 + y^3 ... | Sum of Cubes of Three Indeterminates Minus 3 Times their Product/Proof 2 | https://proofwiki.org/wiki/Sum_of_Cubes_of_Three_Indeterminates_Minus_3_Times_their_Product | https://proofwiki.org/wiki/Sum_of_Cubes_of_Three_Indeterminates_Minus_3_Times_their_Product/Proof_2 | [
"Algebra",
"Cube Roots of Unity",
"Sum of Cubes of Three Indeterminates Minus 3 Times their Product"
] | [
"Definition:Indeterminate"
] | [
"Definition:Determinant",
"Determinant/Examples/Order 3",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Multiple of Row Added to Row of Determinant",
"Determinant/Examples/Order 3",
"Complex Roots of Unity/Examples/Cube Roots",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Mat... |
proofwiki-14764 | Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle | Let $\epsilon_1, \epsilon_2, \epsilon_3$ be complex numbers embedded in the complex plane such that:
:$\epsilon_1, \epsilon_2, \epsilon_3$ all have modulus $1$
:$\epsilon_1 + \epsilon_2 + \epsilon_3 = 0$
Then:
:$\paren {\dfrac {\epsilon_2} {\epsilon_1} }^3 = \paren {\dfrac {\epsilon_3} {\epsilon_2} }^2 = \paren {\dfrac... | We have that:
{{begin-eqn}}
{{eqn | l = \epsilon_1 + \epsilon_2 + \epsilon_3
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \epsilon_1 - \paren {-\epsilon_2}
| r = -\epsilon_3
| c =
}}
{{end-eqn}}
Thus by Geometrical Interpretation of Complex Subtraction, $\epsilon_1$, $\epsilon_2$ and $\ep... | Let $\epsilon_1, \epsilon_2, \epsilon_3$ be [[Definition:Complex Number|complex numbers]] embedded in the [[Definition:Complex Plane|complex plane]] such that:
:$\epsilon_1, \epsilon_2, \epsilon_3$ all have [[Definition:Complex Modulus|modulus]] $1$
:$\epsilon_1 + \epsilon_2 + \epsilon_3 = 0$
Then:
:$\paren {\dfrac ... | We have that:
{{begin-eqn}}
{{eqn | l = \epsilon_1 + \epsilon_2 + \epsilon_3
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \epsilon_1 - \paren {-\epsilon_2}
| r = -\epsilon_3
| c =
}}
{{end-eqn}}
Thus by [[Geometrical Interpretation of Complex Subtraction]], $\epsilon_1$, $\epsilon_2$ an... | Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle | https://proofwiki.org/wiki/Unit_Vectors_in_Complex_Plane_which_are_Vertices_of_Equilateral_Triangle | https://proofwiki.org/wiki/Unit_Vectors_in_Complex_Plane_which_are_Vertices_of_Equilateral_Triangle | [
"Geometry of Complex Plane",
"Equilateral Triangles"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Complex Modulus"
] | [
"Geometrical Interpretation of Complex Subtraction",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Complex Modulus",
"Definition:Triangle (Geometry)/Equilateral",
"File:Equliateral-Triangle-Unit-Sides.png",
"Complex Multiplication as Geometrical Transformation",
"Complex Mul... |
proofwiki-14765 | Combination Theorem for Cauchy Sequences/Sum Rule | :$\sequence {x_n + y_n}$ is a Cauchy sequence. | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
Since $\sequence {x_n}$ is a Cauchy sequence, we can find $N_1$ such that:
:$\forall n, m > N_1: \norm{x_n - x_m} < \dfrac \epsilon 2$
Similarly, $\sequence {y_n}$ is a Cauchy sequence, we can find $N_2$ such that:
: $\forall n, m > N_2: \norm{y_n - y_m} < \dfr... | :$\sequence {x_n + y_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
Since $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]], we can find $N_1$ such that:
:$\forall n, m > N_1: \norm{x_n - x_m} < \dfrac \epsilon 2$
Similarly, $\sequence {y_n}$ is a [[Definition:Cauchy Sequence in No... | Combination Theorem for Cauchy Sequences/Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Sum_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Norm/Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14766 | Power of Complex Number minus 1 | Let $z \in \C$ be a complex number.
Then:
:$z^n - 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k}$
where $\alpha$ is a primitive complex $n$th root of unity. | Follows directly from the corollary to the Polynomial Factor Theorem:
If $\zeta_1, \zeta_2, \ldots, \zeta_n \in \C$ such that all are different, and $\map P {\zeta_1} = \map P {\zeta_2} = \ldots = \map P {\zeta_n} = 0$, then:
:$\ds \map P z = k \prod_{j \mathop = 1}^n \paren {z - \zeta_j}$
where $k \in \C$.
In this con... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$z^n - 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k}$
where $\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $n$th root of unity]]. | Follows directly from the [[Polynomial Factor Theorem/Corollary/Complex Numbers|corollary to the Polynomial Factor Theorem]]:
If $\zeta_1, \zeta_2, \ldots, \zeta_n \in \C$ such that all are different, and $\map P {\zeta_1} = \map P {\zeta_2} = \ldots = \map P {\zeta_n} = 0$, then:
:$\ds \map P z = k \prod_{j \mathop =... | Power of Complex Number minus 1 | https://proofwiki.org/wiki/Power_of_Complex_Number_minus_1 | https://proofwiki.org/wiki/Power_of_Complex_Number_minus_1 | [
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Root of Unity/Complex/Primitive"
] | [
"Polynomial Factor Theorem/Corollary/Complex Numbers",
"Definition:Root of Unity/Complex/Primitive"
] |
proofwiki-14767 | Power of Complex Number minus 1/Corollary | Let $z \in \C$ be a complex number.
Then:
:$\ds \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$
where $\alpha$ is a primitive complex $n$th root of unity. | {{begin-eqn}}
{{eqn | l = z^n - 1
| r = \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z^n - 1} {z - 1}
| r = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}
| c = as $\alpha^k = 1$ when $k = 0$
}}
{{eqn | ll= \leadsto
| l = \sum_{... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\ds \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$
where $\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $n$th root of unity]]. | {{begin-eqn}}
{{eqn | l = z^n - 1
| r = \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {z^n - 1} {z - 1}
| r = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}
| c = as $\alpha^k = 1$ when $k = 0$
}}
{{eqn | ll= \leadsto
| l = \sum_{... | Power of Complex Number minus 1/Corollary | https://proofwiki.org/wiki/Power_of_Complex_Number_minus_1/Corollary | https://proofwiki.org/wiki/Power_of_Complex_Number_minus_1/Corollary | [
"Complex Powers"
] | [
"Definition:Complex Number",
"Definition:Root of Unity/Complex/Primitive"
] | [
"Sum of Geometric Sequence"
] |
proofwiki-14768 | Product of Differences between 1 and Complex Roots of Unity | Let $\alpha$ be a primitive complex $n$th root of unity.
