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-12100 | One Third as Quotient of Sequences of Odd Numbers | :$\dfrac 1 3 = \dfrac {1 + 3} {5 + 7} = \dfrac {1 + 3 + 5} {7 + 9 + 11} = \dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \cdots$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = n + 1}^{2 n} \paren {2 k - 1}
| r = \sum_{k \mathop = 1}^{2 n} \paren {2 k - 1} - \sum_{k \mathop = 1}^n \paren {2 k - 1}
| c =
}}
{{eqn | r = \paren {2 n}^2 - n^2
| c = Odd Number Theorem
}}
{{eqn | r = 3 n^2
| c =
}}
{{eqn | r = 3 \sum_{k \mathop =... | :$\dfrac 1 3 = \dfrac {1 + 3} {5 + 7} = \dfrac {1 + 3 + 5} {7 + 9 + 11} = \dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \cdots$ | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = n + 1}^{2 n} \paren {2 k - 1}
| r = \sum_{k \mathop = 1}^{2 n} \paren {2 k - 1} - \sum_{k \mathop = 1}^n \paren {2 k - 1}
| c =
}}
{{eqn | r = \paren {2 n}^2 - n^2
| c = [[Odd Number Theorem]]
}}
{{eqn | r = 3 n^2
| c =
}}
{{eqn | r = 3 \sum_{k \math... | One Third as Quotient of Sequences of Odd Numbers | https://proofwiki.org/wiki/One_Third_as_Quotient_of_Sequences_of_Odd_Numbers | https://proofwiki.org/wiki/One_Third_as_Quotient_of_Sequences_of_Odd_Numbers | [
"One Third"
] | [] | [
"Odd Number Theorem",
"Odd Number Theorem"
] |
proofwiki-12101 | Euler's Number as Sum of Egyptian Fractions | The reciprocal of Euler's number $e$ can be approximated by the following sequence of Egyptian fractions:
:$\dfrac 1 e = \dfrac 1 3 + \dfrac 1 {29} + \dfrac 1 {15 \, 786} + \dfrac 1 {513 \, 429 \, 610} + \cdots$
{{OEIS|A006526}} | We have by definition of the reciprocal of $e$ that:
:$\dfrac 1 e \approx 0 \cdotp 36787 \, 94411 \, 71442 \, 32159 \, 55237 \, 70161 \, 46086 \, 74458 \, 11131 \, 031 \ldots$
By inspection:
:$\dfrac 1 3 < \dfrac 1 e < \dfrac 1 2$
Thus:
:$\dfrac 1 e - \dfrac 1 3 \approx 0 \cdotp 03454 \, 61078 \, 38109 \, 08826 \, 2190... | The [[Reciprocal of Euler's Number|reciprocal of Euler's number $e$]] can be approximated by the following sequence of [[Definition:Egyptian Fraction|Egyptian fractions]]:
:$\dfrac 1 e = \dfrac 1 3 + \dfrac 1 {29} + \dfrac 1 {15 \, 786} + \dfrac 1 {513 \, 429 \, 610} + \cdots$
{{OEIS|A006526}} | We have by definition of the [[Reciprocal of Euler's Number|reciprocal of $e$]] that:
:$\dfrac 1 e \approx 0 \cdotp 36787 \, 94411 \, 71442 \, 32159 \, 55237 \, 70161 \, 46086 \, 74458 \, 11131 \, 031 \ldots$
By inspection:
:$\dfrac 1 3 < \dfrac 1 e < \dfrac 1 2$
Thus:
:$\dfrac 1 e - \dfrac 1 3 \approx 0 \cdotp 03454... | Euler's Number as Sum of Egyptian Fractions | https://proofwiki.org/wiki/Euler's_Number_as_Sum_of_Egyptian_Fractions | https://proofwiki.org/wiki/Euler's_Number_as_Sum_of_Egyptian_Fractions | [
"Euler's Number"
] | [
"Reciprocal/Examples/Euler's Number",
"Definition:Egyptian Fraction"
] | [
"Reciprocal/Examples/Euler's Number",
"Definition:Integer Sequence",
"Definition:Fraction/Denominator"
] |
proofwiki-12102 | One Half as Pandigital Fraction | There are $12$ ways $\dfrac 1 2$ can be expressed as a pandigital fraction:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6729} {13 \, 458}
}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6792} {13 \, 584}
}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6927} {13 \, 854}
}}
{{eqn | l = \dfrac 1 2
| r = \df... | Can be verified by brute force. | There are $12$ ways $\dfrac 1 2$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6729} {13 \, 458}
}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6792} {13 \, 584}
}}
{{eqn | l = \dfrac 1 2
| r = \dfrac {6927} {13 \, 854}
}}
{{... | Can be verified by brute force. | One Half as Pandigital Fraction | https://proofwiki.org/wiki/One_Half_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Half_as_Pandigital_Fraction | [
"One Half",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12103 | One Seventh as Pandigital Fraction | There are $7$ ways $\dfrac 1 7$ can be expressed as a pandigital fraction:
:$\dfrac 1 7 = \dfrac {2394} {16 \, 758}$
:$\dfrac 1 7 = \dfrac {2637} {18 \, 459}$
:$\dfrac 1 7 = \dfrac {4527} {31 \, 689}$
:$\dfrac 1 7 = \dfrac {5274} {36 \, 918}$
:$\dfrac 1 7 = \dfrac {5418} {37 \, 926}$
:$\dfrac 1 7 = \dfrac {5976} {41 \,... | Can be verified by brute force. | There are $7$ ways $\dfrac 1 7$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 1 7 = \dfrac {2394} {16 \, 758}$
:$\dfrac 1 7 = \dfrac {2637} {18 \, 459}$
:$\dfrac 1 7 = \dfrac {4527} {31 \, 689}$
:$\dfrac 1 7 = \dfrac {5274} {36 \, 918}$
:$\dfrac 1 7 = \dfrac {5418} {37 \, ... | Can be verified by brute force. | One Seventh as Pandigital Fraction | https://proofwiki.org/wiki/One_Seventh_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Seventh_as_Pandigital_Fraction | [
"One Seventh",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12104 | One Third as Pandigital Fraction | There are $2$ ways $\dfrac 1 3$ can be expressed as a pandigital fraction:
{{begin-eqn}}
{{eqn | l = \dfrac 1 3
| r = \dfrac {5823} {17469}
}}
{{eqn | l = \dfrac 1 3
| r = \dfrac {5832} {17496}
}}
{{end-eqn}} | Can be verified by brute force. | There are $2$ ways $\dfrac 1 3$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
{{begin-eqn}}
{{eqn | l = \dfrac 1 3
| r = \dfrac {5823} {17469}
}}
{{eqn | l = \dfrac 1 3
| r = \dfrac {5832} {17496}
}}
{{end-eqn}} | Can be verified by brute force. | One Third as Pandigital Fraction | https://proofwiki.org/wiki/One_Third_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Third_as_Pandigital_Fraction | [
"One Third",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12105 | One Quarter as Pandigital Fraction | There are $4$ ways $\dfrac 1 4$ can be expressed as a pandigital fraction:
:$\dfrac 1 4 = \dfrac {3942} {15768}$
:$\dfrac 1 4 = \dfrac {4392} {17568}$
:$\dfrac 1 4 = \dfrac {5796} {23184}$
:$\dfrac 1 4 = \dfrac {7956} {31824}$ | Can be verified by brute force. | There are $4$ ways $\dfrac 1 4$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 1 4 = \dfrac {3942} {15768}$
:$\dfrac 1 4 = \dfrac {4392} {17568}$
:$\dfrac 1 4 = \dfrac {5796} {23184}$
:$\dfrac 1 4 = \dfrac {7956} {31824}$ | Can be verified by brute force. | One Quarter as Pandigital Fraction | https://proofwiki.org/wiki/One_Quarter_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Quarter_as_Pandigital_Fraction | [
"One Quarter",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12106 | One Fifth as Pandigital Fraction | There are $12$ ways $\dfrac 1 5$ can be expressed as a pandigital fraction:
:$\dfrac 1 5 = \dfrac {2697} {13485}$
:$\dfrac 1 5 = \dfrac {2769} {13845}$
:$\dfrac 1 5 = \dfrac {2937} {14685}$
:$\dfrac 1 5 = \dfrac {2967} {14835}$
:$\dfrac 1 5 = \dfrac {2973} {14865}$
:$\dfrac 1 5 = \dfrac {3297} {16485}$
:$\dfrac 1 5 = \... | Can be verified by brute force. | There are $12$ ways $\dfrac 1 5$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 1 5 = \dfrac {2697} {13485}$
:$\dfrac 1 5 = \dfrac {2769} {13845}$
:$\dfrac 1 5 = \dfrac {2937} {14685}$
:$\dfrac 1 5 = \dfrac {2967} {14835}$
:$\dfrac 1 5 = \dfrac {2973} {14865}$
:$\dfrac 1 5... | Can be verified by brute force. | One Fifth as Pandigital Fraction | https://proofwiki.org/wiki/One_Fifth_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Fifth_as_Pandigital_Fraction | [
"One Fifth",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12107 | One Sixth as Pandigital Fraction | There are $3$ ways $\dfrac 1 6$ can be expressed as a pandigital fraction:
:$\dfrac 1 6 = \dfrac {2943} {17658}$
:$\dfrac 1 6 = \dfrac {4653} {27918}$
:$\dfrac 1 6 = \dfrac {5697} {34182}$ | Can be verified by brute force. | There are $3$ ways $\dfrac 1 6$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 1 6 = \dfrac {2943} {17658}$
:$\dfrac 1 6 = \dfrac {4653} {27918}$
:$\dfrac 1 6 = \dfrac {5697} {34182}$ | Can be verified by brute force. | One Sixth as Pandigital Fraction | https://proofwiki.org/wiki/One_Sixth_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Sixth_as_Pandigital_Fraction | [
"One Sixth",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12108 | One Eighth as Pandigital Fraction | There are $46$ ways $\dfrac 1 8$ can be expressed as a pandigital fraction:
:$\dfrac 1 8 = \dfrac {3187} {25496}$
:$\dfrac 1 8 = \dfrac {4589} {36712}$
:$\dfrac 1 8 = \dfrac {4591} {36728}$
:$\dfrac 1 8 = \dfrac {4689} {37512}$
:$\dfrac 1 8 = \dfrac {4691} {37528}$
:$\dfrac 1 8 = \dfrac {4769} {38152}$
:$\dfrac 1 8 = \... | Can be verified by brute force. | There are $46$ ways $\dfrac 1 8$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
:$\dfrac 1 8 = \dfrac {3187} {25496}$
:$\dfrac 1 8 = \dfrac {4589} {36712}$
:$\dfrac 1 8 = \dfrac {4591} {36728}$
:$\dfrac 1 8 = \dfrac {4689} {37512}$
:$\dfrac 1 8 = \dfrac {4691} {37528}$
:$\dfrac 1 8... | Can be verified by brute force. | One Eighth as Pandigital Fraction | https://proofwiki.org/wiki/One_Eighth_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Eighth_as_Pandigital_Fraction | [
"One Eighth",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12109 | One Ninth as Pandigital Fraction | There are $3$ ways $\dfrac 1 9$ can be expressed as a pandigital fraction:
{{begin-eqn}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {6381} {57429}
}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {6471} {58239}
}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {8361} {75249}
}}
{{end-eqn}} | Can be verified by brute force. | There are $3$ ways $\dfrac 1 9$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]]:
{{begin-eqn}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {6381} {57429}
}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {6471} {58239}
}}
{{eqn | l = \dfrac 1 9
| r = \dfrac {8361} {75249}
}}
{{end-eqn}} | Can be verified by brute force. | One Ninth as Pandigital Fraction | https://proofwiki.org/wiki/One_Ninth_as_Pandigital_Fraction | https://proofwiki.org/wiki/One_Ninth_as_Pandigital_Fraction | [
"One Ninth",
"Pandigital Fractions"
] | [
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12110 | Four Fifths as Pandigital Fraction | The fraction $\dfrac 4 5$ can be expressed as a pandigital fraction in the following interesting way:
:$\dfrac 4 5 = \dfrac {9876} {12 \, 345}$ | Can be found by brute force. | The [[Definition:Fraction|fraction]] $\dfrac 4 5$ can be expressed as a [[Definition:Pandigital Fraction|pandigital fraction]] in the following interesting way:
:$\dfrac 4 5 = \dfrac {9876} {12 \, 345}$ | Can be found by brute force. | Four Fifths as Pandigital Fraction | https://proofwiki.org/wiki/Four_Fifths_as_Pandigital_Fraction | https://proofwiki.org/wiki/Four_Fifths_as_Pandigital_Fraction | [
"One Fifth",
"Pandigital Fractions"
] | [
"Definition:Fraction",
"Definition:Pandigital Fraction"
] | [] |
proofwiki-12111 | Probability of Two Random Integers having no Common Divisor | Let $a$ and $b$ be integers chosen at random.
The probability that $a$ and $b$ are coprime is given by:
:$\map \Pr {a \perp b} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$
where $\zeta$ denotes the zeta function.
The decimal expansion of $\dfrac 1 {\map \zeta 2}$ starts:
:$\dfrac 1 {\map \zeta 2} = 0 \cdotp 60792 \, 7... | Let $a$ and $b$ be two integers chosen at random.
For $a$ and $b$ to be coprime, it is necessary and sufficient that no prime number divides both of them.
The probability that any particular integer is divisible by a prime number $p$ is $\dfrac 1 p$.
Hence the probability that both $a$ and $b$ are divisible by $p$ is $... | Let $a$ and $b$ be [[Definition:Integer|integers]] chosen at random.
The [[Definition:Probability|probability]] that $a$ and $b$ are [[Definition:Coprime Integers|coprime]] is given by:
:$\map \Pr {a \perp b} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|ze... | Let $a$ and $b$ be two [[Definition:Integer|integers]] chosen at random.
For $a$ and $b$ to be [[Definition:Coprime Integers|coprime]], it is [[Definition:Necessary and Sufficient|necessary and sufficient]] that no [[Definition:Prime Number|prime number]] divides both of them.
The [[Definition:Probability|probability... | Probability of Two Random Integers having no Common Divisor | https://proofwiki.org/wiki/Probability_of_Two_Random_Integers_having_no_Common_Divisor | https://proofwiki.org/wiki/Probability_of_Two_Random_Integers_having_no_Common_Divisor | [
"Riemann Zeta Function",
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Probability",
"Definition:Coprime/Integers",
"Definition:Riemann Zeta Function",
"Definition:Decimal Expansion"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:Prime Number",
"Definition:Probability",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Probability",... |
proofwiki-12112 | Probability of Random Integer being Square-Free | Let $a$ be an integer chosen at random.
The probability that $a$ is square-free is given by:
:$\map \Pr {\neg \exists b \in \Z: b^2 \divides a} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$
where $\zeta$ denotes the zeta function.
The decimal expansion of $\dfrac 1 {\map \zeta 2}$ starts:
:$\dfrac 1 {\map \zeta 2} = 0 ... | Let $a$ be an integer chosen at random.
For $a$ to be square-free, it is necessary and sufficient that for all prime numbers $p$, it is not the case that $p^2$ is a divisor of $a$.
The probability that any particular integer is divisible by $p^2$ is $\dfrac 1 {p^2}$.
The probability that $a$ is not divisible by $p^2$ ... | Let $a$ be an [[Definition:Integer|integer]] chosen at random.
The [[Definition:Probability|probability]] that $a$ is [[Definition:Square-Free Integer|square-free]] is given by:
:$\map \Pr {\neg \exists b \in \Z: b^2 \divides a} = \dfrac 1 {\map \zeta 2} = \dfrac 6 {\pi^2}$
where $\zeta$ denotes the [[Definition:Riem... | Let $a$ be an [[Definition:Integer|integer]] chosen at random.
For $a$ to be [[Definition:Square-Free Integer|square-free]], it is [[Definition:Necessary and Sufficient|necessary and sufficient]] that for all [[Definition:Prime Number|prime numbers]] $p$, it is not the case that $p^2$ is a [[Definition:Divisor of Int... | Probability of Random Integer being Square-Free | https://proofwiki.org/wiki/Probability_of_Random_Integer_being_Square-Free | https://proofwiki.org/wiki/Probability_of_Random_Integer_being_Square-Free | [
"Riemann Zeta Function",
"Square-Free Integers"
] | [
"Definition:Integer",
"Definition:Probability",
"Definition:Square-Free Integer",
"Definition:Riemann Zeta Function",
"Definition:Decimal Expansion"
] | [
"Definition:Integer",
"Definition:Square-Free Integer",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Probability",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definiti... |
proofwiki-12113 | Prime Ideal is Prime Element | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a lattice.
Let $I$ be an ideal in $L$.
Then:
:$I$ is a prime ideal
{{iff}}:
:$I$ is a prime element in $\struct {\map {\mathit {Ids} } L, \precsim}$
where:
:$\map {\mathit {Ids} } L$ denotes the set of all ideals in $L$
:$\mathord \precsim := \mathord \subseteq \restricti... | === Sufficient Condition ===
Let $I$ be a prime ideal.
Let $x, y \in \map {\mathit {Ids} } L$ such that
:$x \wedge y \precsim I$
By definition of $\precsim$:
:$x \wedge y \subseteq I$
By Meet is Intersection in Set of Ideals:
:$x \cap y \subseteq I$
{{AimForCont}}:
:$x \not \precsim I$ and $y \not \precsim I$
By defini... | Let $L = \struct {S, \vee, \wedge, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Then:
:$I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]]
{{iff}}:
:$I$ is a [[Definition:Prime Element (Order Theory)|prime element]] in $\st... | === Sufficient Condition ===
Let $I$ be a [[Definition:Prime Ideal (Order Theory)|prime ideal]].
Let $x, y \in \map {\mathit {Ids} } L$ such that
:$x \wedge y \precsim I$
By definition of $\precsim$:
:$x \wedge y \subseteq I$
By [[Meet is Intersection in Set of Ideals]]:
:$x \cap y \subseteq I$
{{AimForCont}}:
:$x... | Prime Ideal is Prime Element | https://proofwiki.org/wiki/Prime_Ideal_is_Prime_Element | https://proofwiki.org/wiki/Prime_Ideal_is_Prime_Element | [
"Prime Ideals (Order Theory)",
"Prime Elements"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Ideal (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Prime Element (Order Theory)",
"Definition:Set of Sets",
"Definition:Ideal (Order Theory)"
] | [
"Definition:Prime Ideal (Order Theory)",
"Meet is Intersection in Set of Ideals",
"Definition:Subset",
"Meet Precedes Operands",
"Definition:Lower Section",
"Definition:Subset",
"Definition:Set Intersection",
"Characterization of Prime Ideal",
"Definition:Contradiction",
"Meet is Intersection in S... |
proofwiki-12114 | Probability of Three Random Integers having no Common Divisor | Let $a, b$ and $c$ be integers chosen at random.