Then:
:$\ds \prod_{k \mathop = 1}^{n - 1} \paren {1 - \alpha^k} = n$ | From {{Corollary|Power of Complex Number minus 1}}:
:$\ds \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$
The result follows by setting $z = 1$.
{{qed}} | Let $\alpha$ be a [[Definition:Primitive Complex Root of Unity|primitive complex $n$th root of unity]].
Then:
:$\ds \prod_{k \mathop = 1}^{n - 1} \paren {1 - \alpha^k} = n$ | From {{Corollary|Power of Complex Number minus 1}}:
:$\ds \sum_{k \mathop = 0}^{n - 1} z^k = \prod_{k \mathop = 1}^{n - 1} \paren {z - \alpha^k}$
The result follows by setting $z = 1$.
{{qed}} | Product of Differences between 1 and Complex Roots of Unity | https://proofwiki.org/wiki/Product_of_Differences_between_1_and_Complex_Roots_of_Unity | https://proofwiki.org/wiki/Product_of_Differences_between_1_and_Complex_Roots_of_Unity | [
"Complex Roots of Unity"
] | [
"Definition:Root of Unity/Complex/Primitive"
] | [] |
proofwiki-14769 | Complex Roots of Unity occur in Conjugate Pairs | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n$ denote the complex $n$th roots of unity:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha \in U_n$ be the first complex $n$th root of unity.
Then:
:$\forall k \in \Z_{>0}, k < \dfrac n 2: \overline {\alpha^k} = \alpha^{n - k}$
That is, each of the complex $n$... | Consider the polynomial equation:
:$(1): \quad z^n - 1 = 0$
The complex $n$th roots of unity are:
:$1, \alpha, \alpha^2, \ldots, \alpha^{n - 1}$
From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs, the roots of $(1)$ occur in conjugate pairs.
Let $k \in \Z$ such that $1 \le k \le n$.
Then:
... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n$ denote the [[Definition:Complex Roots of Unity|complex $n$th roots of unity]]:
:$U_n = \set {z \in \C: z^n = 1}$
Let $\alpha \in U_n$ be the [[Definition:First Complex Root of Unity|first complex $n$th root of uni... | Consider the [[Definition:Polynomial Equation|polynomial equation]]:
:$(1): \quad z^n - 1 = 0$
The [[Definition:Complex Roots of Unity|complex $n$th roots of unity]] are:
:$1, \alpha, \alpha^2, \ldots, \alpha^{n - 1}$
From [[Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs]], the [[Definiti... | Complex Roots of Unity occur in Conjugate Pairs | https://proofwiki.org/wiki/Complex_Roots_of_Unity_occur_in_Conjugate_Pairs | https://proofwiki.org/wiki/Complex_Roots_of_Unity_occur_in_Conjugate_Pairs | [
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Root of Unity/Complex",
"Definition:Root of Unity/Complex/First",
"Definition:Root of Unity/Complex",
"Definition:Complex Conjugate/Conjugate Pair",
"Definition:Odd Integer",
"Definition:Even Integer"
] | [
"Definition:Polynomial Equation",
"Definition:Root of Unity/Complex",
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate/Conjugate Pair",
"Cosine of Angle plus Full Angle",
"Sine of Angle plus Full Angle",
"Cosi... |
proofwiki-14770 | Combination Theorem for Cauchy Sequences/Product Rule | :$\sequence {x_n y_n}$ is a Cauchy sequence. | Because $\sequence {x_n} $ is a Cauchy sequence, it is bounded by Cauchy Sequence is Bounded.
Suppose $\norm {x_n} \le K_1$ for $n = 1, 2, 3, \ldots$.
Because $\sequence {y_n} $ is a is a Cauchy sequence, it is bounded by Cauchy Sequence is Bounded.
Suppose $\norm {y_n} \le K_2$ for $n = 1, 2, 3, \ldots$.
Let $K = \max... | :$\sequence {x_n y_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Because $\sequence {x_n} $ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]], it is [[Definition:Bounded Sequence in Normed Division Ring|bounded]] by [[Cauchy Sequence is Bounded]].
Suppose $\norm {x_n} \le K_1$ for $n = 1, 2, 3, \ldots$.