The probability that $a, b$ and $c$ have no common divisor:
:$\map \Pr {\map \perp {a, b, c} } = \dfrac 1 {\map \zeta 3}$
where $\zeta$ denotes the zeta function:
:$\map \zeta 3 = \dfrac 1 {1^3} + \dfrac 1 {2^3} + \dfrac 1 {3^3} + \dfrac 1 {4^3} + \cdots$
The decimal exp... | Let $a$, $b$ and $c$ be three integers chosen at random.
For $a$, $b$ and $c$ to be coprime, it is necessary and sufficient that no prime number divides all of them.
The probability that any particular integer is divisible by a prime number $p$ is $\dfrac 1 p$.
Hence the probability that $a$, $b$ and $c$ are divisible ... | Let $a, b$ and $c$ be [[Definition:Integer|integers]] chosen at random.
The [[Definition:Probability|probability]] that $a, b$ and $c$ have no [[Definition:Common Divisor of Integers|common divisor]]:
:$\map \Pr {\map \perp {a, b, c} } = \dfrac 1 {\map \zeta 3}$
where $\zeta$ denotes the [[Definition:Riemann Zeta Fu... | Let $a$, $b$ and $c$ be three [[Definition:Integer|integers]] chosen at random.
For $a$, $b$ and $c$ to be [[Definition:Coprime Integers|coprime]], it is [[Definition:Necessary and Sufficient|necessary and sufficient]] that no [[Definition:Prime Number|prime number]] divides all of them.
The probability that any part... | Probability of Three Random Integers having no Common Divisor | https://proofwiki.org/wiki/Probability_of_Three_Random_Integers_having_no_Common_Divisor | https://proofwiki.org/wiki/Probability_of_Three_Random_Integers_having_no_Common_Divisor | [
"Riemann Zeta Function",
"Coprime Integers"
] | [
"Definition:Integer",
"Definition:Probability",
"Definition:Common Divisor/Integers",
"Definition:Riemann Zeta Function",
"Definition:Decimal Expansion"
] | [
"Definition:Integer",
"Definition:Coprime/Integers",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition... |
proofwiki-12115 | Densest Packing of Identical Circles | The densest packing of identical circles in the plane obtains a density of $\dfrac \pi {2 \sqrt 3} = \dfrac \pi {\sqrt {12} }$:
:$\dfrac \pi {2 \sqrt 3} = 0 \cdotp 90689 \, 96821 \ldots$
{{OEIS|A093766}}
This happens when they are packed together in a hexagonal array, with each circle touching $6$ others. | {{ProofWanted|It remains to be proved that the hexagonal packing is in fact the densest that can be achieved.}}
Consider the rectangular area $ABCD$ of the densest packing of circles.
:300px
Let the radius of one circle be $1$.
The length $AB$ is $2$.
The length $AC$ is $2 \sqrt 3$.
Thus, from Area of Rectangle, the ar... | The densest packing of identical [[Definition:Circle|circles]] in [[Definition:The Plane|the plane]] obtains a density of $\dfrac \pi {2 \sqrt 3} = \dfrac \pi {\sqrt {12} }$:
:$\dfrac \pi {2 \sqrt 3} = 0 \cdotp 90689 \, 96821 \ldots$
{{OEIS|A093766}}
This happens when they are packed together in a [[Definition:Regular... | {{ProofWanted|It remains to be proved that the hexagonal packing is in fact the densest that can be achieved.}}
Consider the [[Definition:Rectangle|rectangular area]] $ABCD$ of the densest packing of [[Definition:Circle|circles]].
:[[File:Close-Packed-Circles.png|300px]]
Let the [[Definition:Radius|radius]] of on... | Densest Packing of Identical Circles | https://proofwiki.org/wiki/Densest_Packing_of_Identical_Circles | https://proofwiki.org/wiki/Densest_Packing_of_Identical_Circles | [
"Geometry"
] | [
"Definition:Circle",
"Definition:Plane Surface/The Plane",
"Definition:Hexagon/Regular",
"Definition:Circle"
] | [
"Definition:Quadrilateral/Rectangle",
"Definition:Circle",
"File:Close-Packed-Circles.png",
"Definition:Radius",
"Definition:Circle",
"Definition:Linear Measure/Length",
"Definition:Linear Measure/Length",
"Area of Parallelogram/Rectangle",
"Definition:Area",
"Definition:Circle",
"Definition:Qua... |
proofwiki-12116 | Anning's Theorem | In any base greater than $1$, the fraction:
:$\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$
has the property that if the two $1$'s in the center of the numerator and the denominator are replaced by the same odd number of $1$'s, the value of the fraction remains the same.
For example:
:$\dfrac {101 \, 010 \, 101} {110... | Let $b$ be the base in question.
Let $F = \dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$.
Then:
:$F = \dfrac {1 + b^2 + b^4 + b^6 + b^8} {1 + b + b^4 + b^7 + b^8}$
It is necessary to prove that for all $k \in \Z_{>0}$:
:$F = \dfrac {1 + b^2 + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 4} + b^{2 k + 6} } {1 + b + b^4 ... | In any [[Definition:Number Base|base]] greater than $1$, the [[Definition:Fraction|fraction]]:
:$\dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$
has the property that if the two $1$'s in the center of the [[Definition:Numerator|numerator]] and the [[Definition:Denominator|denominator]] are replaced by the same [[Defin... | Let $b$ be the [[Definition:Number Base|base]] in question.
Let $F = \dfrac {101 \, 010 \, 101} {110 \, 010 \, 011}$.
Then:
:$F = \dfrac {1 + b^2 + b^4 + b^6 + b^8} {1 + b + b^4 + b^7 + b^8}$
It is necessary to prove that for all $k \in \Z_{>0}$:
:$F = \dfrac {1 + b^2 + b^4 + b^5 + \cdots + b^{2 k + 2} + b^{2 k + 4}... | Anning's Theorem | https://proofwiki.org/wiki/Anning's_Theorem | https://proofwiki.org/wiki/Anning's_Theorem | [
"Number Theory"
] | [
"Definition:Number Base",
"Definition:Fraction",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Odd Integer",
"Definition:Fraction",
"Definition:Decimal Notation"
] | [
"Definition:Number Base",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator"
] |
proofwiki-12117 | Meet is Intersection in Set of Ideals | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $\map {\mathit {Ids} } {\mathscr S}$ be the set of all ideals in $\mathscr S$.
Let $P = \struct {\map {\mathit {Ids} } {\mathscr S}, \precsim}$ be an ordered set where $\mathord \precsim = \mathord \subseteq \restriction_{\map {\mathit {Ids} } {... | By Intersection of Semilattice Ideals is Ideal:
:$I_1 \cap I_2 \in \map {\mathit {Ids} } {\mathscr S}$
We will prove that
:$I_1 \cap I_2$ is lower bound for $\set {I_1, I_2}$
Let $x \in \set {I_1, I_2}$.
Then by definition of unordered tuple:
:$x = I_1$ or $x = I_2$
By Intersection is Subset
:$I_1 \cap I_2 \subseteq x$... | Let $\mathscr S = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $\map {\mathit {Ids} } {\mathscr S}$ be the [[Definition:Set of Sets|set]] of all [[Definition:Ideal in Ordered Set|ideals]] in $\mathscr S$.
Let $P = \struct {\map {\mathit {Ids} } {\mathscr S}, \precsim}$ be a... | By [[Intersection of Semilattice Ideals is Ideal]]:
:$I_1 \cap I_2 \in \map {\mathit {Ids} } {\mathscr S}$
We will prove that
:$I_1 \cap I_2$ is [[Definition:Lower Bound of Set|lower bound]] for $\set {I_1, I_2}$
Let $x \in \set {I_1, I_2}$.
Then by definition of [[Definition:Unordered Tuple|unordered tuple]]:
:$x =... | Meet is Intersection in Set of Ideals | https://proofwiki.org/wiki/Meet_is_Intersection_in_Set_of_Ideals | https://proofwiki.org/wiki/Meet_is_Intersection_in_Set_of_Ideals | [
"Join and Meet"
] | [
"Definition:Meet Semilattice",
"Definition:Set of Sets",
"Definition:Ideal in Ordered Set",
"Definition:Ordered Set",
"Definition:Ideal in Ordered Set"
] | [
"Intersection of Semilattice Ideals is Ideal",
"Definition:Lower Bound of Set",
"Definition:Unordered Tuple",
"Intersection is Subset",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Lower Bound of Set",
"Intersection is Largest Subset",
"Definition:Infimum of Set",
... |
proofwiki-12118 | Characterization of Prime Ideal by Finite Infima | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $I$ be an ideal in $L$.
Then
:$I$ is a prime ideal
{{iff}}
:for all non-empty finite subset $A$ of $S: \paren {\inf A \in I \implies \exists a \in A: a \in I}$ | === Sufficient Condition ===
Let $I$ be a prime ideal.
Define:
:$\map P X: \equiv X \ne \O \land \inf X \in I \implies \exists x \in X: x \in I$
where $X$ is subset of $S$.
Let $A$ be a non-empty finite subset of $S$.
By definition of empty set:
:$\map P \O$
We will prove that:
:$\forall x \in A, B \subseteq A: \map P ... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $I$ be an [[Definition:Ideal (Order Theory)|ideal]] in $L$.
Then
:$I$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]]
{{iff}}
:for all [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[... | === Sufficient Condition ===
Let $I$ be a [[Definition:Prime Ideal (Order Theory)|prime ideal]].
Define:
:$\map P X: \equiv X \ne \O \land \inf X \in I \implies \exists x \in X: x \in I$
where $X$ is [[Definition:Subset|subset]] of $S$.
Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|fin... | Characterization of Prime Ideal by Finite Infima | https://proofwiki.org/wiki/Characterization_of_Prime_Ideal_by_Finite_Infima | https://proofwiki.org/wiki/Characterization_of_Prime_Ideal_by_Finite_Infima | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Meet Semilattice",
"Definition:Ideal (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Prime Ideal (Order Theory)",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Empty Set",
"Definition:Induction Hypothesis",
"Union with Empty Set",
"Infimum of Singleton",
"Definition:Singleton",
"Subset of Finite Set is Fini... |
proofwiki-12119 | Unordered Pair is Finite | Let $x, y$ be arbitrary.
Then $\set {x, y}$ is finite. | By Union of Disjoint Singletons is Doubleton:
:$\set {x, y} = \set x \cup \set y$
By Singleton is Finite:
:$\set x$ and $\set y$ are finite.
Thus by Union of Finite Sets is Finite:
:$\set {x, y}$ is finite.
{{qed}} | Let $x, y$ be arbitrary.
Then $\set {x, y}$ is [[Definition:Finite Set|finite]]. | By [[Union of Disjoint Singletons is Doubleton]]:
:$\set {x, y} = \set x \cup \set y$
By [[Singleton is Finite]]:
:$\set x$ and $\set y$ are [[Definition:Finite Set|finite]].
Thus by [[Union of Finite Sets is Finite]]:
:$\set {x, y}$ is [[Definition:Finite Set|finite]].
{{qed}} | Unordered Pair is Finite | https://proofwiki.org/wiki/Unordered_Pair_is_Finite | https://proofwiki.org/wiki/Unordered_Pair_is_Finite | [
"Doubletons",
"Finite Sets"
] | [
"Definition:Finite Set"
] | [
"Union of Disjoint Singletons is Doubleton",
"Singleton is Finite",
"Definition:Finite Set",
"Union of Finite Sets is Finite",
"Definition:Finite Set"
] |
proofwiki-12120 | Length of Diagonal of Unit Square | The length of a diagonal of a square of side length $1$ is $\sqrt 2$) (the square root of $2$). | :300px
Two adjacent sides $AB$, $BC$ and the diagonal $AC$ of square $ABCD$ form a right triangle.
The hypotenuse of triangle $\triangle ABC$ can be found by using Pythagoras's Theorem:
:$AC^2 = AB^2 + BC^2$
from which:
:$AC^2 = 2$
and so:
:$AC = \sqrt 2$
{{qed}} | The [[Definition:Length (Linear Measure)|length]] of a [[Definition:Diagonal of Quadrilateral|diagonal]] of a [[Definition:Square (Geometry)|square]] of [[Definition:Side of Polygon|side]] [[Definition:Length (Linear Measure)|length]] $1$ is $\sqrt 2$) (the [[Definition:Square Root|square root]] of $2$). | :[[File:DiagonalOfSquare.png|300px]]
Two adjacent [[Definition:Side of Polygon|sides]] $AB$, $BC$ and the [[Definition:Diagonal of Quadrilateral|diagonal]] $AC$ of [[Definition:Square (Geometry)|square]] $ABCD$ form a [[Definition:Right Triangle|right triangle]].
The [[Definition:Hypotenuse|hypotenuse]] of [[Definiti... | Length of Diagonal of Unit Square | https://proofwiki.org/wiki/Length_of_Diagonal_of_Unit_Square | https://proofwiki.org/wiki/Length_of_Diagonal_of_Unit_Square | [
"Squares"
] | [
"Definition:Linear Measure/Length",
"Definition:Diameter of Quadrilateral",
"Definition:Quadrilateral/Square",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Square Root"
] | [
"File:DiagonalOfSquare.png",
"Definition:Polygon/Side",
"Definition:Diameter of Quadrilateral",
"Definition:Quadrilateral/Square",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Triangle (Geometry)/Right-Angled/Hypotenuse",
"Definition:Triangle (Geometry)",
"Pythagoras's Theorem"
] |
proofwiki-12121 | Sequence of Best Rational Approximations to Square Root of 2 | A sequence of best rational approximations to the square root of $2$ starts:
:$\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
where:
:the numerators are half of the Pell-Lucas numbers, $\dfrac 1 2 Q_n$
:the denominators are the P... | Let $\tuple {a_0, a_1, \ldots}$ be the continued fraction expansion of $\sqrt 2$.
By Continued Fraction Expansion of Root 2:
:$\sqrt 2 = \sqbrk {1, \sequence 2} = \sqbrk {1, 2, 2, 2, \ldots}$
From Convergents are Best Approximations, the convergents of $\sqbrk {1, \sequence 2}$ are the best rational approximations of $... | A [[Definition:Sequence|sequence]] of [[Definition:Best Rational Approximation|best rational approximations]] to the [[Definition:Square Root|square root]] of $2$ starts:
:$\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
where:
:... | Let $\tuple {a_0, a_1, \ldots}$ be the [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]] of $\sqrt 2$.
By [[Continued Fraction Expansion of Root 2]]:
:$\sqrt 2 = \sqbrk {1, \sequence 2} = \sqbrk {1, 2, 2, 2, \ldots}$
From [[Convergents are Best Approximations]], the [[Definition:... | Sequence of Best Rational Approximations to Square Root of 2 | https://proofwiki.org/wiki/Sequence_of_Best_Rational_Approximations_to_Square_Root_of_2 | https://proofwiki.org/wiki/Sequence_of_Best_Rational_Approximations_to_Square_Root_of_2 | [
"Square Root of 2"
] | [
"Definition:Sequence",
"Definition:Best Rational Approximation",
"Definition:Square Root",
"Definition:Fraction/Numerator",
"Definition:Pell-Lucas Numbers",
"Definition:Fraction/Denominator",
"Definition:Pell Numbers"
] | [
"Definition:Continued Fraction Expansion/Real Number",
"Continued Fraction Expansion of Irrational Square Root/Examples/2",
"Convergents are Best Approximations",
"Definition:Convergent of Continued Fraction",
"Definition:Best Rational Approximation",
"Definition:Numerators and Denominators of Continued F... |
proofwiki-12122 | Difference between Adjacent Convergents of Simple Continued Fraction | Then for $k \ge 1$:
:$p_k q_{k - 1} - p_{k - 1} q_k = \paren {-1}^{k + 1}$
That is:
:$C_k - C_{k - 1} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 1} } {q_{k - 1} } = \dfrac {\paren {-1}^{k + 1} } {q_k q_{k - 1} }$ | Proof by induction:
For all $n \in \Z: n \ge 2$, let $\map P n$ be the proposition:
:$p_n q_{n - 1} - p_{n - 1} q_n = \paren {-1}^{n + 1}$ | Then for $k \ge 1$:
:$p_k q_{k - 1} - p_{k - 1} q_k = \paren {-1}^{k + 1}$
That is:
:$C_k - C_{k - 1} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 1} } {q_{k - 1} } = \dfrac {\paren {-1}^{k + 1} } {q_k q_{k - 1} }$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z: n \ge 2$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$p_n q_{n - 1} - p_{n - 1} q_n = \paren {-1}^{n + 1}$ | Difference between Adjacent Convergents of Simple Continued Fraction | https://proofwiki.org/wiki/Difference_between_Adjacent_Convergents_of_Simple_Continued_Fraction | https://proofwiki.org/wiki/Difference_between_Adjacent_Convergents_of_Simple_Continued_Fraction | [
"Simple Continued Fractions"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12123 | Difference between Adjacent Convergents But One of Simple Continued Fraction | For $k \ge 2$:
:$p_k q_{k - 2} - p_{k - 2} q_k = \paren {-1}^k a_k$
That is:
:$C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} }$ | Let $k \ge 2$.
{{begin-eqn}}
{{eqn | l = p_k q_{k - 2} - p_{k - 2} q_k
| r = \paren {a_k p_{k - 1} + p_{k - 2} } q_{k - 2} - p_{k - 2} \paren {a_k q_{k - 1} + q_{k - 2} }
| c = {{Defof|Numerators and Denominators of Continued Fraction}}
}}
{{eqn | r = a_k \paren {p_{k - 1} q_{k - 2} - p_{k - 2} q_{k - 1} }
... | For $k \ge 2$:
:$p_k q_{k - 2} - p_{k - 2} q_k = \paren {-1}^k a_k$
That is:
:$C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} }$ | Let $k \ge 2$.
{{begin-eqn}}
{{eqn | l = p_k q_{k - 2} - p_{k - 2} q_k
| r = \paren {a_k p_{k - 1} + p_{k - 2} } q_{k - 2} - p_{k - 2} \paren {a_k q_{k - 1} + q_{k - 2} }
| c = {{Defof|Numerators and Denominators of Continued Fraction}}
}}
{{eqn | r = a_k \paren {p_{k - 1} q_{k - 2} - p_{k - 2} q_{k - 1} }... | Difference between Adjacent Convergents But One of Simple Continued Fraction | https://proofwiki.org/wiki/Difference_between_Adjacent_Convergents_But_One_of_Simple_Continued_Fraction | https://proofwiki.org/wiki/Difference_between_Adjacent_Convergents_But_One_of_Simple_Continued_Fraction | [
"Simple Continued Fractions"
] | [] | [
"Difference between Adjacent Convergents of Simple Continued Fraction",
"Category:Simple Continued Fractions"
] |
proofwiki-12124 | Convergents of Simple Continued Fraction are Rationals in Canonical Form | For all $k \ge 1$, $\dfrac {p_k} {q_k}$ is in canonical form:
:$p_k$ and $q_k$ are coprime
:$q_k > 0$. | Let $k \ge 1$.
Let $d = \gcd \set {p_k, q_k}$.
From Common Divisor Divides Integer Combination:
:$p_k q_{k - 1} - p_{k - 1} q_k$ is a multiple of $d$.
From Difference between Adjacent Convergents of Simple Continued Fraction:
:$d \divides \paren {-1}^{k + 1}$
where $\divides$ denotes divisibility.
It follows that:
:$d ... | For all $k \ge 1$, $\dfrac {p_k} {q_k}$ is in [[Definition:Canonical Form of Rational Number|canonical form]]:
:$p_k$ and $q_k$ are [[Definition:Coprime Integers|coprime]]
:$q_k > 0$. | Let $k \ge 1$.
Let $d = \gcd \set {p_k, q_k}$.
From [[Common Divisor Divides Integer Combination]]:
:$p_k q_{k - 1} - p_{k - 1} q_k$ is a [[Definition:Divisor of Integer|multiple]] of $d$.
From [[Difference between Adjacent Convergents of Simple Continued Fraction]]:
:$d \divides \paren {-1}^{k + 1}$
where $\divide... | Convergents of Simple Continued Fraction are Rationals in Canonical Form | https://proofwiki.org/wiki/Convergents_of_Simple_Continued_Fraction_are_Rationals_in_Canonical_Form | https://proofwiki.org/wiki/Convergents_of_Simple_Continued_Fraction_are_Rationals_in_Canonical_Form | [
"Simple Continued Fractions"
] | [
"Definition:Rational Number/Canonical Form",
"Definition:Coprime/Integers"
] | [
"Common Divisor Divides Integer Combination",
"Definition:Divisor (Algebra)/Integer",
"Difference between Adjacent Convergents of Simple Continued Fraction",
"Definition:Divisor (Algebra)/Integer",
"Denominators of Simple Continued Fraction are Strictly Increasing",
"Category:Simple Continued Fractions"
] |
proofwiki-12125 | Characterization of Prime Filter by Finite Suprema | Let $L = \struct {S, \vee, \preceq}$ be a join semilattice.
Let $F$ be a filter in $L$.
Then
:$F$ is a prime filter
{{iff}}:
:for all non-empty finite subset $A$ of $S: \paren {\sup A \in F \implies \exists a \in A: a \in F}$ | === Sufficient Condition ===
Let $F$ be a prime ideal.
Define $\map \PP X: \equiv X \ne \O \land \sup X \in F \implies \exists x \in X: x \in F$
where $X$ is subset of $S$.
Let $A$ be a non-empty finite subset of $S$.
By definition of empty set:
:$\map \PP \O$
We will prove that:
:$\forall x \in A, B \subseteq A: \map ... | Let $L = \struct {S, \vee, \preceq}$ be a [[Definition:Join Semilattice|join semilattice]].
Let $F$ be a [[Definition:Filter|filter]] in $L$.
Then
:$F$ is a [[Definition:Prime Filter (Order Theory)|prime filter]]
{{iff}}:
:for all [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|finite]] [[Definition:S... | === Sufficient Condition ===
Let $F$ be a [[Definition:Prime Filter (Order Theory)|prime ideal]].
Define $\map \PP X: \equiv X \ne \O \land \sup X \in F \implies \exists x \in X: x \in F$
where $X$ is [[Definition:Subset|subset]] of $S$.
Let $A$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Finite Set|fi... | Characterization of Prime Filter by Finite Suprema | https://proofwiki.org/wiki/Characterization_of_Prime_Filter_by_Finite_Suprema | https://proofwiki.org/wiki/Characterization_of_Prime_Filter_by_Finite_Suprema | [
"Prime Ideals (Order Theory)"
] | [
"Definition:Join Semilattice",
"Definition:Filter",
"Definition:Prime Filter (Order Theory)",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset"
] | [
"Definition:Prime Filter (Order Theory)",
"Definition:Subset",
"Definition:Non-Empty Set",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Empty Set",
"Union with Empty Set",
"Supremum of Singleton",
"Definition:Singleton",
"Subset of Finite Set is Finite",
"Definition:Finite Set",
"... |
proofwiki-12126 | Dual of Preordered Set is Preordered Set | Let $P = \struct {S, \preceq}$ be a preordered set.
Then dual of $P$, $P^{-1} = \struct {S, \succeq}$ is also a preordered set.
{{finish|Dual Ordered Set $\ne$ Dual Preordered Set}} | By Inverse of Reflexive Relation is Reflexive:
:$\succeq$ is reflexive.
By Inverse of Transitive Relation is Transitive:
:$\succeq$ is transitive.
Hence $\succeq$ is a preordering.
{{qed}} | Let $P = \struct {S, \preceq}$ be a [[Definition:Preordered Set|preordered set]].
Then [[Definition:Dual Ordered Set|dual]] of $P$, $P^{-1} = \struct {S, \succeq}$ is also a [[Definition:Preordered Set|preordered set]].
{{finish|Dual Ordered Set $\ne$ Dual Preordered Set}} | By [[Inverse of Reflexive Relation is Reflexive]]:
:$\succeq$ is [[Definition:Reflexive Relation|reflexive]].
By [[Inverse of Transitive Relation is Transitive]]:
:$\succeq$ is [[Definition:Transitive Relation|transitive]].
Hence $\succeq$ is a [[Definition:Preordering|preordering]].
{{qed}} | Dual of Preordered Set is Preordered Set | https://proofwiki.org/wiki/Dual_of_Preordered_Set_is_Preordered_Set | https://proofwiki.org/wiki/Dual_of_Preordered_Set_is_Preordered_Set | [
"Dual Orderings"
] | [
"Definition:Preordering/Preordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Preordering/Preordered Set"
] | [
"Inverse of Reflexive Relation is Reflexive",
"Definition:Reflexive Relation",
"Inverse of Transitive Relation is Transitive",
"Definition:Transitive Relation",
"Definition:Preordering"
] |
proofwiki-12127 | Dual Ordered Set is Ordered Set | Let $P = \struct {S, \preceq}$ be an ordered set.
Then its dual, $P^{-1} = \struct {S, \succeq}$, is also an ordered set. | Immediate from Dual Ordering is Ordering.
{{qed}} | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then its [[Definition:Dual Ordered Set|dual]], $P^{-1} = \struct {S, \succeq}$, is also an [[Definition:Ordered Set|ordered set]]. | Immediate from [[Dual Ordering is Ordering]].
{{qed}} | Dual Ordered Set is Ordered Set | https://proofwiki.org/wiki/Dual_Ordered_Set_is_Ordered_Set | https://proofwiki.org/wiki/Dual_Ordered_Set_is_Ordered_Set | [
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Dual Ordering/Dual Ordered Set",
"Definition:Ordered Set"
] | [
"Dual Ordering is Ordering"
] |
proofwiki-12128 | Value of Finite Continued Fraction equals Numerator Divided by Denominator | Let $F$ be a field.
Let $\tuple {a_0, a_1, \ldots, a_n}$ be a finite continued fraction of length $n \ge 0$.
Let $p_n$ and $q_n$ be its $n$th numerator and denominator.
Then the value $\sqbrk {a_0, a_1, \ldots, a_n}$ equals $\dfrac {p_n} {q_n}$. | We will use a proof by induction on the length $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$\sqbrk {a_0, a_1, \ldots, a_n} = \dfrac {p_n} {q_n}$ | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\tuple {a_0, a_1, \ldots, a_n}$ be a [[Definition:Finite Continued Fraction|finite continued fraction]] of [[Definition:Length of Continued Fraction|length]] $n \ge 0$.
Let $p_n$ and $q_n$ be its $n$th [[Definition:Numerator and Denominator of Continued... | We will use a proof by [[Principle of Mathematical Induction|induction]] on the [[Definition:Length of Continued Fraction|length]] $n$.
For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\sqbrk {a_0, a_1, \ldots, a_n} = \dfrac {p_n} {q_n}$ | Value of Finite Continued Fraction equals Numerator Divided by Denominator | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_equals_Numerator_Divided_by_Denominator | https://proofwiki.org/wiki/Value_of_Finite_Continued_Fraction_equals_Numerator_Divided_by_Denominator | [
"Continued Fractions",
"Proofs by Induction"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Continued Fraction/Finite",
"Definition:Length of Continued Fraction",
"Definition:Numerators and Denominators of Continued Fraction",
"Definition:Value of Continued Fraction"
] | [
"Principle of Mathematical Induction",
"Definition:Length of Continued Fraction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12129 | Simple Infinite Continued Fraction Converges | Let $C = (a_0, a_1, \ldots)$ be a simple infinite continued fraction in $\R$.
Then $C$ converges. | We need to show that for any SICF its sequence of convergents $\sequence {C_n}$ always tends to a limit.
Let $\epsilon > 0$.
For $m > n \ge \max \set {5, \dfrac 1 \epsilon}$:
{{begin-eqn}}
{{eqn | l = \size {C_m - C_n}
| o = \le
| r = \size {C_m - C_{m - 1} } + \cdots + \size {C_{n + 1} - C_n}
| c = T... | Let $C = (a_0, a_1, \ldots)$ be a [[Definition:Simple Infinite Continued Fraction|simple infinite continued fraction]] in $\R$.
Then $C$ [[Definition:Convergent Continued Fraction|converges]]. | We need to show that for any [[Definition:Simple Infinite Continued Fraction|SICF]] its [[Definition:Sequence|sequence]] of [[Definition:Convergent of Continued Fraction|convergents]] $\sequence {C_n}$ always tends to a [[Definition:Limit of Sequence (Number Field)|limit]].
Let $\epsilon > 0$.
For $m > n \ge \max \s... | Simple Infinite Continued Fraction Converges | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_Converges | https://proofwiki.org/wiki/Simple_Infinite_Continued_Fraction_Converges | [
"Simple Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Convergent Continued Fraction"
] | [
"Definition:Simple Continued Fraction/Infinite",
"Definition:Sequence",
"Definition:Convergent of Continued Fraction",
"Definition:Limit of Sequence (Number Field)",
"Triangle Inequality",
"Difference between Adjacent Convergents of Simple Continued Fraction",
"Lower Bounds for Denominators of Simple Co... |
proofwiki-12130 | Bound for Difference of Irrational Number with Convergent | Let $x$ be an irrational number.
Let $\sequence {C_n}$ be the sequence of convergents of the continued fraction expansion of $x$.
Then $\forall n \ge 1$:
:$C_n < x < C_{n + 1}$ or $C_{n + 1} < x < C_n$
:$\size {x - C_n} < \dfrac 1 {q_n q_{n + 1} }$ | {{questionable|this needs Correspondence between Irrational Numbers and Simple Infinite Continued Fractions}}
Immediate.
Note that:
:$\size {x - C_n} < \size {C_{n + 1} - C_n} = \dfrac 1 {q_n q_{n + 1} }$
{{qed}}
Category:Continued Fractions
3uolekaum9vp63qh9xj7w2yks1jnw2a | Let $x$ be an [[Definition:Irrational Number|irrational number]].
Let $\sequence {C_n}$ be the [[Definition:Sequence|sequence]] of [[Definition:Convergent of Continued Fraction|convergents]] of the [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]] of $x$.
Then $\forall n \... | {{questionable|this needs [[Correspondence between Irrational Numbers and Simple Infinite Continued Fractions]]}}
Immediate.
Note that:
:$\size {x - C_n} < \size {C_{n + 1} - C_n} = \dfrac 1 {q_n q_{n + 1} }$
{{qed}}
[[Category:Continued Fractions]]
3uolekaum9vp63qh9xj7w2yks1jnw2a | Bound for Difference of Irrational Number with Convergent | https://proofwiki.org/wiki/Bound_for_Difference_of_Irrational_Number_with_Convergent | https://proofwiki.org/wiki/Bound_for_Difference_of_Irrational_Number_with_Convergent | [
"Continued Fractions"
] | [
"Definition:Irrational Number",
"Definition:Sequence",
"Definition:Convergent of Continued Fraction",
"Definition:Continued Fraction Expansion/Real Number"
] | [
"Correspondence between Irrational Numbers and Simple Infinite Continued Fractions",
"Category:Continued Fractions"
] |
proofwiki-12131 | Continued Fraction Identities/First/Infinite | Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple infinite continued fraction.
Then:
:$\left[{a_1, a_2, a_3, \ldots}\right] = a_1 + \dfrac 1 {\left[{a_2, a_3, \ldots}\right]}$ | {{proof wanted|use continuity}}
{{Qed}}
Category:Continued Fractions
5n4u82kpalt0upokwg3op45qyptfyp2 | Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a [[Definition:Simple Infinite Continued Fraction|simple infinite continued fraction]].
Then:
:$\left[{a_1, a_2, a_3, \ldots}\right] = a_1 + \dfrac 1 {\left[{a_2, a_3, \ldots}\right]}$ | {{proof wanted|use continuity}}
{{Qed}}
[[Category:Continued Fractions]]
5n4u82kpalt0upokwg3op45qyptfyp2 | Continued Fraction Identities/First/Infinite | https://proofwiki.org/wiki/Continued_Fraction_Identities/First/Infinite | https://proofwiki.org/wiki/Continued_Fraction_Identities/First/Infinite | [
"Continued Fractions"
] | [
"Definition:Simple Continued Fraction/Infinite"
] | [
"Category:Continued Fractions"
] |
proofwiki-12132 | Relation between Adjacent Best Rational Approximations to Root 2 | Consider the Sequence of Best Rational Approximations to Square Root of 2:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Let $\dfrac {p_n} {q_n}$ and $\dfrac {p_{n + 1} } {q_{n + 1} }$ be adjacent terms of $\seq... | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\dfrac {p_{n + 1} } {q_{n + 1} } = \dfrac {p_n + 2 q_n} {p_n + q_n}$ | Consider the [[Sequence of Best Rational Approximations to Square Root of 2]]:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Let $\dfrac {p_n} {q_n}$ and $\dfrac {p_{n + 1} } {q_{n + 1} }$ be adjacent [[Definit... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\dfrac {p_{n + 1} } {q_{n + 1} } = \dfrac {p_n + 2 q_n} {p_n + q_n}$ | Relation between Adjacent Best Rational Approximations to Root 2 | https://proofwiki.org/wiki/Relation_between_Adjacent_Best_Rational_Approximations_to_Root_2 | https://proofwiki.org/wiki/Relation_between_Adjacent_Best_Rational_Approximations_to_Root_2 | [
"Square Root of 2"
] | [
"Sequence of Best Rational Approximations to Square Root of 2",
"Definition:Term of Sequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12133 | Parity of Pell Numbers | Consider the Pell numbers $P_0, P_1, P_2, \ldots$
:$0, 1, 2, 5, 12, 29, \ldots$
$P_n$ has the same parity as $n$.
That is:
:if $n$ is odd then $P_n$ is odd
:if $n$ is even then $P_n$ is even. | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$P_n$ has the same parity as $n$. | Consider the [[Definition:Pell Numbers|Pell numbers]] $P_0, P_1, P_2, \ldots$
:$0, 1, 2, 5, 12, 29, \ldots$
$P_n$ has the same [[Definition:Parity of Integer|parity]] as $n$.
That is:
:if $n$ is [[Definition:Odd Integer|odd]] then $P_n$ is [[Definition:Odd Integer|odd]]
:if $n$ is [[Definition:Even Integer|even]] th... | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$P_n$ has the same [[Definition:Parity of Integer|parity]] as $n$. | Parity of Pell Numbers | https://proofwiki.org/wiki/Parity_of_Pell_Numbers | https://proofwiki.org/wiki/Parity_of_Pell_Numbers | [
"Pell Numbers"
] | [
"Definition:Pell Numbers",
"Definition:Parity of Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Even Integer"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"Definition:Parity of Integer",
"D... |
proofwiki-12134 | Parity of Best Rational Approximations to Root 2 | Consider the Sequence of Best Rational Approximations to Square Root of 2:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
where $S_1 := \dfrac 1 1$.
The numerators of the terms of $\sequence S$ are all odd.
For a... | First the parity of the numerators of the terms of $\sequence S$ is established.
Let $\dfrac {p_n} {q_n}$ be a general term of $\sequence S$.
By Relation between Adjacent Best Rational Approximations to Root 2:
:$p_{n + 1} = p_n + 2 q_n$
Thus if $p_n$ is odd then so is $p_{n + 1}$.
But $p_1 = 1$ is odd.
So $p_n$ is odd... | Consider the [[Sequence of Best Rational Approximations to Square Root of 2]]:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
where $S_1 := \dfrac 1 1$.
The [[Definition:Numerator|numerators]] of the [[Definitio... | First the [[Definition:Parity of Integer|parity]] of the [[Definition:Numerator|numerators]] of the [[Definition:Term of Sequence|terms]] of $\sequence S$ is established.
Let $\dfrac {p_n} {q_n}$ be a general [[Definition:Term of Sequence|term]] of $\sequence S$.
By [[Relation between Adjacent Best Rational Approxim... | Parity of Best Rational Approximations to Root 2 | https://proofwiki.org/wiki/Parity_of_Best_Rational_Approximations_to_Root_2 | https://proofwiki.org/wiki/Parity_of_Best_Rational_Approximations_to_Root_2 | [
"Square Root of 2"
] | [
"Sequence of Best Rational Approximations to Square Root of 2",
"Definition:Fraction/Numerator",
"Definition:Term of Sequence",
"Definition:Odd Integer",
"Definition:Parity of Integer",
"Definition:Fraction/Denominator",
"Definition:Term of Sequence",
"Definition:Parity of Integer"
] | [
"Definition:Parity of Integer",
"Definition:Fraction/Numerator",
"Definition:Term of Sequence",
"Definition:Term of Sequence",
"Relation between Adjacent Best Rational Approximations to Root 2",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Principle of Mathematical ... |
proofwiki-12135 | Best Rational Approximations to Root 2 generate Pythagorean Triples | Consider the Sequence of Best Rational Approximations to Square Root of 2:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Every odd term of $\sequence S$ can be expressed as:
:$\dfrac {2 a + 1} b$
such that:
:$a^... | From Parity of Best Rational Approximations to Root 2:
:The numerators of the terms of $\sequence S$ are all odd.
:For all $n$, the parity of the denominator of term $S_n$ is the same as the parity of $n$.
Thus it follows that every other term of $\sequence S$ has a numerator and a denominator which are both odd.
This ... | Consider the [[Sequence of Best Rational Approximations to Square Root of 2]]:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Every [[Definition:Odd Integer|odd]] [[Definition:Term of Sequence|term]] of $\sequen... | From [[Parity of Best Rational Approximations to Root 2]]:
:The [[Definition:Numerator|numerators]] of the [[Definition:Term of Sequence|terms]] of $\sequence S$ are all [[Definition:Odd Integer|odd]].