Because $\sequence {y_n} $ is a is a [[Definition:Ca... | Combination Theorem for Cauchy Sequences/Product Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Product_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Product_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring",
"Cauchy Sequence is Bounded",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring",
"Cauchy Sequence is Bounded",
"Definition:Bounded Sequence/Normed Di... |
proofwiki-14771 | Factorisation of x^(2n+1)-1 in Real Domain | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\ds z^{2 n + 1} - 1 = \paren {z - 1} \prod_{k \mathop = 1}^n \paren {z^2 - 2 z \cos \dfrac {2 \pi k} {2 n + 1} + 1}$ | From Power of Complex Number minus 1:
:$\ds z^{2 n + 1} - 1 = \prod_{k \mathop = 0}^{2 n} \paren {z - \alpha^k}$
where:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n + 1} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \pi} {2 n + 1} + i \sin \dfrac {2 \pi} {2 n + 1}
| c =
}}
{{end-eqn}}
Fro... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds z^{2 n + 1} - 1 = \paren {z - 1} \prod_{k \mathop = 1}^n \paren {z^2 - 2 z \cos \dfrac {2 \pi k} {2 n + 1} + 1}$ | From [[Power of Complex Number minus 1]]:
:$\ds z^{2 n + 1} - 1 = \prod_{k \mathop = 0}^{2 n} \paren {z - \alpha^k}$
where:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n + 1} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \pi} {2 n + 1} + i \sin \dfrac {2 \pi} {2 n + 1}
| c =
}}
{{end-eqn... | Factorisation of x^(2n+1)-1 in Real Domain | https://proofwiki.org/wiki/Factorisation_of_x^(2n+1)-1_in_Real_Domain | https://proofwiki.org/wiki/Factorisation_of_x^(2n+1)-1_in_Real_Domain | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Power of Complex Number minus 1",
"Complex Roots of Unity occur in Conjugate Pairs",
"Definition:Root of Unity/Complex",
"Definition:Multiplication/Complex Numbers",
"Complex Roots of Unity occur in Conjugate Pairs",
"Modulus in Terms of Conjugate",
"Modulus of Complex Root of Unity equals 1"
] |
proofwiki-14772 | Factorisation of x^(2n)-1 in Real Domain | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\ds z^{2 n} - 1 = \paren {z - 1} \paren {z + 1} \prod_{k \mathop = 1}^n \paren {z^2 - 2 \cos \dfrac {k \pi} n z + 1}$ | From Power of Complex Number minus 1:
:$\ds z^{2 n} - 1 = \prod_{k \mathop = 0}^{2 n - 1} \paren {z - \alpha^k}$
where:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \pi} {2 n} + i \sin \dfrac {2 \pi} {2 n}
| c =
}}
{{eqn | r = \cos \dfrac \pi... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds z^{2 n} - 1 = \paren {z - 1} \paren {z + 1} \prod_{k \mathop = 1}^n \paren {z^2 - 2 \cos \dfrac {k \pi} n z + 1}$ | From [[Power of Complex Number minus 1]]:
:$\ds z^{2 n} - 1 = \prod_{k \mathop = 0}^{2 n - 1} \paren {z - \alpha^k}$
where:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \pi} {2 n} + i \sin \dfrac {2 \pi} {2 n}
| c =
}}
{{eqn | r = \cos \dfr... | Factorisation of x^(2n)-1 in Real Domain | https://proofwiki.org/wiki/Factorisation_of_x^(2n)-1_in_Real_Domain | https://proofwiki.org/wiki/Factorisation_of_x^(2n)-1_in_Real_Domain | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Power of Complex Number minus 1",
"Complex Roots of Unity occur in Conjugate Pairs",
"Definition:Root of Unity/Complex",
"Definition:Multiplication/Complex Numbers",
"Complex Roots of Unity occur in Conjugate Pairs",
"Modulus in Terms of Conjugate",
"Modulus of Complex Root of Unity equals 1"
] |
proofwiki-14773 | Factorisation of z^n-a | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $a \in \C$ be a complex number.
Then:
:$z^n - a = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k b}$
where:
:$\alpha$ is a primitive complex $n$th root of unity
:$b$ is any complex number such that $b^n = a$. | From $z^n - a = 0$ we have that:
:$a = z^n$
Let $b = a^{1 / n}$, hence $b^n = a$, with $a, b \in \C$.
From Roots of Complex Number:
{{begin-eqn}}
{{eqn | l = z^{1 / n}
| r = \set {a^{1 / n} e^{i \paren {\theta + 2 k \pi} / n}: k \in \set {0, 1, 2, \ldots, n - 1}, \theta = \arg a}
| c =
}}
{{eqn | r = \set {... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $a \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$z^n - a = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k b}$
where:
:$\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive ... | From $z^n - a = 0$ we have that:
:$a = z^n$
Let $b = a^{1 / n}$, hence $b^n = a$, with $a, b \in \C$.
From [[Roots of Complex Number]]:
{{begin-eqn}}
{{eqn | l = z^{1 / n}
| r = \set {a^{1 / n} e^{i \paren {\theta + 2 k \pi} / n}: k \in \set {0, 1, 2, \ldots, n - 1}, \theta = \arg a}
| c =
}}
{{eqn | r = ... | Factorisation of z^n-a | https://proofwiki.org/wiki/Factorisation_of_z^n-a | https://proofwiki.org/wiki/Factorisation_of_z^n-a | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Complex Number",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Complex Number"
] | [
"Roots of Complex Number",
"Definition:Root of Polynomial",
"First Complex Root of Unity is Primitive",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Root of Polynomial",
"Polynomial Factor Theorem/Corollary/Complex Numbers",
"Definition:Monic Polynomial"
] |
proofwiki-14774 | Triple Angle Formulas/Cosine/2 cos 3 theta + 1 | :$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$ | {{begin-eqn}}
{{eqn | l = z^6 + z^3 + 1
| r = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}
| c = Complex Algebra Examples: $z^6 + z^3 + 1$
}}
{{eqn | ll= \leadsto
| l = z^3 + z^0 + z^{-3}
| r = \paren {z - ... | :$2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$ | {{begin-eqn}}
{{eqn | l = z^6 + z^3 + 1