:For all $n$, the [[Definition:Parity of Integer|parity]] of the [[Definition:Denominator|denominator]] of [[Definit... | Best Rational Approximations to Root 2 generate Pythagorean Triples/Proof 1 | https://proofwiki.org/wiki/Best_Rational_Approximations_to_Root_2_generate_Pythagorean_Triples | https://proofwiki.org/wiki/Best_Rational_Approximations_to_Root_2_generate_Pythagorean_Triples/Proof_1 | [
"Square Root of 2",
"Pythagorean Triples",
"Pell Numbers"
] | [
"Sequence of Best Rational Approximations to Square Root of 2",
"Definition:Odd Integer",
"Definition:Term of Sequence"
] | [
"Parity of Best Rational Approximations to Root 2",
"Definition:Fraction/Numerator",
"Definition:Term of Sequence",
"Definition:Odd Integer",
"Definition:Parity of Integer",
"Definition:Fraction/Denominator",
"Definition:Term of Sequence",
"Definition:Parity of Integer",
"Definition:Term of Sequence... |
proofwiki-12136 | Best Rational Approximations to Root 2 generate Pythagorean Triples | Consider the Sequence of Best Rational Approximations to Square Root of 2:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Every odd term of $\sequence S$ can be expressed as:
:$\dfrac {2 a + 1} b$
such that:
:$a^... | From Pell Number as Sum of Squares, we have:
:$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$
Therefore:
{{begin-eqn}}
{{eqn | l = P_{2 n + 1}^2
| r = \paren {P_{n + 1}^2 + P_n^2}^2
}}
{{eqn | r = P_{n + 1}^4 + 2 P_{n + 1}^2 P_n^2 + P_n^4
| c = Square of Sum
}}
{{eqn | r = P_{n + 1}^4 + 2 P_{n + 1}^2 P_n^2 + P_n^4 + 2 ... | Consider the [[Sequence of Best Rational Approximations to Square Root of 2]]:
:$\sequence S := \dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \ldots$
Every [[Definition:Odd Integer|odd]] [[Definition:Term of Sequence|term]] of $\sequen... | From [[Pell Number as Sum of Squares]], we have:
:$P_{2 n + 1} = P_{n + 1}^2 + P_n^2$
Therefore:
{{begin-eqn}}
{{eqn | l = P_{2 n + 1}^2
| r = \paren {P_{n + 1}^2 + P_n^2}^2
}}
{{eqn | r = P_{n + 1}^4 + 2 P_{n + 1}^2 P_n^2 + P_n^4
| c = [[Square of Sum]]
}}
{{eqn | r = P_{n + 1}^4 + 2 P_{n + 1}^2 P_n^2 +... | Best Rational Approximations to Root 2 generate Pythagorean Triples/Proof 2 | https://proofwiki.org/wiki/Best_Rational_Approximations_to_Root_2_generate_Pythagorean_Triples | https://proofwiki.org/wiki/Best_Rational_Approximations_to_Root_2_generate_Pythagorean_Triples/Proof_2 | [
"Square Root of 2",
"Pythagorean Triples",
"Pell Numbers"
] | [
"Sequence of Best Rational Approximations to Square Root of 2",
"Definition:Odd Integer",
"Definition:Term of Sequence"
] | [
"Pell Number as Sum of Squares",
"Square of Sum",
"Definition:Zero (Number)"
] |
proofwiki-12137 | Prime Ideal is Prime Filter in Dual Ordered Set | Let $P = \struct {S, \preceq}$ be a ordered set.
Let $X$ be a subset of $S$.
Then
:$X$ is a prime ideal of $P$
{{iff}}:
:$X$ is a prime filter of $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the dual of $P$. | === Sufficient Condition ===
Let $X$ be a prime ideal in $P$.
Then
:$X$ is an ideal of $P$.
:$S \setminus X$ is a filter of $P$.
by definition of a prime ideal.
By Ideal is Filter in Dual Ordered Set:
:$X$ is a filter of $P^{-1}$.
and by Filter is Ideal in Dual Ordered Set:
:$S \setminus X$ is an ideal of $P^{-1}$.
Hen... | Let $P = \struct {S, \preceq}$ be a [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then
:$X$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] of $P$
{{iff}}:
:$X$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] of $P^{-1}$
where $P^{-1} = \struct {S, ... | === Sufficient Condition ===
Let $X$ be a [[Definition:Prime Ideal (Order Theory)|prime ideal]] in $P$.
Then
:$X$ is an [[Definition:Ideal (Order Theory)|ideal]] of $P$.
:$S \setminus X$ is a [[Definition:Filter|filter]] of $P$.
by definition of a [[Definition:Prime Ideal (Order Theory)|prime ideal]].
By [[Ideal is ... | Prime Ideal is Prime Filter in Dual Ordered Set | https://proofwiki.org/wiki/Prime_Ideal_is_Prime_Filter_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Prime_Ideal_is_Prime_Filter_in_Dual_Ordered_Set | [
"Prime Ideals (Order Theory)",
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Prime Ideal (Order Theory)",
"Definition:Prime Filter (Order Theory)",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Definition:Prime Ideal (Order Theory)",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Prime Ideal (Order Theory)",
"Ideal is Filter in Dual Ordered Set",
"Definition:Filter",
"Filter is Ideal in Dual Ordered Set",
"Definition:Ideal (Order Theory)",
"Definition:Prime Filter (Or... |
proofwiki-12138 | Ideal is Filter in Dual Ordered Set | Let $P = \struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then
:$X$ is ideal in $P$
{{iff}}
:$X$ is filter in $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the dual of $P$. | === Sufficient Condition ===
Let $X$ be ideal in $P$.
By definition of ideal in ordered set:
:$X$ is non-empty directed lower.
By definition of directed:
:$\forall x, y \in X: \exists z \in X: x \preceq z \land y \preceq z$
Then
:$\forall x, y \in X: \exists z \in X: z \succeq x \land z \succeq y$
By definition
:$X$ is... | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then
:$X$ is [[Definition:Ideal (Order Theory)|ideal]] in $P$
{{iff}}
:$X$ is [[Definition:Filter|filter]] in $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the [[Definition:Dual Ord... | === Sufficient Condition ===
Let $X$ be [[Definition:Ideal (Order Theory)|ideal]] in $P$.
By definition of [[Definition:Ideal in Ordered Set|ideal in ordered set]]:
:$X$ is [[Definition:Non-Empty Set|non-empty]] [[Definition:Directed Subset|directed]] [[Definition:Lower Section|lower]].
By definition of [[Definition... | Ideal is Filter in Dual Ordered Set | https://proofwiki.org/wiki/Ideal_is_Filter_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Ideal_is_Filter_in_Dual_Ordered_Set | [
"Order Theory",
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Ideal (Order Theory)",
"Definition:Filter",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Definition:Ideal (Order Theory)",
"Definition:Ideal in Ordered Set",
"Definition:Non-Empty Set",
"Definition:Directed Subset",
"Definition:Lower Section",
"Definition:Directed Subset",
"Definition:Filtered Subset",
"Definition:Lower Section",
"Definition:Upper Section",
"Definition:Filter in Orde... |
proofwiki-12139 | Square Root of 2 as Sum of Egyptian Fractions | The square root of $2$ can be approximated by the following sequence of Egyptian fractions:
:$\sqrt 2 = 1 + \dfrac 1 3 + \dfrac 1 {13} + \dfrac 1 {253} + \dfrac 1 {218 \, 201} + \dfrac 1 {61 \, 323 \, 543 \, 802} + \cdots$
{{OEIS|A006487}} | We have by definition of the square root of $2$ that:
:$\sqrt 2 - 1 \approx 0 \cdotp 41421 \, 35623 \, 73095 \, 04880 \ldots$
By inspection:
:$\dfrac 1 3 < \sqrt 2 - 1 < \dfrac 1 2$
Thus:
:$\sqrt 2 - 1 - \dfrac 1 3 \approx 0 \cdotp 08088 \, 02290 \, 39761 \, 71546 \ldots$
Then:
:$\dfrac 1 {13} = 0 \cdotp 07692 \, 3 \ld... | The [[Definition:Square Root|square root]] of $2$ can be approximated by the following sequence of [[Definition:Egyptian Fraction|Egyptian fractions]]:
:$\sqrt 2 = 1 + \dfrac 1 3 + \dfrac 1 {13} + \dfrac 1 {253} + \dfrac 1 {218 \, 201} + \dfrac 1 {61 \, 323 \, 543 \, 802} + \cdots$
{{OEIS|A006487}} | We have by definition of the [[Square Root of 2|square root of $2$]] that:
:$\sqrt 2 - 1 \approx 0 \cdotp 41421 \, 35623 \, 73095 \, 04880 \ldots$
By inspection:
:$\dfrac 1 3 < \sqrt 2 - 1 < \dfrac 1 2$
Thus:
:$\sqrt 2 - 1 - \dfrac 1 3 \approx 0 \cdotp 08088 \, 02290 \, 39761 \, 71546 \ldots$
Then:
:$\dfrac 1 {13} =... | Square Root of 2 as Sum of Egyptian Fractions | https://proofwiki.org/wiki/Square_Root_of_2_as_Sum_of_Egyptian_Fractions | https://proofwiki.org/wiki/Square_Root_of_2_as_Sum_of_Egyptian_Fractions | [
"Square Root of 2"
] | [
"Definition:Square Root",
"Definition:Egyptian Fraction"
] | [
"Square Root/Examples/2",
"Definition:Integer Sequence",
"Definition:Fraction/Denominator"
] |
proofwiki-12140 | Sprague's Property of Root 2 | Let $S = \sequence {s_n}$ be the sequence of fractions defined as follows:
Let the numerator of $s_n$ be:
:$\floor {n \sqrt 2}$
where $\floor x$ denotes the floor of $x$.
Let the denominators of the terms of $S$ be the (strictly) positive integers missing from the numerators of $S$:
:$S := \dfrac 1 3, \dfrac 2 6, \dfra... | Denote the numerators of the terms of $S$ as $\sequence {N_n}$.
Denote the denominators of the terms of $S$ as $\sequence {D_n}$.
From the definition:
:$\sequence {N_n}$ is a Beatty sequence, where $\sequence {N_n} = \BB_{\sqrt 2} = \sequence{\floor{n \sqrt 2} }_{n \mathop \in \Z_{> 0} }$
:$\sequence {N_n}$ and $\seque... | Let $S = \sequence {s_n}$ be the [[Definition:Sequence|sequence]] of [[Definition:Fraction|fractions]] defined as follows:
Let the [[Definition:Numerator|numerator]] of $s_n$ be:
:$\floor {n \sqrt 2}$
where $\floor x$ denotes the [[Definition:Floor Function|floor]] of $x$.
Let the [[Definition:Denominator|denominator... | Denote the [[Definition:Numerator|numerators]] of the [[Definition:Term of Sequence|terms]] of $S$ as $\sequence {N_n}$.
Denote the [[Definition:Denominator|denominators]] of the [[Definition:Term of Sequence|terms]] of $S$ as $\sequence {D_n}$.
From the definition:
:$\sequence {N_n}$ is a [[Definition:Beatty Sequen... | Sprague's Property of Root 2 | https://proofwiki.org/wiki/Sprague's_Property_of_Root_2 | https://proofwiki.org/wiki/Sprague's_Property_of_Root_2 | [
"Square Root of 2",
"Beatty Sequences"
] | [
"Definition:Sequence",
"Definition:Fraction",
"Definition:Fraction/Numerator",
"Definition:Floor Function",
"Definition:Fraction/Denominator",
"Definition:Term of Sequence",
"Definition:Strictly Positive/Integer",
"Definition:Fraction/Numerator",
"Definition:Fraction/Numerator",
"Definition:Fracti... | [
"Definition:Fraction/Numerator",
"Definition:Term of Sequence",
"Definition:Fraction/Denominator",
"Definition:Term of Sequence",
"Definition:Beatty Sequence",
"Definition:Beatty Sequence/Complementary",
"Beatty's Theorem",
"Definition:Beatty Sequence",
"Beatty's Theorem",
"Definition:Fraction/Num... |
proofwiki-12141 | Steiner's Calculus Problem | Let $f: \R_{>0} \to \R$ be the real function defined as:
:$\forall x \in \R_{>0}: \map f x = x^{1/x}$
Then $\map f x$ reaches its maximum at $x = e$ where $e$ is Euler's number. | {{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \frac \d {\d x} x^{1/x}
}}
{{eqn | r = \frac \d {\d x} e^{\ln x / x}
}}
{{eqn | r = e^{\ln x / x} \paren {\frac 1 {x^2} - \frac {\ln x} {x^2} }
}}
{{eqn | r = \frac {x^{1/x} } {x^2} \paren {1 - \ln x}
}}
{{end-eqn}}
$\dfrac {x^{1/x} } {x^2}$ is always greater than $0$.
... | Let $f: \R_{>0} \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\forall x \in \R_{>0}: \map f x = x^{1/x}$
Then $\map f x$ reaches its maximum at $x = e$ where $e$ is [[Definition:Euler's Number|Euler's number]]. | {{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \frac \d {\d x} x^{1/x}
}}
{{eqn | r = \frac \d {\d x} e^{\ln x / x}
}}
{{eqn | r = e^{\ln x / x} \paren {\frac 1 {x^2} - \frac {\ln x} {x^2} }
}}
{{eqn | r = \frac {x^{1/x} } {x^2} \paren {1 - \ln x}
}}
{{end-eqn}}
$\dfrac {x^{1/x} } {x^2}$ is always greater than $0$.... | Steiner's Calculus Problem | https://proofwiki.org/wiki/Steiner's_Calculus_Problem | https://proofwiki.org/wiki/Steiner's_Calculus_Problem | [
"Euler's Number",
"Steiner's Problem"
] | [
"Definition:Real Function",
"Definition:Euler's Number"
] | [
"Interior Extremum Theorem"
] |
proofwiki-12142 | Ratio of Lengths of Arms of Pentagram | Consider a pentagram.
:400px
Let $AC$ be the length of one of the lines which span the pentagram and define it.
Let $B$ be one of the points where $AC$ intersects one of the other such lines such that $AB > AC$.
Then:
:$\dfrac {AC} {AB} = \phi$
where $\phi$ denotes the golden mean. | Follows directly from Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio.
{{qed}}
{{ProofWanted|Flesh this out}} | Consider a [[Definition:Pentagram|pentagram]].
:[[File:PentagramArmLengths.png|400px]]
Let $AC$ be the [[Definition:Length (Linear Measure)|length]] of one of the [[Definition:Line Segment|lines]] which span the [[Definition:Pentagram|pentagram]] and define it.
Let $B$ be one of the [[Definition:Point|points]] where... | Follows directly from [[Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio]].
{{qed}}
{{ProofWanted|Flesh this out}} | Ratio of Lengths of Arms of Pentagram | https://proofwiki.org/wiki/Ratio_of_Lengths_of_Arms_of_Pentagram | https://proofwiki.org/wiki/Ratio_of_Lengths_of_Arms_of_Pentagram | [
"Golden Mean",
"Pentagrams"
] | [
"Definition:Pentagram",
"File:PentagramArmLengths.png",
"Definition:Linear Measure/Length",
"Definition:Line/Segment",
"Definition:Pentagram",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Golden Mean"
] | [
"Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio"
] |
proofwiki-12143 | Filter is Ideal in Dual Ordered Set | Let $P = \struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then
:$X$ is filter in $P$
{{iff}}
:$X$ is ideal in $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the dual of $P$. | By Dual of Dual Ordering:
:dual of $P^{-1}$ is $P$.
Hence by Ideal is Filter in Dual Ordered Set:
:the result follows.
{{qed}} | Let $P = \struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then
:$X$ is [[Definition:Filter|filter]] in $P$
{{iff}}
:$X$ is [[Definition:Ideal (Order Theory)|ideal]] in $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the [[Definition:Dual Ord... | By [[Dual of Dual Ordering]]:
:[[Definition:Dual Ordered Set|dual]] of $P^{-1}$ is $P$.
Hence by [[Ideal is Filter in Dual Ordered Set]]:
:the result follows.
{{qed}} | Filter is Ideal in Dual Ordered Set | https://proofwiki.org/wiki/Filter_is_Ideal_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Filter_is_Ideal_in_Dual_Ordered_Set | [
"Order Theory",
"Dual Orderings"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Filter",
"Definition:Ideal (Order Theory)",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Dual of Dual Ordering",
"Definition:Dual Ordering/Dual Ordered Set",
"Ideal is Filter in Dual Ordered Set"
] |
proofwiki-12144 | Prime Filter is Prime Ideal in Dual Ordered Set | Let $P = \struct {S, \preceq}$ be a lattice.
Let $X$ be a subset of $S$.
Then
:$X$ is a prime filter of $P$
{{iff}}:
:$X$ is a prime ideal of $P^{-1}$
where $P^{-1} = \struct {S, \succeq}$ denotes the dual of $P$. | By Dual of Dual Ordering:
:dual of $P^{-1}$ is $P$.
Hence by Prime Ideal is Prime Filter in Dual Ordered Set:
:the result follows.
{{qed}} | Let $P = \struct {S, \preceq}$ be a [[Definition:Lattice (Order Theory)|lattice]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then
:$X$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] of $P$
{{iff}}:
:$X$ is a [[Definition:Prime Ideal (Order Theory)|prime ideal]] of $P^{-1}$
where $P^{-1} = \stru... | By [[Dual of Dual Ordering]]:
:[[Definition:Dual Ordered Set|dual]] of $P^{-1}$ is $P$.
Hence by [[Prime Ideal is Prime Filter in Dual Ordered Set]]:
:the result follows.
{{qed}} | Prime Filter is Prime Ideal in Dual Ordered Set | https://proofwiki.org/wiki/Prime_Filter_is_Prime_Ideal_in_Dual_Ordered_Set | https://proofwiki.org/wiki/Prime_Filter_is_Prime_Ideal_in_Dual_Ordered_Set | [
"Prime Ideals (Order Theory)",
"Dual Orderings"
] | [
"Definition:Lattice (Order Theory)",
"Definition:Subset",
"Definition:Prime Filter (Order Theory)",
"Definition:Prime Ideal (Order Theory)",
"Definition:Dual Ordering/Dual Ordered Set"
] | [
"Dual of Dual Ordering",
"Definition:Dual Ordering/Dual Ordered Set",
"Prime Ideal is Prime Filter in Dual Ordered Set"
] |
proofwiki-12145 | Continued Fraction Expansion of Golden Mean/Rate of Convergence | This continued fraction expansion has the slowest rate of convergence of all simple infinite continued fractions. | {{ProofWanted|"Rate of convergence" of a CFE needs to be formally defined}} | This [[Definition:Continued Fraction Expansion of Irrational Number|continued fraction expansion]] has the slowest rate of convergence of all [[Definition:Simple Infinite Continued Fraction|simple infinite continued fractions]]. | {{ProofWanted|"Rate of convergence" of a CFE needs to be formally defined}} | Continued Fraction Expansion of Golden Mean/Rate of Convergence | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean/Rate_of_Convergence | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean/Rate_of_Convergence | [
"Continued Fraction Expansion of Golden Mean"
] | [
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Simple Continued Fraction/Infinite"
] | [] |
proofwiki-12146 | Continued Fraction Expansion of Golden Mean/Successive Convergents | The $n$th convergent is given by:
:$C_n = \dfrac {F_{n + 1} } {F_n}$
where $F_n$ denotes the $n$th Fibonacci number. | The proof proceeds by induction.
Listing the first few convergents, which can be calculated:
:$C_1 = \dfrac 1 1$
:$C_2 = \dfrac 2 1$
:$C_3 = \dfrac 3 2$
:$C_4 = \dfrac 5 3$
and so on.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$C_n = \dfrac {F_{n + 1} } {F_n}$
$\map P 1$ is the case:
:$C_1 = \dfrac {... | The $n$th [[Definition:Convergent of Continued Fraction|convergent]] is given by:
:$C_n = \dfrac {F_{n + 1} } {F_n}$
where $F_n$ denotes the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]]. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
Listing the first few [[Definition:Convergent of Continued Fraction|convergents]], which can be calculated:
:$C_1 = \dfrac 1 1$
:$C_2 = \dfrac 2 1$
:$C_3 = \dfrac 3 2$
:$C_4 = \dfrac 5 3$
and so on.
For all $n \in \Z_{> 0}$, let $\map P n$ be... | Continued Fraction Expansion of Golden Mean/Successive Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean/Successive_Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Golden_Mean/Successive_Convergents | [
"Continued Fraction Expansion of Golden Mean"
] | [
"Definition:Convergent of Continued Fraction",
"Definition:Fibonacci Number"
] | [
"Principle of Mathematical Induction",
"Definition:Convergent of Continued Fraction",
"Definition:Proposition",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-12147 | Continued Fraction Expansion of Pi/Convergents | The convergents of the continued fraction expansion to $\pi$ (pi) are:
:$3, \dfrac {22} 7, \dfrac {333} {106}, \dfrac {355} {113}, \dfrac {103993} {33102}, \dfrac {104348} {33215}$
{{OEIS-Numerators|A002485}}
{{OEIS-Denominators|A002486}}
These best rational approximations are accurate to $0, 2, 4, 6, 9, 9, 9, 10, 11, ... | {{ProofWanted|Calculation needed}} | The [[Definition:Convergent of Continued Fraction|convergents]] of the [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]] to [[Definition:Pi|$\pi$ (pi)]] are:
:$3, \dfrac {22} 7, \dfrac {333} {106}, \dfrac {355} {113}, \dfrac {103993} {33102}, \dfrac {104348} {33215}$
{{OEIS-Numerat... | {{ProofWanted|Calculation needed}} | Continued Fraction Expansion of Pi/Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Pi/Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Pi/Convergents | [
"Pi",
"Examples of Continued Fractions"
] | [
"Definition:Convergent of Continued Fraction",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Pi",
"Definition:Best Rational Approximation",
"Definition:Decimal Notation",
"Definition:Zu Chongzhi Fraction"
] | [] |
proofwiki-12148 | Filter is Prime iff For Every Element Element either Negation Belongs to Filter in Boolean Lattice | Let $B = \struct {S, \vee, \wedge, \neg, \preceq}$ be a Boolean lattice.
Let $F$ be a filter in $B$.
Then
:$F$ is prime
{{iff}}
:$\forall x \in S: x \in F \lor \paren {\neg x} \in F$ | === Sufficient Condition ===
Let $F$ be prime.
Let $x \in S$.
By definition of Boolean lattice:
:$x \vee \neg x = \top$
where $\top$ denotes the top of $B$.
By definition of non-empty set:
:$\exists y: y \in F$
By definition of greatest element:
:$y \preceq \top$
By definition of upper section:
:$\top \in F$
Thus by de... | Let $B = \struct {S, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $F$ be a [[Definition:Filter|filter]] in $B$.
Then
:$F$ is [[Definition:Prime Filter (Order Theory)|prime]]
{{iff}}
:$\forall x \in S: x \in F \lor \paren {\neg x} \in F$ | === Sufficient Condition ===
Let $F$ be [[Definition:Prime Filter (Order Theory)|prime]].
Let $x \in S$.
By definition of [[Definition:Boolean Lattice|Boolean lattice]]:
:$x \vee \neg x = \top$
where $\top$ denotes the [[Definition:Top of Lattice|top]] of $B$.
By definition of [[Definition:Non-Empty Set|non-empty s... | Filter is Prime iff For Every Element Element either Negation Belongs to Filter in Boolean Lattice | https://proofwiki.org/wiki/Filter_is_Prime_iff_For_Every_Element_Element_either_Negation_Belongs_to_Filter_in_Boolean_Lattice | https://proofwiki.org/wiki/Filter_is_Prime_iff_For_Every_Element_Element_either_Negation_Belongs_to_Filter_in_Boolean_Lattice | [
"Prime Ideals (Order Theory)",
"Boolean Algebras"
] | [
"Definition:Boolean Lattice",
"Definition:Filter",
"Definition:Prime Filter (Order Theory)"
] | [
"Definition:Prime Filter (Order Theory)",
"Definition:Boolean Lattice",
"Definition:Top of Lattice",
"Definition:Non-Empty Set",
"Definition:Greatest Element",
"Definition:Upper Section",
"Definition:Prime Filter (Order Theory)",
"Definition:Boolean Lattice",
"Definition:Upper Section"
] |
proofwiki-12149 | Beatty's Theorem | Let $r, s \in \R \setminus \Q$ be an irrational number such that $r > 1$ and $s > 1$.
Let $\BB_r$ and $\BB_s$ be the Beatty sequences on $r$ and $s$ respectively.
Then $\BB_r$ and $\BB_s$ are complementary Beatty sequences {{iff}}:
:$\dfrac 1 r + \dfrac 1 s = 1$ | We have been given that $r > 1$.
Let $\dfrac 1 r + \dfrac 1 s = 1$.
Then:
:$s = \dfrac r {r - 1}$
It is to be shown that every positive integer lies in exactly one of the two Beatty sequences $\BB_r$ and $\BB_s$.
Consider the ordinal positions occupied by all the fractions $\dfrac j r$ and $\dfrac k s$ when they are jo... | Let $r, s \in \R \setminus \Q$ be an [[Definition:Irrational Number|irrational number]] such that $r > 1$ and $s > 1$.
Let $\BB_r$ and $\BB_s$ be the [[Definition:Beatty Sequence|Beatty sequences]] on $r$ and $s$ respectively.
Then $\BB_r$ and $\BB_s$ are [[Definition:Complementary Beatty Sequence|complementary Beat... | We have been given that $r > 1$.
Let $\dfrac 1 r + \dfrac 1 s = 1$.
Then:
:$s = \dfrac r {r - 1}$
It is to be shown that every [[Definition:Positive Integer|positive integer]] lies in exactly one of the two [[Definition:Beatty Sequence|Beatty sequences]] $\BB_r$ and $\BB_s$.
Consider the [[Definition:Ordinal Number... | Beatty's Theorem/Proof 1 | https://proofwiki.org/wiki/Beatty's_Theorem | https://proofwiki.org/wiki/Beatty's_Theorem/Proof_1 | [
"Beatty Sequences"
] | [
"Definition:Irrational Number",
"Definition:Beatty Sequence",
"Definition:Beatty Sequence/Complementary"
] | [
"Definition:Positive/Integer",
"Definition:Beatty Sequence",
"Definition:Ordinal",
"Definition:Fraction",
"Definition:Positive/Integer",
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Positive/Integer",
"Definition:Real Number",
"Definition:Positive/Integer",
"Definition... |
proofwiki-12150 | Beatty's Theorem | Let $r, s \in \R \setminus \Q$ be an irrational number such that $r > 1$ and $s > 1$.
Let $\BB_r$ and $\BB_s$ be the Beatty sequences on $r$ and $s$ respectively.
Then $\BB_r$ and $\BB_s$ are complementary Beatty sequences {{iff}}:
:$\dfrac 1 r + \dfrac 1 s = 1$ | === Collisions ===
{{AimForCont}} there exist integers $j > 0, k, m$ such that:
:$j = \floor {k \cdot r} = \floor {m \cdot s}$
This is equivalent to the inequalities:
:$j \le k \cdot r < j + 1$
and:
:$j \le m \cdot s < j + 1$
As $r$ and $s$ are irrational, equality cannot happen.
So:
:$j < k \cdot r < j + 1$
and:
:$j <... | Let $r, s \in \R \setminus \Q$ be an [[Definition:Irrational Number|irrational number]] such that $r > 1$ and $s > 1$.
Let $\BB_r$ and $\BB_s$ be the [[Definition:Beatty Sequence|Beatty sequences]] on $r$ and $s$ respectively.
Then $\BB_r$ and $\BB_s$ are [[Definition:Complementary Beatty Sequence|complementary Beat... | === Collisions ===
{{AimForCont}} there exist [[Definition:Integer|integers]] $j > 0, k, m$ such that:
:$j = \floor {k \cdot r} = \floor {m \cdot s}$
This is equivalent to the inequalities:
:$j \le k \cdot r < j + 1$
and:
:$j \le m \cdot s < j + 1$
As $r$ and $s$ are [[Definition:Irrational Number|irrational]], equa... | Beatty's Theorem/Proof 2 | https://proofwiki.org/wiki/Beatty's_Theorem | https://proofwiki.org/wiki/Beatty's_Theorem/Proof_2 | [
"Beatty Sequences"
] | [
"Definition:Irrational Number",
"Definition:Beatty Sequence",
"Definition:Beatty Sequence/Complementary"
] | [
"Definition:Integer",
"Definition:Irrational Number",
"Definition:By Hypothesis",
"Definition:Integer",
"Definition:Integer",
"Definition:Integer",
"Definition:Irrational Number"
] |
proofwiki-12151 | Proper and Prime iff Ultrafilter in Boolean Lattice | Let $B = \struct {S, \vee, \wedge, \neg, \preceq}$ be a Boolean lattice.
Let $F$ be a filter in $B$.
Then
:$F$ is a proper subset of $S$ and $F$ is a prime filter in $B$
{{iff}}:
:$F$ is ultrafilter on $B$ | === Sufficient Condition ===
Let us assume
:$F$ is a proper subset of $S$ and $F$ is a prime filter in $B$.
Thus
:$F$ is a proper subset of $S$.
Let $G$ be a filter in $B$ such that
:$F \subseteq G$ and $F \ne G$.
By definitions of subset and set equality:
:$\exists x: x \in G \land x \notin F$
By definition of Boolean... | Let $B = \struct {S, \vee, \wedge, \neg, \preceq}$ be a [[Definition:Boolean Lattice|Boolean lattice]].
Let $F$ be a [[Definition:Filter|filter]] in $B$.
Then
:$F$ is a [[Definition:Proper Subset|proper subset]] of $S$ and $F$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] in $B$
{{iff}}:
:$F$ is [[Defi... | === Sufficient Condition ===
Let us assume
:$F$ is a [[Definition:Proper Subset|proper subset]] of $S$ and $F$ is a [[Definition:Prime Filter (Order Theory)|prime filter]] in $B$.
Thus
:$F$ is a [[Definition:Proper Subset|proper subset]] of $S$.
Let $G$ be a [[Definition:Filter|filter]] in $B$ such that
:$F \subsete... | Proper and Prime iff Ultrafilter in Boolean Lattice | https://proofwiki.org/wiki/Proper_and_Prime_iff_Ultrafilter_in_Boolean_Lattice | https://proofwiki.org/wiki/Proper_and_Prime_iff_Ultrafilter_in_Boolean_Lattice | [
"Prime Ideals (Order Theory)",
"Boolean Lattices"
] | [
"Definition:Boolean Lattice",
"Definition:Filter",
"Definition:Proper Subset",
"Definition:Prime Filter (Order Theory)",
"Definition:Ultrafilter (Order Theory)"
] | [
"Definition:Proper Subset",
"Definition:Prime Filter (Order Theory)",
"Definition:Proper Subset",
"Definition:Filter",
"Definition:Subset",
"Definition:Set Equality",
"Definition:Boolean Algebra",
"Definition:Top of Lattice",
"Top in Filter",
"Definition:Prime Filter (Order Theory)",
"Definition... |
proofwiki-12152 | Equivalence of Definitions of Upper Wythoff Sequence | The following definitions of the upper Wythoff sequence are equivalent: | From Beatty's Theorem, the Beatty sequences $\BB_r$ and $\BB_s$ are complementary {{iff}}:
:$\dfrac 1 r + \dfrac 1 s = 1$
It remains to be demonstrated that this holds for $r = \phi$ and $s = \phi^2$.
Thus:
{{begin-eqn}}
{{eqn | l = \dfrac 1 \phi + \dfrac 1 {\phi^2}
| r = \dfrac {\phi + 1} {\phi^2}
| c =
}... | The following definitions of the [[Definition:Upper Wythoff Sequence|upper Wythoff sequence]] are [[Definition:Logical Equivalence|equivalent]]: | From [[Beatty's Theorem]], the [[Definition:Beatty Sequence|Beatty sequences]] $\BB_r$ and $\BB_s$ are [[Definition:Complementary Beatty Sequence|complementary]] {{iff}}:
:$\dfrac 1 r + \dfrac 1 s = 1$
It remains to be demonstrated that this holds for $r = \phi$ and $s = \phi^2$.
Thus:
{{begin-eqn}}
{{eqn | l = \df... | Equivalence of Definitions of Upper Wythoff Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Upper_Wythoff_Sequence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Upper_Wythoff_Sequence | [
"Beatty Sequences"
] | [
"Definition:Upper Wythoff Sequence",
"Definition:Logical Equivalence"
] | [
"Beatty's Theorem",
"Definition:Beatty Sequence",
"Definition:Beatty Sequence/Complementary",
"Square of Golden Mean equals One plus Golden Mean",
"Category:Beatty Sequences"
] |
proofwiki-12153 | Difference between Terms of Wythoff Pair | Let $\tuple {\floor {k \phi}, \floor {k \phi^2} }$ be a Wythoff pair.
The difference between the coordinates of this Wythoff pair is $k$.
That is:
:$\floor {k \phi^2} - \floor {k \phi} = k$ | {{begin-eqn}}
{{eqn | l = \floor {k \phi^2}
| r = \floor {k \paren {1 + \phi} }
| c = Square of Golden Mean equals One plus Golden Mean
}}
{{eqn | r = \floor {k + k \phi}
| c =
}}
{{eqn | r = k + \floor {k \phi}
| c =
}}
{{end-eqn}}
{{qed}} | Let $\tuple {\floor {k \phi}, \floor {k \phi^2} }$ be a [[Definition:Wythoff Pair|Wythoff pair]].
The [[Definition:Integer Subtraction|difference]] between the [[Definition:Coordinate of Ordered Tuple|coordinates]] of this [[Definition:Wythoff Pair|Wythoff pair]] is $k$.
That is:
:$\floor {k \phi^2} - \floor {k \phi... | {{begin-eqn}}
{{eqn | l = \floor {k \phi^2}
| r = \floor {k \paren {1 + \phi} }
| c = [[Square of Golden Mean equals One plus Golden Mean]]
}}
{{eqn | r = \floor {k + k \phi}
| c =
}}
{{eqn | r = k + \floor {k \phi}
| c =
}}
{{end-eqn}}
{{qed}} | Difference between Terms of Wythoff Pair | https://proofwiki.org/wiki/Difference_between_Terms_of_Wythoff_Pair | https://proofwiki.org/wiki/Difference_between_Terms_of_Wythoff_Pair | [
"Beatty Sequences"
] | [
"Definition:Wythoff Pair",
"Definition:Subtraction/Integers",
"Definition:Cartesian Product/Coordinate",
"Definition:Wythoff Pair"
] | [
"Square of Golden Mean equals One plus Golden Mean"
] |
proofwiki-12154 | Finite Infima Set and Upper Closure is Smallest Filter | Let $L = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $X$ be a non-empty subset of $S$.
Then
:$X \subseteq \map {\operatorname {fininfs} } X^\succeq$ and
:for every a filter $F$ in $L$: $\paren {X \subseteq F \implies \map {\operatorname {fininfs} } X^\succeq \subseteq F}$
where
:$\map {\operatorname {finin... | By Set is Subset of Finite Infima Set:
:$X \subseteq \map {\operatorname {fininfs} } X$
By Upper Closure of Subset is Subset of Upper Closure:
:$X^\succeq \subseteq \map {\operatorname {fininfs} } X^\succeq$
By Set is Subset of Upper Closure:
:$X \subseteq X^\succeq$
Thus by Subset Relation is Transitive:
:$X \subseteq... | Let $L = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then
:$X \subseteq \map {\operatorname {fininfs} } X^\succeq$ and
:for every a [[Definition:Filter|filter]] $F$ in $L$: $\paren {X \su... | By [[Set is Subset of Finite Infima Set]]:
:$X \subseteq \map {\operatorname {fininfs} } X$
By [[Upper Closure of Subset is Subset of Upper Closure]]:
:$X^\succeq \subseteq \map {\operatorname {fininfs} } X^\succeq$
By [[Set is Subset of Upper Closure]]:
:$X \subseteq X^\succeq$
Thus by [[Subset Relation is Transiti... | Finite Infima Set and Upper Closure is Smallest Filter | https://proofwiki.org/wiki/Finite_Infima_Set_and_Upper_Closure_is_Smallest_Filter | https://proofwiki.org/wiki/Finite_Infima_Set_and_Upper_Closure_is_Smallest_Filter | [
"Order Theory"
] | [
"Definition:Meet Semilattice",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Filter",
"Definition:Finite Infima Set",
"Definition:Upper Closure/Set"
] | [
"Set is Subset of Finite Infima Set",
"Upper Closure of Subset is Subset of Upper Closure",
"Set is Subset of Upper Closure",
"Subset Relation is Transitive",
"Definition:Filter",
"Definition:Upper Closure/Set",
"Definition:Finite Infima Set",
"Definition:Infimum of Set",
"Definition:Set of Sets",
... |
proofwiki-12155 | Smaller Number of Wythoff Pair is Smallest Number not yet in Sequence | Consider the sequence of Wythoff pairs arranged in sequential order:
:$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$
The first coordinate of each Wythoff pair is the smallest positive integer which has not yet appeared in the sequence. | From Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each, every positive integer can be found in the sequence of Wythoff pairs.
From Difference between Terms of Wythoff Pair, the first coordinate is the smaller of the coordinates of the Wythoff pair.
So consider a given Wythoff pair.
Let $p$ be t... | Consider the [[Definition:Sequence|sequence]] of [[Definition:Wythoff Pair|Wythoff pairs]] arranged in sequential order:
:$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$
The [[Definition:Coordinate of Ordered Tuple|first coordinate]] of each [[Definition:Wythoff Pa... | From [[Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each]], every [[Definition:Positive Integer|positive integer]] can be found in the [[Definition:Sequence|sequence]] of [[Definition:Wythoff Pair|Wythoff pairs]].