| r = \paren {z^2 - 2 z \cos \dfrac {2 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {4 \pi} 9 + 1} \paren {z^2 - 2 z \cos \dfrac {8 \pi} 9 + 1}
| c = [[Complex Algebra/Examples/z^6 + z^3 + 1|Complex Algebra Examples: $z^6 + z^3 + 1$]]
}}
{{eqn | ll= \leadsto
| l = ... | Triple Angle Formulas/Cosine/2 cos 3 theta + 1 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine/2_cos_3_theta_+_1 | https://proofwiki.org/wiki/Triple_Angle_Formulas/Cosine/2_cos_3_theta_+_1 | [
"Triple Angle Formula for Cosine"
] | [] | [
"Complex Algebra/Examples/z^6 + z^3 + 1",
"Euler's Cosine Identity"
] |
proofwiki-14775 | Cosine of 144 Degrees | :$\cos 144 \degrees = \cos \dfrac {4 \pi} 5 = -\dfrac \phi 2 = -\dfrac {1 + \sqrt 5} 4$ | {{begin-eqn}}
{{eqn | l = \cos 144 \degrees
| r = \map \cos {180 \degrees - 36 \degrees}
| c =
}}
{{eqn | r = -\cos 36 \degrees
| c = Cosine of Supplementary Angle
}}
{{eqn | r = -\dfrac {1 + \sqrt 5} 4
| c = {{cos|36}}
}}
{{end-eqn}}
{{qed}}
Category:Cosine Function
ge3xtunhsntyfh7t8q3jydddxp8... | :$\cos 144 \degrees = \cos \dfrac {4 \pi} 5 = -\dfrac \phi 2 = -\dfrac {1 + \sqrt 5} 4$ | {{begin-eqn}}
{{eqn | l = \cos 144 \degrees
| r = \map \cos {180 \degrees - 36 \degrees}
| c =
}}
{{eqn | r = -\cos 36 \degrees
| c = [[Cosine of Supplementary Angle]]
}}
{{eqn | r = -\dfrac {1 + \sqrt 5} 4
| c = {{cos|36}}
}}
{{end-eqn}}
{{qed}}
[[Category:Cosine Function]]
ge3xtunhsntyfh7t8q... | Cosine of 144 Degrees | https://proofwiki.org/wiki/Cosine_of_144_Degrees | https://proofwiki.org/wiki/Cosine_of_144_Degrees | [
"Cosine Function"
] | [] | [
"Cosine of Supplementary Angle",
"Category:Cosine Function"
] |
proofwiki-14776 | Sine of 144 Degrees | :$\sin 144 \degrees = \cos \dfrac {4 \pi} 5 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$ | {{begin-eqn}}
{{eqn | l = \sin 144 \degrees
| r = \sin \paren {180 \degrees - 36 \degrees}
| c =
}}
{{eqn | r = \sin 36 \degrees
| c = Sine of Supplementary Angle
}}
{{eqn | r = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}
| c = Sine of $36 \degrees$
}}
{{end-eqn}}
{{qed}}
Category:Sine Function
4fz... | :$\sin 144 \degrees = \cos \dfrac {4 \pi} 5 = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}$ | {{begin-eqn}}
{{eqn | l = \sin 144 \degrees
| r = \sin \paren {180 \degrees - 36 \degrees}
| c =
}}
{{eqn | r = \sin 36 \degrees
| c = [[Sine of Supplementary Angle]]
}}
{{eqn | r = \sqrt {\dfrac 5 8 - \dfrac {\sqrt 5} 8}
| c = [[Sine of 36 Degrees|Sine of $36 \degrees$]]
}}
{{end-eqn}}
{{qed}}... | Sine of 144 Degrees | https://proofwiki.org/wiki/Sine_of_144_Degrees | https://proofwiki.org/wiki/Sine_of_144_Degrees | [
"Sine Function"
] | [] | [
"Sine of Supplementary Angle",
"Sine of 36 Degrees",
"Category:Sine Function"
] |
proofwiki-14777 | Cube Root of Unity if Modulus is 1 and Real Part is Minus Half | Let $z \in \C$ be a complex number such that:
:$\cmod z = 1$
:$\Re \paren z = -\dfrac 1 2$
where:
:$\cmod z$ denotes the complex modulus of $z$
:$\Re \paren z$ denotes the real part of $z$.
Then:
:$z^3 = 1$ | Let $z = x + i y$.
From $\Re \paren z = -\dfrac 1 2$:
:$x = -\dfrac 1 2$
by definition of the real part of $z$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r = 1
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll= \leadsto
| l = \paren ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] such that:
:$\cmod z = 1$
:$\Re \paren z = -\dfrac 1 2$
where:
:$\cmod z$ denotes the [[Definition:Complex Modulus|complex modulus]] of $z$
:$\Re \paren z$ denotes the [[Definition:Real Part|real part]] of $z$.
Then:
:$z^3 = 1$ | Let $z = x + i y$.
From $\Re \paren z = -\dfrac 1 2$:
:$x = -\dfrac 1 2$
by definition of the [[Definition:Real Part|real part]] of $z$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod z
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = x^2 + y^2
| r = 1
| c = {{Defof|Complex Modulus}}
}}
{{eqn | ll=... | Cube Root of Unity if Modulus is 1 and Real Part is Minus Half | https://proofwiki.org/wiki/Cube_Root_of_Unity_if_Modulus_is_1_and_Real_Part_is_Minus_Half | https://proofwiki.org/wiki/Cube_Root_of_Unity_if_Modulus_is_1_and_Real_Part_is_Minus_Half | [
"Cube Roots of Unity"
] | [
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Complex Number/Real Part"
] | [
"Definition:Complex Number/Real Part",
"Complex Roots of Unity/Examples/Cube Roots"
] |
proofwiki-14778 | Sum of Two Cubes in Complex Domain | :$a^3 + b^3 = \paren {a + b} \paren {a \omega + b \omega^2} \paren {a \omega^2 + b \omega}$
where:
: $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | From Sum of Cubes of Three Indeterminates Minus 3 Times their Product:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
Setting $x \gets 0, y \gets a, z \gets b$:
:$0^3 + a^3 + b^3 - 3 \times 0 \times a b = \paren {0 + a + b} \paren {0 + \omega a + ... | :$a^3 + b^3 = \paren {a + b} \paren {a \omega + b \omega^2} \paren {a \omega^2 + b \omega}$
where:
: $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | From [[Sum of Cubes of Three Indeterminates Minus 3 Times their Product]]:
:$x^3 + y^3 + z^3 - 3 x y z = \paren {x + y + z} \paren {x + \omega y + \omega^2 z} \paren {x + \omega^2 y + \omega z}$
Setting $x \gets 0, y \gets a, z \gets b$:
:$0^3 + a^3 + b^3 - 3 \times 0 \times a b = \paren {0 + a + b} \paren {0 + \ome... | Sum of Two Cubes in Complex Domain | https://proofwiki.org/wiki/Sum_of_Two_Cubes_in_Complex_Domain | https://proofwiki.org/wiki/Sum_of_Two_Cubes_in_Complex_Domain | [
"Cube Roots of Unity",
"Algebra"
] | [] | [
"Sum of Cubes of Three Indeterminates Minus 3 Times their Product"
] |
proofwiki-14779 | Three Times Sum of Cubes of Three Indeterminates Plus 6 Times their Product | :$3 \paren {a^3 + b^3 + c^3 + 6 a b c} = \paren {a + b + c}^3 + \paren {a + b \omega + c \omega^2}^3 + \paren {a + b \omega^2 + c \omega}^3$
where:
: $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | Multiplying out:
{{begin-eqn}}
{{eqn | l = \paren {a + b + c}^3
| r = \paren {a + b + c} \paren {a^2 + b^2 + c^2 + 2 a b + 2 a c + 2 b c}
| c =
}}
{{eqn | n = 1
| r = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 a b^2 + 3 b^2 c + 3 a c^2 + 3 b c^2 + 6 a b c
| c =
}}
{{end-eqn}}
Replacing $b$ with $... | :$3 \paren {a^3 + b^3 + c^3 + 6 a b c} = \paren {a + b + c}^3 + \paren {a + b \omega + c \omega^2}^3 + \paren {a + b \omega^2 + c \omega}^3$
where:
: $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$ | Multiplying out:
{{begin-eqn}}
{{eqn | l = \paren {a + b + c}^3
| r = \paren {a + b + c} \paren {a^2 + b^2 + c^2 + 2 a b + 2 a c + 2 b c}
| c =
}}
{{eqn | n = 1
| r = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 a b^2 + 3 b^2 c + 3 a c^2 + 3 b c^2 + 6 a b c
| c =
}}
{{end-eqn}}
Replacing $b$ wit... | Three Times Sum of Cubes of Three Indeterminates Plus 6 Times their Product | https://proofwiki.org/wiki/Three_Times_Sum_of_Cubes_of_Three_Indeterminates_Plus_6_Times_their_Product | https://proofwiki.org/wiki/Three_Times_Sum_of_Cubes_of_Three_Indeterminates_Plus_6_Times_their_Product | [
"Cube Roots of Unity",
"Algebra"
] | [] | [
"Sum of Cube Roots of Unity"
] |
proofwiki-14780 | Roots of Complex Number/Examples/z^8 + 1 = 0 | The roots of the polynomial:
:$z^8 + 1 = 0$
are:
:$\set {\cos \dfrac {\paren {2 k + 1} \pi} 8 + i \sin \dfrac {\paren {2 k + 1} \pi} 8: k \in \set {0, 1, \ldots, 7} }$ | {{begin-eqn}}
{{eqn | l = z^8 + 1
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = z
| r = \paren {-1}^{1/8}
| c =
}}
{{end-eqn}}
From Euler's Identity:
:$-1 = e^{i \pi}$
Let $b$ be defined as:
{{begin-eqn}}
{{eqn | l = b
| r = \sqrt [8] 1 \paren {\cos \dfrac \pi 8 + i \sin \dfrac \pi 8... | The [[Definition:Root of Polynomial|roots]] of the [[Definition:Polynomial over Complex Numbers|polynomial]]:
:$z^8 + 1 = 0$
are:
:$\set {\cos \dfrac {\paren {2 k + 1} \pi} 8 + i \sin \dfrac {\paren {2 k + 1} \pi} 8: k \in \set {0, 1, \ldots, 7} }$ | {{begin-eqn}}
{{eqn | l = z^8 + 1
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = z
| r = \paren {-1}^{1/8}
| c =
}}
{{end-eqn}}
From [[Euler's Identity]]:
:$-1 = e^{i \pi}$
Let $b$ be defined as:
{{begin-eqn}}
{{eqn | l = b
| r = \sqrt [8] 1 \paren {\cos \dfrac \pi 8 + i \sin \df... | Roots of Complex Number/Examples/z^8 + 1 = 0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^8_+_1_=_0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^8_+_1_=_0 | [
"Examples of Complex Roots"
] | [
"Definition:Root of Polynomial",
"Definition:Polynomial/Complex Numbers"
] | [
"Euler's Identity",
"Roots of Complex Number/Corollary"
] |
proofwiki-14781 | Quadruple Angle Formulas/Cosine/Factor Form | :$\cos 4 \theta = \paren {\cos \theta - \cos \dfrac \pi 8} \paren {\cos \theta - \cos \dfrac {3 \pi} 8} \paren {\cos \theta - \cos \dfrac {5 \pi} 8} \paren {\cos \theta - \cos \dfrac {7 \pi} 8}$ | {{begin-eqn}}
{{eqn | l = z^8 + 1
| r = \prod_{k \mathop = 0}^3 \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} 8 + 1}
| c = Complex Algebra Examples: $z^8 + 1$
}}
{{eqn | ll= \leadsto
| l = z^4 + z^{-4}
| r = \prod_{k \mathop = 0}^3 \paren {z - 2 \cos \dfrac {\paren {2 k + 1} \pi} 8 + z^{-... | :$\cos 4 \theta = \paren {\cos \theta - \cos \dfrac \pi 8} \paren {\cos \theta - \cos \dfrac {3 \pi} 8} \paren {\cos \theta - \cos \dfrac {5 \pi} 8} \paren {\cos \theta - \cos \dfrac {7 \pi} 8}$ | {{begin-eqn}}
{{eqn | l = z^8 + 1
| r = \prod_{k \mathop = 0}^3 \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} 8 + 1}
| c = [[Complex Algebra/Examples/z^8 + 1|Complex Algebra Examples: $z^8 + 1$]]
}}
{{eqn | ll= \leadsto
| l = z^4 + z^{-4}
| r = \prod_{k \mathop = 0}^3 \paren {z - 2 \cos \... | Quadruple Angle Formulas/Cosine/Factor Form | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine/Factor_Form | https://proofwiki.org/wiki/Quadruple_Angle_Formulas/Cosine/Factor_Form | [
"Quadruple Angle Formula for Cosine",
"Cosine Function"
] | [] | [
"Complex Algebra/Examples/z^8 + 1",
"Euler's Cosine Identity"
] |
proofwiki-14782 | Factorisation of z^n+1 | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$z^n + 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} n}$ | From Factorisation of $z^n - a$, setting $a = -1$:
:$z^n + 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k b}$
where:
:$\alpha$ is a primitive complex $n$th root of unity
:$b$ is any complex number such that $b^n = a$.
From Euler's Identity:
:$-1 = e^{i \pi}$
From Exponential of Product:
:$\paren {\exp \dfra... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$z^n + 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} n}$ | From [[Factorisation of z^n-a|Factorisation of $z^n - a$]], setting $a = -1$:
:$z^n + 1 = \ds \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha^k b}$
where:
:$\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $n$th root of unity]]
:$b$ is any [[Definition:Complex Number|complex number]] such ... | Factorisation of z^n+1 | https://proofwiki.org/wiki/Factorisation_of_z^n+1 | https://proofwiki.org/wiki/Factorisation_of_z^n+1 | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n-a",
"Definition:Root of Unity/Complex/Primitive",
"Definition:Complex Number",
"Euler's Identity",
"Exponential of Product",
"Definition:Root of Unity/Complex/First",
"First Complex Root of Unity is Primitive",
"Exponential of Product",
"Exponential of Sum",
"Category:Algebra... |
proofwiki-14783 | Combination Theorem for Cauchy Sequences/Quotient Rule | Suppose $\sequence {y_n}$ does not converge to $0$.