From [[Difference between Terms of Wythoff Pair]], the [[Definition:Coordinate o... | Smaller Number of Wythoff Pair is Smallest Number not yet in Sequence | https://proofwiki.org/wiki/Smaller_Number_of_Wythoff_Pair_is_Smallest_Number_not_yet_in_Sequence | https://proofwiki.org/wiki/Smaller_Number_of_Wythoff_Pair_is_Smallest_Number_not_yet_in_Sequence | [
"Beatty Sequences"
] | [
"Definition:Sequence",
"Definition:Wythoff Pair",
"Definition:Cartesian Product/Coordinate",
"Definition:Wythoff Pair",
"Definition:Positive/Integer",
"Definition:Sequence"
] | [
"Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each",
"Definition:Positive/Integer",
"Definition:Sequence",
"Definition:Wythoff Pair",
"Difference between Terms of Wythoff Pair",
"Definition:Cartesian Product/Coordinate",
"Definition:Cartesian Product/Coordinate",
"Definition:W... |
proofwiki-12156 | Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each | Consider the sequence of Wythoff pairs arranged in sequential order:
:$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$
Apart from the first Wythoff pair $\tuple {0, 0}$, every positive integer appears in this sequence exactly once. | By definition, the $n$th Wythoff pair is $\tuple {\floor {n \phi}, \floor {n \phi^2} }$.
Thus the coordinates of the sequence of Wythoff pairs are the terms of the lower and upper Wythoff sequences.
By definition:
:the lower Wythoff sequence is the Beatty sequence $\BB_x$ on the golden section $\phi$.
:the upper Wythof... | Consider the [[Definition:Sequence|sequence]] of [[Definition:Wythoff Pair|Wythoff pairs]] arranged in sequential order:
:$\tuple {0, 0}, \tuple {1, 2}, \tuple {3, 5}, \tuple {4, 7}, \tuple {6, 10}, \tuple {8, 13}, \ldots$
Apart from the first [[Definition:Wythoff Pair|Wythoff pair]] $\tuple {0, 0}$, every [[Definiti... | By definition, the $n$th [[Definition:Wythoff Pair|Wythoff pair]] is $\tuple {\floor {n \phi}, \floor {n \phi^2} }$.
Thus the [[Definition:Coordinate of Ordered Tuple|coordinates]] of the [[Definition:Sequence|sequence]] of [[Definition:Wythoff Pair|Wythoff pairs]] are the [[Definition:Term of Sequence|terms]] of the ... | Sequence of Wythoff Pairs contains all Positive Integers exactly Once Each | https://proofwiki.org/wiki/Sequence_of_Wythoff_Pairs_contains_all_Positive_Integers_exactly_Once_Each | https://proofwiki.org/wiki/Sequence_of_Wythoff_Pairs_contains_all_Positive_Integers_exactly_Once_Each | [
"Beatty Sequences"
] | [
"Definition:Sequence",
"Definition:Wythoff Pair",
"Definition:Wythoff Pair",
"Definition:Positive/Integer",
"Definition:Sequence"
] | [
"Definition:Wythoff Pair",
"Definition:Cartesian Product/Coordinate",
"Definition:Sequence",
"Definition:Wythoff Pair",
"Definition:Term of Sequence",
"Definition:Lower Wythoff Sequence",
"Definition:Upper Wythoff Sequence",
"Definition:Lower Wythoff Sequence",
"Definition:Beatty Sequence",
"Defin... |
proofwiki-12157 | Zeta of 2 as Product of Fractions with Prime Numerators | {{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }
| c =
}}
{{eqn | r = \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {... | From Sum of Reciprocals of Powers as Euler Product:
:$\ds \map \zeta z = \prod_p \frac 1 {1 - p^{-z} }$
where $p$ ranges over the prime numbers.
Thus:
{{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \prod_p \frac 1 {1 - p^{-2} }
| c =
}}
{{eqn | r = \prod_p \frac {p^2} {p^2 - 1}
| c = multiplying top an... | {{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }
| c =
}}
{{eqn | r = \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {... | From [[Sum of Reciprocals of Powers as Euler Product]]:
:$\ds \map \zeta z = \prod_p \frac 1 {1 - p^{-z} }$
where $p$ ranges over the [[Definition:Prime Number|prime numbers]].
Thus:
{{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \prod_p \frac 1 {1 - p^{-2} }
| c =
}}
{{eqn | r = \prod_p \frac {p^2} {p^2 -... | Zeta of 2 as Product of Fractions with Prime Numerators | https://proofwiki.org/wiki/Zeta_of_2_as_Product_of_Fractions_with_Prime_Numerators | https://proofwiki.org/wiki/Zeta_of_2_as_Product_of_Fractions_with_Prime_Numerators | [
"Riemann Zeta Function at Even Integers"
] | [
"Definition:Riemann Zeta Function",
"Definition:Continued Product",
"Definition:Prime Number"
] | [
"Sum of Reciprocals of Powers as Euler Product",
"Definition:Prime Number",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares"
] |
proofwiki-12158 | Top in Filter | Let $\struct {S, \preceq}$ be a bounded above ordered set.
Let $F$ be a filter on $S$.
Then $\top \in F$
where $\top$ denotes the greatest element of $S$. | By definition of filter in ordered set:
:$F$ is non-empty and upper.
By definition of non-empty set:
:$\exists x: x \in F$
By definition of greatest element:
:$x \preceq \top$
Thus by definition of upper section:
:$\top \in F$
{{qed}} | Let $\struct {S, \preceq}$ be a [[Definition:Bounded Above Set|bounded above]] [[Definition:Ordered Set|ordered set]].
Let $F$ be a [[Definition:Filter|filter]] on $S$.
Then $\top \in F$
where $\top$ denotes the [[Definition:Greatest Element|greatest element]] of $S$. | By definition of [[Definition:Filter in Ordered Set|filter in ordered set]]:
:$F$ is [[Definition:Non-Empty Set|non-empty]] and [[Definition:Upper Section|upper]].
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\exists x: x \in F$
By definition of [[Definition:Greatest Element|greatest element]]:
:$x ... | Top in Filter | https://proofwiki.org/wiki/Top_in_Filter | https://proofwiki.org/wiki/Top_in_Filter | [
"Order Theory"
] | [
"Definition:Bounded Above Set",
"Definition:Ordered Set",
"Definition:Filter",
"Definition:Greatest Element"
] | [
"Definition:Filter in Ordered Set",
"Definition:Non-Empty Set",
"Definition:Upper Section",
"Definition:Non-Empty Set",
"Definition:Greatest Element",
"Definition:Upper Section"
] |
proofwiki-12159 | Finite Infima Set and Upper Closure is Filter | Let $P = \struct {S, \wedge, \preceq}$ be a meet semilattice.
Let $X$ be a non-empty subset of $S$.
Then
:$\map {\operatorname {fininfs} } X^\succeq$ is filter in $P$.
where
:$\map {\operatorname {fininfs} } X$ denotes the finite infima set of $X$,
:$X^\succeq$ denotes the upper closure of $X$. | By Finite Infima Set and Upper Closure is Smallest Filter:
:$X \subseteq \map {\operatorname {fininfs} } X^\succeq$
By definition of non-empty set:
:$\map {\operatorname {fininfs} } X^\succeq$ is a non-empty set.
We will prove that
:$\map {\operatorname {fininfs} } X$ is filtered.
Let $x, y \in \map {\operatorname {fin... | Let $P = \struct {S, \wedge, \preceq}$ be a [[Definition:Meet Semilattice|meet semilattice]].
Let $X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then
:$\map {\operatorname {fininfs} } X^\succeq$ is [[Definition:Filter|filter]] in $P$.
where
:$\map {\operatorname {fininfs} } X$ d... | By [[Finite Infima Set and Upper Closure is Smallest Filter]]:
:$X \subseteq \map {\operatorname {fininfs} } X^\succeq$
By definition of [[Definition:Non-Empty Set|non-empty set]]:
:$\map {\operatorname {fininfs} } X^\succeq$ is a [[Definition:Non-Empty Set|non-empty set]].
We will prove that
:$\map {\operatorname {f... | Finite Infima Set and Upper Closure is Filter | https://proofwiki.org/wiki/Finite_Infima_Set_and_Upper_Closure_is_Filter | https://proofwiki.org/wiki/Finite_Infima_Set_and_Upper_Closure_is_Filter | [
"Order Theory"
] | [
"Definition:Meet Semilattice",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Filter",
"Definition:Finite Infima Set",
"Definition:Upper Closure/Set"
] | [
"Finite Infima Set and Upper Closure is Smallest Filter",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Filtered Subset",
"Definition:Finite Infima Set",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Set of Sets",
"Definition:Finite Set",
"Definition:S... |
proofwiki-12160 | Divisibility by 2 | An integer $N$ expressed in decimal notation is divisible by $2$ {{iff}} the {{LSD}} of $N$ is divisible by $2$.
That is:
:$N = [a_n \ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $2$
{{iff}}:
:$a_0$ is divisible by $2$. | Let $N$ be divisible by $2$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 2
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 2
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{k \mathop = 1}^n a_k 10^{k - 1}
| o ... | An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $2$ {{iff}} the {{LSD}} of $N$ is [[Definition:Divisor of Integer|divisible]] by $2$.
That is:
:$N = [a_n \ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + ... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $2$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 2
}}
{{eqn | ll= \leadstoandfrom
| l = \sum_{k \mathop = 0}^n a_k 10^k
| o = \equiv
| r = 0 \pmod 2
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + 10 \sum_{k \math... | Divisibility by 2 | https://proofwiki.org/wiki/Divisibility_by_2 | https://proofwiki.org/wiki/Divisibility_by_2 | [
"Divisibility Tests",
"2"
] | [
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12161 | Divisibility by Power of 2 | Let $r \in \Z_{\ge 1}$ be a strictly positive integer.
An integer $N$ expressed in decimal notation is divisible by $2^r$ {{iff}} the last $r$ digits of $N$ form an integer divisible by $2^r$.
That is:
:$N = [a_n \ldots a_2 a_1 a_0]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $2^r$
{{iff}}:
:$a... | First note that:
:$10^r = 2^r 5^r$
and so:
:$2^r \divides 10^r$
where $\divides$ denotes divisibility.
Thus:
:$\forall s \in \Z: s \ge r: 2^r \divides 10^s$
but:
:$\forall s \in \Z: s < r: 2^r \nmid 10^s$
Thus let $N$ be divisible by $2^r$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod {2^r}
... | Let $r \in \Z_{\ge 1}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
An [[Definition:Integer|integer]] $N$ expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $2^r$ {{iff}} the last $r$ [[Definition:Digit|digits]] of $N$ form an [[D... | First note that:
:$10^r = 2^r 5^r$
and so:
:$2^r \divides 10^r$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Thus:
:$\forall s \in \Z: s \ge r: 2^r \divides 10^s$
but:
:$\forall s \in \Z: s < r: 2^r \nmid 10^s$
Thus let $N$ be [[Definition:Divisor of Integer|divisible]] by $2^r$.
Then:
... | Divisibility by Power of 2 | https://proofwiki.org/wiki/Divisibility_by_Power_of_2 | https://proofwiki.org/wiki/Divisibility_by_Power_of_2 | [
"Divisibility Tests",
"2"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer",
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-12162 | Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2 | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then $n$ can be expressed as the sum of $2$ or more consecutive (strictly) positive integers {{iff}} $n$ is not a power of $2$. | === Necessary Condition ===
Let $a$ be the smallest of $m$ consecutive (strictly) positive integers, where $m \ge 2$.
From Sum of Arithmetic Sequence, their sum is $\dfrac {m \paren {2 a + m - 1} } 2$.
{{AimForCont}} $\dfrac {m \paren {2 a + m - 1} } 2$ is a power of $2$.
Then $m$ and $2 a + m - 1$ must also be powers... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then $n$ can be expressed as the [[Definition:Integer Addition|sum]] of $2$ or more consecutive [[Definition:Strictly Positive Integer|(strictly) positive integers]] {{iff}} $n$ is not a [[Definition:Integer Power|power]] of... | === Necessary Condition ===
Let $a$ be the smallest of $m$ consecutive [[Definition:Strictly Positive Integer|(strictly) positive integers]], where $m \ge 2$.
From [[Sum of Arithmetic Sequence]], their [[Definition:Integer Addition|sum]] is $\dfrac {m \paren {2 a + m - 1} } 2$.
{{AimForCont}} $\dfrac {m \paren {2 ... | Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2 | https://proofwiki.org/wiki/Positive_Integer_is_Sum_of_Consecutive_Positive_Integers_iff_not_Power_of_2 | https://proofwiki.org/wiki/Positive_Integer_is_Sum_of_Consecutive_Positive_Integers_iff_not_Power_of_2 | [
"Number Theory",
"2",
"Positive Integer is Sum of Consecutive Positive Integers iff not Power of 2"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Addition/Integers",
"Definition:Strictly Positive/Integer",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Strictly Positive/Integer",
"Sum of Arithmetic Sequence",
"Definition:Addition/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Power (Algebra)/Integer",
"Definition:Contradiction",
"... |
proofwiki-12163 | Equivalence of Definitions of Deficient Number | The following definitions of a deficient number are equivalent: | By definition of abundance:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
:$\map {\sigma_1} n$ is the sum of all the divisors of $n$.
Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.
Hence the result.
{{qed}}
Category:Deficient Numbers
6yqynr752q7u4eoehiw1j4hp7sgqe6f | The following definitions of a [[Definition:Deficient Number|deficient number]] are [[Definition:Logical Equivalence|equivalent]]: | By definition of [[Definition:Abundance|abundance]]:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of [[Definition:Divisor Sum Function|divisor sum function]]:
:$\map {\sigma_1} n$ is the [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of $n$.
Thus $\map {\sigma_1} n -... | Equivalence of Definitions of Deficient Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Deficient_Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Deficient_Number | [
"Deficient Numbers"
] | [
"Definition:Deficient Number",
"Definition:Logical Equivalence"
] | [
"Definition:Abundance",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Aliquot Sum",
"Category:Deficient Numbers"
] |
proofwiki-12164 | Equivalence of Definitions of Abundant Number | The following definitions of a abundant number are equivalent: | By definition of abundance:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
:$\map {\sigma_1} n$ is the sum of all the divisors of $n$.
Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.
The result follows.
{{qed}}
Category:Abundant Numbers
c39jf7s0hwcrkg4tdw3re6eyrwnctgd | The following definitions of a [[Definition:Abundant Number|abundant number]] are [[Definition:Logical Equivalence|equivalent]]: | By definition of [[Definition:Abundance|abundance]]:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of [[Definition:Divisor Sum Function|divisor sum function]]:
:$\map {\sigma_1} n$ is the [[Definition:Integer Addition|sum]] of all the [[Definition:Divisor of Integer|divisors]] of $n$.
Thus $\map {\sigma_1} n -... | Equivalence of Definitions of Abundant Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Abundant_Number | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Abundant_Number | [
"Abundant Numbers"
] | [
"Definition:Abundant Number",
"Definition:Logical Equivalence"
] | [
"Definition:Abundance",
"Definition:Divisor Sum Function",
"Definition:Addition/Integers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Aliquot Sum",
"Category:Abundant Numbers"
] |
proofwiki-12165 | Power of Prime is Deficient | Let $n \in \Z_{>0}$ be a power of a prime number $p$:
:$n = p^k$
for some $k \in \Z_{>0}$.
Then $n$ is deficient. | From Divisor Sum of Power of Prime:
:$\map {\sigma_1} n = \dfrac {p^{k + 1} - 1} {p - 1}$
Thus:
{{begin-eqn}}
{{eqn | l = \map A n
| r = \map {\sigma_1} n - 2 n
| c = {{Defof|Abundance}}
}}
{{eqn | r = \dfrac {p^{k + 1} - 1} {p - 1} - 2 p^k
| c =
}}
{{eqn | r = \dfrac {p^{k + 1} - 1 - \paren {2 p^{k ... | Let $n \in \Z_{>0}$ be a [[Definition:Integer Power|power]] of a [[Definition:Prime Number|prime number]] $p$:
:$n = p^k$
for some $k \in \Z_{>0}$.
Then $n$ is [[Definition:Deficient Number|deficient]]. | From [[Divisor Sum of Power of Prime]]:
:$\map {\sigma_1} n = \dfrac {p^{k + 1} - 1} {p - 1}$
Thus:
{{begin-eqn}}
{{eqn | l = \map A n
| r = \map {\sigma_1} n - 2 n
| c = {{Defof|Abundance}}
}}
{{eqn | r = \dfrac {p^{k + 1} - 1} {p - 1} - 2 p^k
| c =
}}
{{eqn | r = \dfrac {p^{k + 1} - 1 - \paren {2 ... | Power of Prime is Deficient | https://proofwiki.org/wiki/Power_of_Prime_is_Deficient | https://proofwiki.org/wiki/Power_of_Prime_is_Deficient | [
"Deficient Numbers"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Deficient Number"
] | [
"Divisor Sum of Power of Prime",
"Definition:Deficient Number"
] |
proofwiki-12166 | Power of 2 is Almost Perfect | Let $n \in \Z_{>0}$ be a power of $2$:
:$n = 2^k$
for some $k \in \Z_{>0}$.
Then $n$ is almost perfect. | {{begin-eqn}}
{{eqn | l = \map A n
| r = \dfrac {2^k} {2 - 1} - 2^k - \dfrac 1 {2 - 1}
| c = Power of Prime is Deficient
}}
{{eqn | r = 2^k - 2^k - 1
| c =
}}
{{eqn | r = - 1
| c =
}}
{{end-eqn}}
and so $n = p^k$ is almost perfect by definition.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Integer Power|power]] of $2$:
:$n = 2^k$
for some $k \in \Z_{>0}$.
Then $n$ is [[Definition:Almost Perfect Number|almost perfect]]. | {{begin-eqn}}
{{eqn | l = \map A n
| r = \dfrac {2^k} {2 - 1} - 2^k - \dfrac 1 {2 - 1}
| c = [[Power of Prime is Deficient]]
}}
{{eqn | r = 2^k - 2^k - 1
| c =
}}
{{eqn | r = - 1
| c =
}}
{{end-eqn}}
and so $n = p^k$ is [[Definition:Almost Perfect Number|almost perfect]] by definition.
{{qed... | Power of 2 is Almost Perfect | https://proofwiki.org/wiki/Power_of_2_is_Almost_Perfect | https://proofwiki.org/wiki/Power_of_2_is_Almost_Perfect | [
"Almost Perfect Numbers",
"2"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Almost Perfect Number"
] | [
"Power of Prime is Deficient",
"Definition:Almost Perfect Number"
] |
proofwiki-12167 | Set is Subset of Finite Infima Set | Let $\struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then $X \subseteq \map {\operatorname{fininfs} } X$
where $\map {\operatorname{fininfs} } X$ denotes the finite infima set of $X$. | Let $x \in X$.