Then:
:$\exists K \in \N: \forall n > K : y_n \ne 0$
and the sequences:
:$\sequence { {x_{K + n} } \paren {y_{K + n} }^{-1} }_{n \mathop \in \N}$ and $\sequence {\paren {y_{K + n} }^{-1} {x_{K + n} } }_{n \mathop \in \N}$ are well-defined and Cauchy sequences. | By the Inverse Rule for Normed Division Ring:
:$\exists K \in \N : \forall n > K : y_n \ne 0$.
and the sequence:
:$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a Cauchy sequence.
By Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence, $\sequence {x_{K + n} }_{n \ma... | Suppose $\sequence {y_n}$ does not [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $0$.
Then:
:$\exists K \in \N: \forall n > K : y_n \ne 0$
and the [[Definition:Sequence|sequences]]:
:$\sequence { {x_{K + n} } \paren {y_{K + n} }^{-1} }_{n \mathop \in \N}$ and $\sequence {\paren {y_{K + n} }^{-... | By the [[Combination Theorem for Cauchy Sequences/Inverse Rule|Inverse Rule for Normed Division Ring]]:
:$\exists K \in \N : \forall n > K : y_n \ne 0$.
and the [[Definition:Sequence|sequence]]:
:$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a [[Definition:Cauchy Sequence in Normed Divi... | Combination Theorem for Cauchy Sequences/Quotient Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Quotient_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Quotient_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Combination Theorem for Cauchy Sequences/Inverse Rule",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Combination Theorem for Cauchy Sequences/Produ... |
proofwiki-14784 | Factorisation of z^(2n)+1 in Real Domain | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\ds z^{2 n} + 1 = \prod_{k \mathop = 1}^n \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n} + 1}$ | From Factorisation of $z^n + 1$:
:$(1): \ds \quad z^{2 n} + 1 = \prod_{k \mathop = 0}^{2 n - 1} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} {2 n} }$
From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs, the roots of $(1)$ occur in conjugate pairs.
Hence we can express $(1)$ as:
{{begin-... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds z^{2 n} + 1 = \prod_{k \mathop = 1}^n \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n} + 1}$ | From [[Factorisation of z^n+1|Factorisation of $z^n + 1$]]:
:$(1): \ds \quad z^{2 n} + 1 = \prod_{k \mathop = 0}^{2 n - 1} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} {2 n} }$
From [[Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs]], the [[Definition:Root of Polynomial|roots]] of $(1... | Factorisation of z^(2n)+1 in Real Domain | https://proofwiki.org/wiki/Factorisation_of_z^(2n)+1_in_Real_Domain | https://proofwiki.org/wiki/Factorisation_of_z^(2n)+1_in_Real_Domain | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n+1",
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate/Conjugate Pair",
"Euler's Formula"
] |
proofwiki-14785 | Factorisation of z^(2n+1)+1 in Real Domain | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\ds z^{2 n + 1} + 1 = \paren {z + 1} \prod_{k \mathop = 0}^{n - 1} \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n + 1} + 1}$ | From Factorisation of $z^n + 1$:
:$(1): \quad \ds z^{2 n + 1} + 1 = \prod_{k \mathop = 0}^{2 n} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} {2 n + 1} }$
From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs, the roots of $(1)$ occur in conjugate pairs.
Hence we can express $(1)$ as:
{{be... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds z^{2 n + 1} + 1 = \paren {z + 1} \prod_{k \mathop = 0}^{n - 1} \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n + 1} + 1}$ | From [[Factorisation of z^n+1|Factorisation of $z^n + 1$]]:
:$(1): \quad \ds z^{2 n + 1} + 1 = \prod_{k \mathop = 0}^{2 n} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} {2 n + 1} }$
From [[Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs]], the [[Definition:Root of Polynomial|roots]] of... | Factorisation of z^(2n+1)+1 in Real Domain | https://proofwiki.org/wiki/Factorisation_of_z^(2n+1)+1_in_Real_Domain | https://proofwiki.org/wiki/Factorisation_of_z^(2n+1)+1_in_Real_Domain | [
"Algebra",
"Complex Roots"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n+1",
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Definition:Root of Polynomial",
"Definition:Complex Conjugate/Conjugate Pair",
"Euler's Formula"
] |
proofwiki-14786 | Roots of Complex Number/Examples/z^5 + 1 = 0 | The roots of the polynomial:
:$z^5 + 1 = 0$
are:
:$\set {\cos \dfrac \pi 5 \pm i \sin \dfrac \pi 5, \cos \dfrac {3 \pi} 5 \pm i \sin \dfrac {3 \pi} 5, -1}$ | From Factorisation of $z^n + 1$:
:$z^5 + 1 = \ds \prod_{k \mathop = 0}^4 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 5}$
Thus:
:$z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 5}$
{{begin-eqn}}
{{eqn | n = k = 0
| l = z
| r = \cos \dfrac \pi 5 + i \sin \dfrac \pi 5
| c =
}}
{{eqn | n = k = 1
... | The [[Definition:Root of Polynomial|roots]] of the [[Definition:Polynomial over Complex Numbers|polynomial]]:
:$z^5 + 1 = 0$
are:
:$\set {\cos \dfrac \pi 5 \pm i \sin \dfrac \pi 5, \cos \dfrac {3 \pi} 5 \pm i \sin \dfrac {3 \pi} 5, -1}$ | From [[Factorisation of z^n+1|Factorisation of $z^n + 1$]]:
:$z^5 + 1 = \ds \prod_{k \mathop = 0}^4 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 5}$
Thus:
:$z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 5}$
{{begin-eqn}}
{{eqn | n = k = 0
| l = z
| r = \cos \dfrac \pi 5 + i \sin \dfrac \pi 5
... | Roots of Complex Number/Examples/z^5 + 1 = 0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^5_+_1_=_0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^5_+_1_=_0 | [
"Examples of Complex Roots"
] | [
"Definition:Root of Polynomial",
"Definition:Polynomial/Complex Numbers"
] | [
"Factorisation of z^n+1",
"Euler's Identity"
] |
proofwiki-14787 | 4 Sine Pi over 10 by Cosine Pi over 5 | :$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$ | Rewrite the {{LHS}}:
{{begin-eqn}}
{{eqn | l = 4 \sin \dfrac \pi {10} \cos \dfrac \pi 5
| r = 1
| c = {{hypothesis}}
}}