By Infimum of Singleton:
:$\set x$ admits an infimum and $\inf \set x = x$
By definitions of subset and singleton:
:$\set x \subseteq X$
By Singleton is Finite:
:$\set x$ is a finite set.
Thus by definition of finite infima set:
:$x \in \map {\operatorname{fininfs} } X$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then $X \subseteq \map {\operatorname{fininfs} } X$
where $\map {\operatorname{fininfs} } X$ denotes the [[Definition:Finite Infima Set|finite infima set]] of $X$. | Let $x \in X$.
By [[Infimum of Singleton]]:
:$\set x$ admits an [[Definition:Infimum of Set|infimum]] and $\inf \set x = x$
By definitions of [[Definition:Subset|subset]] and [[Definition:Singleton|singleton]]:
:$\set x \subseteq X$
By [[Singleton is Finite]]:
:$\set x$ is a [[Definition:Finite Set|finite set]].
Th... | Set is Subset of Finite Infima Set | https://proofwiki.org/wiki/Set_is_Subset_of_Finite_Infima_Set | https://proofwiki.org/wiki/Set_is_Subset_of_Finite_Infima_Set | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Finite Infima Set"
] | [
"Infimum of Singleton",
"Definition:Infimum of Set",
"Definition:Subset",
"Definition:Singleton",
"Singleton is Finite",
"Definition:Finite Set",
"Definition:Finite Infima Set"
] |
proofwiki-12168 | Set is Subset of Upper Closure | Let $\struct {S, \preceq}$ be an ordered set.
Let $X$ be a subset of $S$.
Then $X \subseteq X^\succeq$
where $X^\succeq$ denotes the upper closure of $X$. | Let $x \in X$.
By definition of reflexivity:
:$x \preceq x$
Thus by definition of upper closure:
:$x \in X^\succeq$
{{qed}} | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $X$ be a [[Definition:Subset|subset]] of $S$.
Then $X \subseteq X^\succeq$
where $X^\succeq$ denotes the [[Definition:Upper Closure of Subset|upper closure]] of $X$. | Let $x \in X$.
By definition of [[Definition:Reflexivity|reflexivity]]:
:$x \preceq x$
Thus by definition of [[Definition:Upper Closure of Subset|upper closure]]:
:$x \in X^\succeq$
{{qed}} | Set is Subset of Upper Closure | https://proofwiki.org/wiki/Set_is_Subset_of_Upper_Closure | https://proofwiki.org/wiki/Set_is_Subset_of_Upper_Closure | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Upper Closure/Set"
] | [
"Definition:Reflexivity",
"Definition:Upper Closure/Set"
] |
proofwiki-12169 | Tamref's Last Theorem | The Diophantine equation:
:$n^x + n^y = n^z$
has exactly one form of solutions in integers:
:$2^x + 2^x = 2^{x + 1}$
for all $x \in \Z$. | Since $n^z = n^x + n^y > n^x$ and $n^y$, $z > x,y$.
{{WLOG}} assume that $x \le y < z$.
{{begin-eqn}}
{{eqn | l = n^x + n^y
| r = n^z
}}
{{eqn | ll = \leadsto
| l = 1 + n^{y - x}
| r = n^{z - x}
| c = Divide both sides by $n^x$
}}
{{eqn | ll = \leadsto
| l = 1
| r = n^{y - x} \paren... | The [[Definition:Diophantine Equation|Diophantine equation]]:
:$n^x + n^y = n^z$
has exactly one form of solutions in [[Definition:Integer|integers]]:
:$2^x + 2^x = 2^{x + 1}$
for all $x \in \Z$. | Since $n^z = n^x + n^y > n^x$ and $n^y$, $z > x,y$.
{{WLOG}} assume that $x \le y < z$.
{{begin-eqn}}
{{eqn | l = n^x + n^y
| r = n^z
}}
{{eqn | ll = \leadsto
| l = 1 + n^{y - x}
| r = n^{z - x}
| c = Divide both sides by $n^x$
}}
{{eqn | ll = \leadsto
| l = 1
| r = n^{y - x} \par... | Tamref's Last Theorem | https://proofwiki.org/wiki/Tamref's_Last_Theorem | https://proofwiki.org/wiki/Tamref's_Last_Theorem | [
"Number Theory"
] | [
"Definition:Diophantine Equation",
"Definition:Integer"
] | [
"Definition:Positive/Integer",
"Definition:Integer"
] |
proofwiki-12170 | 3^x + 4^y equals 5^z has Unique Solution | The Diophantine equation:
:$3^x + 4^y = 5^z$
has exactly one solution in (strictly) positive integers:
:$3^2 + 4^2 = 5^2$ | Rewriting our Diophantine equation modulo $4$ we have:
:$\paren {-1}^x + 0^y \equiv 1^z \pmod 4$
Therefore $x$ is even.
Rewriting our Diophantine equation modulo 3 we have:
:$0^x + 1^y \equiv \paren {-1}^z \pmod 3$
Therefore $z$ is even.
Since $x$ and $z$ must both be even, we will rewrite $x$ as $2 r$, $z$ as $2 s$ an... | The [[Definition:Diophantine Equation|Diophantine equation]]:
:$3^x + 4^y = 5^z$
has exactly one solution in [[Definition:Strictly Positive Integer|(strictly) positive integers]]:
:$3^2 + 4^2 = 5^2$ | Rewriting our [[Definition:Diophantine Equation|Diophantine equation]] [[Definition:Congruence Modulo Integer|modulo $4$]] we have:
:$\paren {-1}^x + 0^y \equiv 1^z \pmod 4$
Therefore $x$ is [[Definition:Even Integer|even]].
Rewriting our [[Definition:Diophantine Equation|Diophantine equation]] [[Congruence (Number T... | 3^x + 4^y equals 5^z has Unique Solution | https://proofwiki.org/wiki/3^x_+_4^y_equals_5^z_has_Unique_Solution | https://proofwiki.org/wiki/3^x_+_4^y_equals_5^z_has_Unique_Solution | [
"Number Theory"
] | [
"Definition:Diophantine Equation",
"Definition:Strictly Positive/Integer"
] | [
"Definition:Diophantine Equation",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Even Integer",
"Definition:Diophantine Equation",
"Congruence (Number Theory)/Integers/Examples/Modulo 3",
"Definition:Even Integer",
"Definition:Even Integer",
"Difference of Two Squares",
"Definition:O... |
proofwiki-12171 | 5^x + 12^y equals 13^z has Unique Solution | The Diophantine equation:
:$5^x + 12^y = 13^z$
has exactly one solution in (strictly) positive integers:
:$5^2 + 12^2 = 13^2$ | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.
Suppose $x,y,z$ are positive integers with $5^x+12^y=13^z$.
Reducing modulo $3$ gives us $(-1)^x+0^y\equiv1^z$.
Thus $x$ is even.
Say $x=2a$ for postive integer $a$, so we have $(1):\quad 25^a+12^y=13^z$.
Now reducing modulo $5$ gives us $0^x... | The [[Definition:Diophantine Equation|Diophantine equation]]:
:$5^x + 12^y = 13^z$
has exactly one solution in [[Definition:Strictly Positive Integer|(strictly) positive integers]]:
:$5^2 + 12^2 = 13^2$ | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the [[Definition:P-adic Valuation on Rational Numbers|$p$-adic valuation]] on $\Q$.
Suppose $x,y,z$ are positive integers with $5^x+12^y=13^z$.
Reducing modulo $3$ gives us $(-1)^x+0^y\equiv1^z$.
Thus $x$ is even.
Say $x=2a$ for postive integer $a$, so we have $(1):\qua... | 5^x + 12^y equals 13^z has Unique Solution | https://proofwiki.org/wiki/5^x_+_12^y_equals_13^z_has_Unique_Solution | https://proofwiki.org/wiki/5^x_+_12^y_equals_13^z_has_Unique_Solution | [
"Number Theory"
] | [
"Definition:Diophantine Equation",
"Definition:Strictly Positive/Integer"
] | [
"Definition:P-adic Valuation/Rational Numbers"
] |
proofwiki-12172 | Upper Closure of Subset is Subset of Upper Closure | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $X, Y$ be subsets of $S$.
Then
:$X \subseteq Y \implies X^\succeq \subseteq Y^\succeq$
where $X^\succeq$ denotes the upper closure of $X$. | Let $X \subseteq Y$.
Let $x \in X^\succeq$.
By definition of upper closure of subset:
:$\exists y \in X: y \preceq x$
By definition of subset:
:$y \in Y$
Thus by definition of upper closure of subset:
:$x \in Y^\succeq$
{{qed}} | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $X, Y$ be [[Definition:Subset|subsets]] of $S$.
Then
:$X \subseteq Y \implies X^\succeq \subseteq Y^\succeq$
where $X^\succeq$ denotes the [[Definition:Upper Closure of Subset|upper closure]] of $X$. | Let $X \subseteq Y$.
Let $x \in X^\succeq$.
By definition of [[Definition:Upper Closure of Subset|upper closure of subset]]:
:$\exists y \in X: y \preceq x$
By definition of [[Definition:Subset|subset]]:
:$y \in Y$
Thus by definition of [[Definition:Upper Closure of Subset|upper closure of subset]]:
:$x \in Y^\succ... | Upper Closure of Subset is Subset of Upper Closure | https://proofwiki.org/wiki/Upper_Closure_of_Subset_is_Subset_of_Upper_Closure | https://proofwiki.org/wiki/Upper_Closure_of_Subset_is_Subset_of_Upper_Closure | [
"Order Theory",
"Upper Closures"
] | [
"Definition:Ordered Set",
"Definition:Subset",
"Definition:Upper Closure/Set"
] | [
"Definition:Upper Closure/Set",
"Definition:Subset",
"Definition:Upper Closure/Set"
] |
proofwiki-12173 | Euler's Number as Limit of n over nth Root of n Factorial | :$\ds e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$
where:
:$e$ denotes Euler's number
:$n!$ denotes $n$ factorial. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} {n!} {n^n \sqrt n e^{-n} }
| r = \sqrt {2 \pi}
| c = Lemma for Stirling's Formula
}}
{{eqn | ll= \leadsto
| l = e
| r = \lim_{n \mathop \to \infty} \dfrac {n \paren {2 \pi n}^{1 / 2 n} } {\sqrt [n] {n!} }
| c =
}}
{{eqn | ll= \leadsto
... | :$\ds e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$
where:
:$e$ denotes [[Definition:Euler's Number|Euler's number]]
:$n!$ denotes [[Definition:Factorial|$n$ factorial]]. | {{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} {n!} {n^n \sqrt n e^{-n} }
| r = \sqrt {2 \pi}
| c = [[Stirling's Formula/Proof 2/Lemma 5|Lemma for Stirling's Formula]]
}}
{{eqn | ll= \leadsto
| l = e
| r = \lim_{n \mathop \to \infty} \dfrac {n \paren {2 \pi n}^{1 / 2 n} } {\sqrt [n] {n!} ... | Euler's Number as Limit of n over nth Root of n Factorial | https://proofwiki.org/wiki/Euler's_Number_as_Limit_of_n_over_nth_Root_of_n_Factorial | https://proofwiki.org/wiki/Euler's_Number_as_Limit_of_n_over_nth_Root_of_n_Factorial | [
"Euler's Number"
] | [
"Definition:Euler's Number",
"Definition:Factorial"
] | [
"Stirling's Formula/Proof 2/Lemma 5",
"Limit of Root of Positive Real Number",
"Limit of Integer to Reciprocal Power"
] |
proofwiki-12174 | Continued Fraction Expansion of Euler's Number | The constant Euler's number $e$ has the continued fraction expansion:
{{begin-eqn}}
{{eqn | l = e
| r = \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }
| c =
}}
{{eqn | l =
| r = \sqbrk {1, 0, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }
| c =
}}
{{end-eqn}} | From the recursive definition of continued fractions, we have:
{{begin-eqn}}
{{eqn | l = p_i
| r = a_i p_{i - 1} + p_{i - 2}
| c =
}}
{{eqn | l = q_i
| r = a_i q_{i - 1} + q_{i - 2}
| c =
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn | l = \sqbrk {a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, \ldo... | The constant [[Definition:Euler's Number|Euler's number $e$]] has the [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]]:
{{begin-eqn}}
{{eqn | l = e
| r = \sqbrk {2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \ldots }
| c =
}}
{{eqn | l =
| r = \sqbrk {1, 0, 1, 1, 2, 1, 1, 4,... | From the [[Definition:Numerators and Denominators of Continued Fraction|recursive definition of continued fractions]], we have:
{{begin-eqn}}
{{eqn | l = p_i
| r = a_i p_{i - 1} + p_{i - 2}
| c =
}}
{{eqn | l = q_i
| r = a_i q_{i - 1} + q_{i - 2}
| c =
}}
{{end-eqn}}
Let:
{{begin-eqn}}
{{eqn... | Continued Fraction Expansion of Euler's Number/Proof 1 | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Euler's_Number | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Euler's_Number/Proof_1 | [
"Euler's Number",
"Examples of Continued Fractions",
"Continued Fraction Expansion of Euler's Number"
] | [
"Definition:Euler's Number",
"Definition:Continued Fraction Expansion/Real Number"
] | [
"Definition:Numerators and Denominators of Continued Fraction",
"Definition:Recursive Sequence",
"Definition:Definite Integral",
"Continued Fraction Expansion of Euler's Number/Proof 1/Lemma",
"Radius of Convergence from Limit of Sequence/Real Case",
"Radius of Convergence from Limit of Sequence/Real Case... |
proofwiki-12175 | Continued Fraction Expansion of Euler's Number/Convergents | The convergents of the continued fraction expansion to Euler's number $e$ are:
:$2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}, \dfrac {1264} {465}, \dfrac {1457} {536}, \dfrac {2721} {1001}, \ldots$
{{OEIS-Numerators|A007676}}
{{OEIS-Denominators|A007677}}
These... | {{ProofWanted|Calculation needed}} | The [[Definition:Convergent of Continued Fraction|convergents]] of the [[Definition:Continued Fraction Expansion of Real Number|continued fraction expansion]] to [[Definition:Euler's Number|Euler's number $e$]] are:
:$2, 3, \dfrac 8 3, \dfrac {11} 4, \dfrac {19} 7, \dfrac {87} {32}, \dfrac {106} {39}, \dfrac {193} {71}... | {{ProofWanted|Calculation needed}} | Continued Fraction Expansion of Euler's Number/Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Euler's_Number/Convergents | https://proofwiki.org/wiki/Continued_Fraction_Expansion_of_Euler's_Number/Convergents | [
"Continued Fraction Expansion of Euler's Number"
] | [
"Definition:Convergent of Continued Fraction",
"Definition:Continued Fraction Expansion/Real Number",
"Definition:Euler's Number",
"Definition:Best Rational Approximation",
"Definition:Decimal Notation",
"Definition:Fraction",
"Definition:Continued Fraction Expansion/Real Number"
] | [] |
proofwiki-12176 | Sine of X over X as Infinite Product | Let $z \in \C$ be a non-zero complex number.
Then:
:$\ds \frac {\sin z} z = \cos \frac z 2 \cos \frac z 4 \cos \frac z 8 \cdots = \prod_{i \mathop = 1}^{\infty} \cos \frac z {2^i}$
where $\sin$ denotes the sine function and $\cos$ denotes the cosine function. | First we prove that:
:$\ds \frac {\sin z} z = \paren {\frac {2^n} z} \sin \frac z {2^n} \prod_{i \mathop = 1}^n \cos \frac z {2^i}$
for $n \in \N$.
The proof proceeds by induction.
For all $n \in \N$, let $\map P n$ be the proposition:
:$\ds \frac {\sin z} z = \paren {\frac {2^n} z} \sin \frac z {2^n} \prod_{i \mathop ... | Let $z \in \C$ be a non-[[Definition:Zero|zero]] [[Definition:Complex Number|complex number]].
Then:
:$\ds \frac {\sin z} z = \cos \frac z 2 \cos \frac z 4 \cos \frac z 8 \cdots = \prod_{i \mathop = 1}^{\infty} \cos \frac z {2^i}$
where $\sin$ denotes the [[Definition:Complex Cosine Function|sine function]] and $\c... | First we prove that:
:$\ds \frac {\sin z} z = \paren {\frac {2^n} z} \sin \frac z {2^n} \prod_{i \mathop = 1}^n \cos \frac z {2^i}$
for $n \in \N$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \frac {\s... | Sine of X over X as Infinite Product | https://proofwiki.org/wiki/Sine_of_X_over_X_as_Infinite_Product | https://proofwiki.org/wiki/Sine_of_X_over_X_as_Infinite_Product | [
"Proofs by Induction",
"Trigonometric Identities",
"Examples of Infinite Products"
] | [
"Definition:Zero",
"Definition:Complex Number",
"Definition:Cosine/Complex Function",
"Definition:Cosine/Complex Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-12177 | Maximum of Three Mutually Perpendicular Lines in Ordinary Space | In ordinary space, there can be no more than $3$ straight lines which are pairwise perpendicular.
Thus, in a configuration of $4$ straight lines in space, at least one pair will not be perpendicular to each other. | By assumption and popular belief, ordinary space (on the measurable local level) is an instance of a $3$-dimensional Euclidean space.
Hence the results of {{ElementsLink}} can be applied.
Let there be $4$ straight lines $a$, $b$, $c$ and $d$ in space.
Let $a$ be perpendicular to each of $b$, $c$ and $d$.
From Three Int... | In [[Definition:Ordinary Space|ordinary space]], there can be no more than $3$ [[Definition:Straight Line|straight lines]] which are pairwise [[Definition:Perpendicular|perpendicular]].
Thus, in a configuration of $4$ [[Definition:Straight Line|straight lines]] in [[Definition:Ordinary Space|space]], at least one pair... | By assumption and popular belief, [[Definition:Ordinary Space|ordinary space]] (on the measurable local level) is an instance of a [[Definition:Dimension (Geometry)|$3$-dimensional]] [[Definition:Euclidean Space|Euclidean space]].
Hence the results of {{ElementsLink}} can be applied.
Let there be $4$ [[Definition:Str... | Maximum of Three Mutually Perpendicular Lines in Ordinary Space | https://proofwiki.org/wiki/Maximum_of_Three_Mutually_Perpendicular_Lines_in_Ordinary_Space | https://proofwiki.org/wiki/Maximum_of_Three_Mutually_Perpendicular_Lines_in_Ordinary_Space | [
"Solid Geometry"
] | [
"Definition:Ordinary Space",
"Definition:Line/Straight Line",
"Definition:Right Angle/Perpendicular",
"Definition:Line/Straight Line",
"Definition:Ordinary Space",
"Definition:Right Angle/Perpendicular"
] | [
"Definition:Ordinary Space",
"Definition:Dimension (Geometry)",
"Definition:Euclidean Space",
"Definition:Line/Straight Line",
"Definition:Ordinary Space",
"Definition:Right Angle/Perpendicular",
"Three Intersecting Lines Perpendicular to Another Line are in One Plane",
"Definition:Plane Surface",
"... |
proofwiki-12178 | Complex Logarithm of 1 | :$\ln 1 = \set {2 k \pi i: k \in \Z}$ | Denote as $\ln_\R$ the real natural logarithm.