{{eqn | l = 4 \sin \dfrac \pi {10} \cos \dfrac {2 \pi} {10}
| r = 1
| c = multiplying the angle inside the cosine by $\dfrac 2 2$
}}
{{eqn | l = 4 \sin \dfrac \pi {10} \paren {... | :$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$ | Rewrite the {{LHS}}:
{{begin-eqn}}
{{eqn | l = 4 \sin \dfrac \pi {10} \cos \dfrac \pi 5
| r = 1
| c = {{hypothesis}}
}}
{{eqn | l = 4 \sin \dfrac \pi {10} \cos \dfrac {2 \pi} {10}
| r = 1
| c = multiplying the angle inside the cosine by $\dfrac 2 2$
}}
{{eqn | l = 4 \sin \dfrac \pi {10} \paren ... | 4 Sine Pi over 10 by Cosine Pi over 5/Proof 2 | https://proofwiki.org/wiki/4_Sine_Pi_over_10_by_Cosine_Pi_over_5 | https://proofwiki.org/wiki/4_Sine_Pi_over_10_by_Cosine_Pi_over_5/Proof_2 | [
"4 Sine Pi over 10 by Cosine Pi over 5",
"Sine Function",
"Cosine Function"
] | [] | [
"Cosine of Right Angle"
] |
proofwiki-14788 | 4 Sine Pi over 10 by Cosine Pi over 5 | :$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$ | {{begin-eqn}}
{{eqn | l = 4 \sin \theta \cos 2 \theta
| r = 1
| c = Solve for $\theta$
}}
{{eqn | l = 4 \sin \theta \cos \theta \cos 2\theta
| r = \cos \theta
| c = multiplying both sides by $\cos \theta$
}}
{{eqn | l = 2 \paren {2 \sin \theta \cos \theta } \cos 2\theta
| r = \cos \theta
... | :$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$ | {{begin-eqn}}
{{eqn | l = 4 \sin \theta \cos 2 \theta
| r = 1
| c = Solve for $\theta$
}}
{{eqn | l = 4 \sin \theta \cos \theta \cos 2\theta
| r = \cos \theta
| c = multiplying both sides by $\cos \theta$
}}
{{eqn | l = 2 \paren {2 \sin \theta \cos \theta } \cos 2\theta
| r = \cos \theta
... | 4 Sine Pi over 10 by Cosine Pi over 5/Proof 3 | https://proofwiki.org/wiki/4_Sine_Pi_over_10_by_Cosine_Pi_over_5 | https://proofwiki.org/wiki/4_Sine_Pi_over_10_by_Cosine_Pi_over_5/Proof_3 | [
"4 Sine Pi over 10 by Cosine Pi over 5",
"Sine Function",
"Cosine Function"
] | [] | [
"Double Angle Formulas/Sine",
"Double Angle Formulas/Sine",
"Sine of Complement equals Cosine"
] |
proofwiki-14789 | Convergence of Modulus of Convergent Complex Sequence | Let $\sequence {z_n}$ be a sequence in $\C$.
Let $\sequence {z_n}$ converge to a value $c \in \C$.
Let $\cmod z$ denote the modulus of a complex number $z$.
Then:
:$\sequence {\cmod {z_n} }$ converges to a value $\cmod c$. | By definition of convergent complex sequence:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$
From the Reverse Triangle Inequality:
:$\size {\cmod x - \cmod y} \le \cmod {x - y}$
and the result follows.
{{qed}} | Let $\sequence {z_n}$ be a [[Definition:Complex Sequence|sequence in $\C$]].
Let $\sequence {z_n}$ [[Definition:Convergent Complex Sequence|converge]] to a value $c \in \C$.
Let $\cmod z$ denote the [[Definition:Complex Modulus|modulus]] of a [[Definition:Complex Number|complex number]] $z$.
Then:
:$\sequence {\cmo... | By definition of [[Definition:Convergent Complex Sequence|convergent complex sequence]]:
:$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$
From the [[Reverse Triangle Inequality]]:
:$\size {\cmod x - \cmod y} \le \cmod {x - y}$
and the result follows.
{{qed}} | Convergence of Modulus of Convergent Complex Sequence | https://proofwiki.org/wiki/Convergence_of_Modulus_of_Convergent_Complex_Sequence | https://proofwiki.org/wiki/Convergence_of_Modulus_of_Convergent_Complex_Sequence | [
"Convergent Complex Sequences"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Complex Modulus",
"Definition:Complex Number",
"Definition:Convergent Sequence/Complex Numbers"
] | [
"Definition:Convergent Sequence/Complex Numbers",
"Reverse Triangle Inequality"
] |
proofwiki-14790 | Combination Theorem for Sequences/Real/Difference Rule | :$\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$ | From Sum Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
From Multiple Rule for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + \paren {-y_n} } = l + \paren {-m}$
The result follows.
{{qed}} | :$\ds \lim_{n \mathop \to \infty} \paren {x_n - y_n} = l - m$ | From [[Sum Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
From [[Multiple Rule for Real Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {x_n + \paren {-y_n} } = l + \paren {-m}$
The result follows.
{{qed}} | Combination Theorem for Sequences/Real/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Real/Difference_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Combination Theorem for Sequences/Real/Sum Rule",
"Combination Theorem for Sequences/Real/Multiple Rule"
] |
proofwiki-14791 | Combination Theorem for Sequences/Complex/Sum Rule/Proof 1 | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
{{:Sum Rule for Complex Sequences}} | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the limit of a complex sequence, we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c} < \dfrac \epsilon 2$
where $\cmod {z_n - c... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the [[Definition:Limit of Complex Sequence|limit of a complex sequence]], we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_1 | [
"Sum Rule for Complex Sequences"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers"
] | [
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Complex Modulus",
"Triangle Inequality/Complex Numbers"
] |
proofwiki-14792 | Combination Theorem for Sequences/Complex/Sum Rule | :$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the limit of a complex sequence, we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c} < \dfrac \epsilon 2$
where $\cmod {z_n - c... | :$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the [[Definition:Limit of Complex Sequence|limit of a complex sequence]], we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_1 | [
"Combination Theorems for Sequences",
"Sum Rule for Complex Sequences"
] | [] | [
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Complex Modulus",
"Triangle Inequality/Complex Numbers"
] |
proofwiki-14793 | Combination Theorem for Sequences/Complex/Sum Rule | :$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are real.
By definition:
... | :$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$ | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are [[Definition:Re... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_2 | [
"Combination Theorems for Sequences",
"Sum Rule for Complex Sequences"
] | [] | [
"Definition:Real Number",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Absolute Value"
] |
proofwiki-14794 | Combination Theorem for Sequences/Complex/Sum Rule/Proof 2 | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Then:
{{:Combination Theorem for Sequences/Complex/Sum Rule}} | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are real.
By definition:
... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are [[Definition:Re... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_2 | [
"Sum Rule for Complex Sequences"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers"
] | [
"Definition:Real Number",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Absolute Value"
] |
proofwiki-14795 | Combination Theorem for Sequences/Complex | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then the following results hold:
=== Sum Rule ===
{{:Combi... | Because $\sequence {z_n}$ converges, it is bounded by Convergent Sequence is Bounded.
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_n w_n - c d}
| r = \cmod {z_n w_n - z_n d + z_n d - c d}
| c =
}}
{{eqn | o = \le
| r = \cmod {z_n w_n - z_n d} + \cmod... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Because $\sequence {z_n}$ [[Definition:Convergent Real Sequence|converges]], it is bounded by [[Convergent Sequence is Bounded]].
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_n w_n - c d}
| r = \cmod {z_n w_n - z_n d + z_n d - c d}
| c =
}}
{{eqn | o = ... | Combination Theorem for Sequences/Complex/Product Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule/Proof_1 | [
"Combination Theorems for Sequences",
"Complex Analysis"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Combination Theorem for Sequences/Complex/Difference Rule",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"... | [
"Definition:Convergent Sequence/Real Numbers",
"Convergent Sequence in Metric Space is Bounded",
"Triangle Inequality/Complex Numbers",
"Complex Modulus of Product of Complex Numbers",
"Convergent Sequence Minus Limit",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem/Sequenc... |
proofwiki-14796 | Combination Theorem for Sequences/Complex | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then the following results hold:
=== Sum Rule ===
{{:Combi... | Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of convergent complex sequence:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} z_n
| r = c
| c =
}}
{{eqn | ll= \leadsto
| l = \lim_{n \mathop \to \infty} x_n + i \lim_{n \mathop \to \infty} y_... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of [[Definition:Convergent Complex Sequence|convergent complex sequence]]:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} z_n
| r = c
| c =
}}
{{eqn | ll= \leadsto
| l = \lim_{n \mathop \... | Combination Theorem for Sequences/Complex/Product Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Product_Rule/Proof_2 | [
"Combination Theorems for Sequences",
"Complex Analysis"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Combination Theorem for Sequences/Complex/Difference Rule",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"... | [
"Definition:Convergent Sequence/Complex Numbers",
"Combination Theorem for Sequences/Real/Sum Rule",
"Combination Theorem for Sequences/Real/Product Rule"
] |
proofwiki-14797 | Combination Theorem for Sequences/Complex | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then the following results hold:
=== Sum Rule ===
{{:Combi... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the limit of a complex sequence, we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c} < \dfrac \epsilon 2$
where $\cmod {z_n - c... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
By definition of the [[Definition:Limit of Complex Sequence|limit of a complex sequence]], we can find $N_1$ such that:
:$\forall n > N_1: \cmod {z_n - c... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 1 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_1 | [
"Combination Theorems for Sequences",
"Complex Analysis"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Combination Theorem for Sequences/Complex/Difference Rule",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"... | [
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Complex Modulus",
"Triangle Inequality/Complex Numbers"
] |
proofwiki-14798 | Combination Theorem for Sequences/Complex | Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.
Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let $\lambda, \mu \in \C$.
Then the following results hold:
=== Sum Rule ===
{{:Combi... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are real.
By definition:
... | Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Complex Sequence|sequences in $\C$]].
Let $\sequence {z_n}$ and $\sequence {w_n}$ be [[Definition:Convergent Complex Sequence|convergent]] to the following [[Definition:Limit of Complex Sequence|limits]]:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim... | Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
:$\ds \lim_{n \mathop \to \infty} z_n = c$
:$\ds \lim_{n \mathop \to \infty} w_n = d$
Let:
:$z_n = x_n + i y_n$
:$w_n = r_n + i s_n$
:$c = a + i b$
:$d = l + i m$
where each of $x_n, y_n, r_n, s_n, a, b, l, m \in \R$ are [[Definition:Re... | Combination Theorem for Sequences/Complex/Sum Rule/Proof 2 | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Sum_Rule/Proof_2 | [
"Combination Theorems for Sequences",
"Complex Analysis"
] | [
"Definition:Complex Sequence",
"Definition:Convergent Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Combination Theorem for Sequences/Complex/Sum Rule",
"Combination Theorem for Sequences/Complex/Difference Rule",
"Combination Theorem for Sequences/Complex/Multiple Rule",
"... | [
"Definition:Real Number",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Limit of Sequence/Complex Numbers",
"Definition:Absolute Value"
] |
proofwiki-14799 | Combination Theorem for Sequences/Complex/Difference Rule | :$\ds \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$ | From Sum Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
From Multiple Rule for Complex Sequences:
:$\ds \lim_{n \mathop \to \infty} \paren {-w_n} = -d$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + \paren {-w_n} } = c + \paren {-d}$
The result follows.
{{qed}} | :$\ds \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$ | From [[Sum Rule for Complex Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + w_n} = c + d$
From [[Multiple Rule for Complex Sequences]]:
:$\ds \lim_{n \mathop \to \infty} \paren {-w_n} = -d$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {z_n + \paren {-w_n} } = c + \paren {-d}$
The result follows.
{{q... | Combination Theorem for Sequences/Complex/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Sequences/Complex/Difference_Rule | [
"Combination Theorems for Sequences"
] | [] | [
"Combination Theorem for Sequences/Complex/Sum Rule",
"Combination Theorem for Sequences/Complex/Multiple Rule"
] |
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