{{begin-eqn}}
{{eqn | l = \map \ln {r e^{i \theta} }
| r = \set {\map {\ln_\R} r + i \theta + 2 k \pi i: k \in \Z}
| c = {{Defof|Natural Logarithm/Complex|index = 1|Complex Natural Logarithm}}
}}
{{eqn | ll= \leadsto
| l = \map \ln {1 e^{i \times 0} }
... | :$\ln 1 = \set {2 k \pi i: k \in \Z}$ | Denote as $\ln_\R$ the [[Definition:Real Natural Logarithm|real natural logarithm]].
{{begin-eqn}}
{{eqn | l = \map \ln {r e^{i \theta} }
| r = \set {\map {\ln_\R} r + i \theta + 2 k \pi i: k \in \Z}
| c = {{Defof|Natural Logarithm/Complex|index = 1|Complex Natural Logarithm}}
}}
{{eqn | ll= \leadsto
... | Complex Logarithm of 1 | https://proofwiki.org/wiki/Complex_Logarithm_of_1 | https://proofwiki.org/wiki/Complex_Logarithm_of_1 | [
"Logarithms"
] | [] | [
"Definition:Natural Logarithm/Positive Real",
"Real Multiplication Identity is One",
"Natural Logarithm of 1 is 0",
"Complex Addition Identity is Zero",
"Category:Logarithms"
] |
proofwiki-12179 | Derivative of Constant/Complex | Let $\map {f_c} z$ be the constant function on an open domain $D \in \C$, where $c \in \C$.
Then:
:$\forall z \in D : \map { {f_c}'} z = 0$ | The function $f_c: D \to \C$ is defined as:
:$\forall z \in D: \map {f_c} z = c$
Thus:
{{begin-eqn}}
{{eqn | l = \map { {f_c}'} z
| r = \lim_{h \mathop \to 0} \frac {\map {f_c} {z + h} - \map {f_c} z} h
| c = {{Defof|Derivative of Complex Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {c - c} h
... | Let $\map {f_c} z$ be the [[Definition:Constant Mapping|constant function]] on an [[Definition:Open Set (Complex Analysis)|open]] domain $D \in \C$, where $c \in \C$.
Then:
:$\forall z \in D : \map { {f_c}'} z = 0$ | The function $f_c: D \to \C$ is defined as:
:$\forall z \in D: \map {f_c} z = c$
Thus:
{{begin-eqn}}
{{eqn | l = \map { {f_c}'} z
| r = \lim_{h \mathop \to 0} \frac {\map {f_c} {z + h} - \map {f_c} z} h
| c = {{Defof|Derivative of Complex Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {c - c} h
... | Derivative of Constant/Complex | https://proofwiki.org/wiki/Derivative_of_Constant/Complex | https://proofwiki.org/wiki/Derivative_of_Constant/Complex | [
"Complex Differential Calculus",
"Constant Mappings"
] | [
"Definition:Constant Mapping",
"Definition:Open Set/Complex Analysis"
] | [
"Category:Complex Differential Calculus",
"Category:Constant Mappings"
] |
proofwiki-12180 | Derivative of Identity Function/Real | Let $I_\R: \R \to \R$ be the identity mapping on the real numbers $\R$.
Then:
:$\map {I_\R'} x = 1$ | The identity mapping is defined as:
:$\forall x \in \R: \map {I_\R} x = x$
Thus:
{{begin-eqn}}
{{eqn | l = \map {I_\R'} x
| r = \lim_{\delta x \mathop \to 0} \frac {\map {I_\R} {x + \delta x} - \map {I_\R} x} {\delta x}
| c = {{Defof|Derivative of Real Function}}
}}
{{eqn | r = \lim_{\delta x \mathop \to 0}... | Let $I_\R: \R \to \R$ be the [[Definition:Identity Mapping|identity mapping]] on the [[Definition:Real Number|real numbers]] $\R$.
Then:
:$\map {I_\R'} x = 1$ | The [[Definition:Identity Mapping|identity mapping]] is defined as:
:$\forall x \in \R: \map {I_\R} x = x$
Thus:
{{begin-eqn}}
{{eqn | l = \map {I_\R'} x
| r = \lim_{\delta x \mathop \to 0} \frac {\map {I_\R} {x + \delta x} - \map {I_\R} x} {\delta x}
| c = {{Defof|Derivative of Real Function}}
}}
{{eqn |... | Derivative of Identity Function/Real | https://proofwiki.org/wiki/Derivative_of_Identity_Function/Real | https://proofwiki.org/wiki/Derivative_of_Identity_Function/Real | [
"Derivatives",
"Identity Mappings"
] | [
"Definition:Identity Mapping",
"Definition:Real Number"
] | [
"Definition:Identity Mapping",
"Category:Derivatives",
"Category:Identity Mappings"
] |
proofwiki-12181 | Derivative of Identity Function/Complex | Let $I_\C: \C \to \C$ be the identity function.
Then:
:$\map {I_\C'} z = 1$ | The identity function is defined as $\forall x \in \C: \map {I_\C} z = z$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {I_\C'} z
| r = \lim_{h \mathop \to 0} \frac {\map {I_\C} {z + h} - \map {I_\C} z} h
| c = {{Defof|Derivative of Complex Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\paren {z + h} - z}... | Let $I_\C: \C \to \C$ be the [[Definition:Identity Mapping|identity function]].
Then:
:$\map {I_\C'} z = 1$ | The identity function is defined as $\forall x \in \C: \map {I_\C} z = z$.
Thus:
{{begin-eqn}}
{{eqn | l = \map {I_\C'} z
| r = \lim_{h \mathop \to 0} \frac {\map {I_\C} {z + h} - \map {I_\C} z} h
| c = {{Defof|Derivative of Complex Function}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\paren {z + h} - ... | Derivative of Identity Function/Complex | https://proofwiki.org/wiki/Derivative_of_Identity_Function/Complex | https://proofwiki.org/wiki/Derivative_of_Identity_Function/Complex | [
"Complex Differential Calculus",
"Identity Mappings"
] | [
"Definition:Identity Mapping"
] | [
"Category:Complex Differential Calculus",
"Category:Identity Mappings"
] |
proofwiki-12182 | Chain Rule for Real-Valued Functions/Corollary | Let $\Psi$ represent a differentiable function of $x$ and $y$.
Let $y$ represent a differentiable function of $x$.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\d \Psi} {\d x}
| r = \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac {\d \Psi} {\d x}
| r = \frac {\partial \Psi} {\partial x} \frac {\d x} {\d x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}
| c = Chain Rule for Real-Valued Functions
}}
{{eqn | r = \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {\d y... | Let $\Psi$ represent a [[Definition:Differentiable Real Function|differentiable function]] of $x$ and $y$.
Let $y$ represent a [[Definition:Differentiable Real Function|differentiable function]] of $x$.
Then:
{{begin-eqn}}
{{eqn | l = \frac {\d \Psi} {\d x}
| r = \frac {\partial \Psi} {\partial x} + \frac {\pa... | {{begin-eqn}}
{{eqn | l = \frac {\d \Psi} {\d x}
| r = \frac {\partial \Psi} {\partial x} \frac {\d x} {\d x} + \frac {\partial \Psi} {\partial y} \frac {\d y} {\d x}
| c = [[Chain Rule for Real-Valued Functions]]
}}
{{eqn | r = \frac {\partial \Psi} {\partial x} + \frac {\partial \Psi} {\partial y} \frac {... | Chain Rule for Real-Valued Functions/Corollary | https://proofwiki.org/wiki/Chain_Rule_for_Real-Valued_Functions/Corollary | https://proofwiki.org/wiki/Chain_Rule_for_Real-Valued_Functions/Corollary | [
"Derivative of Composite Function"
] | [
"Definition:Differentiable Mapping/Real Function",
"Definition:Differentiable Mapping/Real Function"
] | [
"Chain Rule for Real-Valued Functions",
"Derivative of Identity Function"
] |
proofwiki-12183 | Laplace Transform of Periodic Function | Let $f$ be periodic, that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform. | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \sum_{k \mathop = 0}^{\infty} \int_{k T}^{\paren {k + 1} T} e^{-s t} \map f t \rd t
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions: Corol... | Let $f$ be [[Definition:Real Periodic Function|periodic]], that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the [[Definition:Laplace Transform|Laplace ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \sum_{k \mathop = 0}^{\infty} \int_{k T}^{\paren {k + 1} T} e^{-s t} \map f t \rd t
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions/Coro... | Laplace Transform of Periodic Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function/Proof_1 | [
"Laplace Transform of Periodic Function",
"Laplace Transforms",
"Periodic Functions"
] | [
"Definition:Periodic Function/Real",
"Definition:Laplace Transform"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary",
"Integration by Substitution/Corollary",
"General Periodicity Property",
"Real Multiplication Distributes over Addition",
"Exponent Combination Laws/Product of Powers",
"Primitive of Constant Multiple of Function",
"Summation i... |
proofwiki-12184 | Laplace Transform of Periodic Function | Let $f$ be periodic, that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform. | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^T e^{-s t} \map f t \rd t + \int_T^{\infty} e^{-s t} \map f t \rd t
| c = Sum of Integrals on Adjacent Intervals for Integrable Functions
}}
{{eqn | r = ... | Let $f$ be [[Definition:Real Periodic Function|periodic]], that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the [[Definition:Laplace Transform|Laplace ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^T e^{-s t} \map f t \rd t + \int_T^{\infty} e^{-s t} \map f t \rd t
| c = [[Sum of Integrals on Adjacent Intervals for Integrable Functions]]
}}
{{eqn | ... | Laplace Transform of Periodic Function/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function/Proof_2 | [
"Laplace Transform of Periodic Function",
"Laplace Transforms",
"Periodic Functions"
] | [
"Definition:Periodic Function/Real",
"Definition:Laplace Transform"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Integration by Substitution/Corollary",
"Real Multiplication Distributes over Addition",
"Exponent Combination Laws/Product of Powers",
"Primitive of Constant Multiple of Function"
] |
proofwiki-12185 | Laplace Transform of Periodic Function | Let $f$ be periodic, that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the Laplace transform. | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^T e^{-s t} \map f t \rd t + \int_T^{2 T} e^{-s t} \map f t \rd t + \int_{2 T}^{3 T} e^{-s t} \map f t \rd t + \dotsb
| c = Sum of Integrals on Adjacent I... | Let $f$ be [[Definition:Real Periodic Function|periodic]], that is:
:$\exists T \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + T}$
Then:
:$\laptrans {\map f t} = \dfrac 1 {1 - e^{-s T} } \ds \int_0^T e^{-s t} \map f t \rd t$
where $\laptrans {\map f t}$ denotes the [[Definition:Laplace Transform|Laplace ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^{\infty} e^{-s t} \map f t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^T e^{-s t} \map f t \rd t + \int_T^{2 T} e^{-s t} \map f t \rd t + \int_{2 T}^{3 T} e^{-s t} \map f t \rd t + \dotsb
| c = [[Sum of Integrals on Adjacent... | Laplace Transform of Periodic Function/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Periodic_Function/Proof_3 | [
"Laplace Transform of Periodic Function",
"Laplace Transforms",
"Periodic Functions"
] | [
"Definition:Periodic Function/Real",
"Definition:Laplace Transform"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Integration by Substitution",
"Second Translation Property of Laplace Transforms",
"Definition:Definite Integral/Limits of Integration",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-12186 | Triangle is Medial Triangle of Larger Triangle | Let $\triangle ABC$ be a triangle.
$\triangle ABC$ is the medial triangle of a larger triangle. | By {{EuclidPostulateLink|Fifth}}, it is possible to construct exactly one straight line parallel to each of $AC$, $BC$ and $AC$.
:500px
From Parallelism implies Equal Corresponding Angles:
:$\angle ABC = \angle ECB = \angle DAB$
:$\angle ACB = \angle CBE = \angle CAF$
:$\angle CAB = \angle ABD = \angle ACF$
By Triangle... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
$\triangle ABC$ is the [[Definition:Medial Triangle|medial triangle]] of a larger [[Definition:Triangle (Geometry)|triangle]]. | By {{EuclidPostulateLink|Fifth}}, it is possible to construct exactly one [[Definition:Straight Line|straight line]] [[Definition:Parallel Lines|parallel]] to each of $AC$, $BC$ and $AC$.
:[[File:TriangleIsMedial.png|500px]]
From [[Parallelism implies Equal Corresponding Angles]]:
:$\angle ABC = \angle ECB = \angle... | Triangle is Medial Triangle of Larger Triangle | https://proofwiki.org/wiki/Triangle_is_Medial_Triangle_of_Larger_Triangle | https://proofwiki.org/wiki/Triangle_is_Medial_Triangle_of_Larger_Triangle | [
"Medial Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Medial Triangle",
"Definition:Triangle (Geometry)"
] | [
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"File:TriangleIsMedial.png",
"Parallelism implies Equal Corresponding Angles",
"Triangle Angle-Side-Angle Congruence",
"Definition:Medial Triangle"
] |
proofwiki-12187 | Altitudes of Triangle Meet at Point | Let $\triangle ABC$ be a triangle.
The altitudes of $\triangle ABC$ all intersect at the same point. | 300px
In $\triangle ABC$ construct the altitude from vertex $A$ to side $BC$ at $P$.
Also draw the altitude from vertex $B$ to side $AC$ at $Q$.
By Two Straight Lines make Equal Opposite Angles:
:$\angle AOQ = \angle BOP$
Given:
:$\angle AQO = \angle BPO = $ one right angle.
By Triangles with Two Equal Angles are Simil... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
The [[Definition:Altitude of Triangle|altitudes]] of $\triangle ABC$ all [[Definition:Intersection (Geometry)|intersect]] at the same [[Definition:Point|point]]. | [[File:Newton altitude.png|300px]]
In $\triangle ABC$ construct the [[Definition:Altitude of Triangle|altitude]] from [[Definition:Vertex of Polygon|vertex]] $A$ to [[Definition:Side of Polygon|side]] $BC$ at $P$.
Also draw the [[Definition:Altitude of Triangle|altitude]] from [[Definition:Vertex of Polygon|vertex]] ... | Altitudes of Triangle Meet at Point/Proof 2 | https://proofwiki.org/wiki/Altitudes_of_Triangle_Meet_at_Point | https://proofwiki.org/wiki/Altitudes_of_Triangle_Meet_at_Point/Proof_2 | [
"Altitudes of Triangle Meet at Point",
"Orthocenters of Triangles",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle",
"Definition:Intersection (Geometry)",
"Definition:Point"
] | [
"File:Newton altitude.png",
"Definition:Altitude of Triangle",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side",
"Definition:Altitude of Triangle",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side",
"Two Straight Lines make Equal Opposite Angles",
"Definition:Right Angle",
"Triangles wi... |
proofwiki-12188 | Perpendicular Bisectors of Triangle Meet at Point | Let $\triangle ABC$ be a triangle.
The perpendicular bisectors of $AB$, $BC$ and $AC$ all intersect at the same point. | Let the perpendicular bisectors of $AC$ and $AB$ be constructed through $D$ and $E$ respectively to meet at $F$.
:500px
By definition of perpendicular bisector:
:$AE = EB$
and:
:$\angle AEF = \angle BEF$ are right angles.
From Triangle Side-Angle-Side Congruence:
:$\triangle AEF = \triangle BEF$
and so $AF = BF$.
Simil... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
The [[Definition:Perpendicular Bisector|perpendicular bisectors]] of $AB$, $BC$ and $AC$ all [[Definition:Intersection (Geometry)|intersect]] at the same [[Definition:Point|point]]. | Let the [[Definition:Perpendicular Bisector|perpendicular bisectors]] of $AC$ and $AB$ be constructed through $D$ and $E$ respectively to meet at $F$.
:[[File:PerpendicularBisectorsMeetAtPoint.png|500px]]
By definition of [[Definition:Perpendicular Bisector|perpendicular bisector]]:
:$AE = EB$
and:
:$\angle AEF = \... | Perpendicular Bisectors of Triangle Meet at Point | https://proofwiki.org/wiki/Perpendicular_Bisectors_of_Triangle_Meet_at_Point | https://proofwiki.org/wiki/Perpendicular_Bisectors_of_Triangle_Meet_at_Point | [
"Perpendicular Bisectors",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Perpendicular Bisector",
"Definition:Intersection (Geometry)",
"Definition:Point"
] | [
"Definition:Perpendicular Bisector",
"File:PerpendicularBisectorsMeetAtPoint.png",
"Definition:Perpendicular Bisector",
"Definition:Right Angle",
"Triangle Side-Angle-Side Congruence",
"Definition:Perpendicular Bisector",
"Definition:Right Angle",
"Triangle Side-Angle-Side Congruence",
"Definition:A... |
proofwiki-12189 | Centroid of Triangle is Centroid of Medial | Let $\triangle ABC$ be a triangle.
Let $\triangle DEF$ be the medial triangle of $\triangle ABC$.
Let $G$ be the centroid of $\triangle ABC$.
Then $G$ is also the centroid of $\triangle DEF$. | :500px
By definition of centroid and medial triangle, the lines $AE$, $BF$ and $CD$ intersect at $G$.
It remains to be shown that $AE$, $BF$ and $CD$ bisect the sides of $DF$, $DE$ and $EF$ respectively.
{{WLOG}}, let $AE$ intersect $DF$ at $H$.
From the working of Triangle is Medial Triangle of Larger Triangle, we hav... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle DEF$ be the [[Definition:Medial Triangle|medial triangle]] of $\triangle ABC$.
Let $G$ be the [[Definition:Centroid of Triangle|centroid]] of $\triangle ABC$.
Then $G$ is also the [[Definition:Centroid of Triangle|centroid]] of $\t... | :[[File:CentroidOfMedial.png|500px]]
By definition of [[Definition:Centroid of Triangle|centroid]] and [[Definition:Medial Triangle|medial triangle]], the [[Definition:Line Segment|lines]] $AE$, $BF$ and $CD$ [[Definition:Intersection (Geometry)|intersect]] at $G$.
It remains to be shown that $AE$, $BF$ and $CD$ [[D... | Centroid of Triangle is Centroid of Medial | https://proofwiki.org/wiki/Centroid_of_Triangle_is_Centroid_of_Medial | https://proofwiki.org/wiki/Centroid_of_Triangle_is_Centroid_of_Medial | [
"Medial Triangles",
"Centroids of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Medial Triangle",
"Definition:Centroid/Triangle",
"Definition:Centroid/Triangle"
] | [
"File:CentroidOfMedial.png",
"Definition:Centroid/Triangle",
"Definition:Medial Triangle",
"Definition:Line/Segment",
"Definition:Intersection (Geometry)",
"Definition:Bisection",
"Definition:Polygon/Side",
"Definition:Intersection (Geometry)",
"Triangle is Medial Triangle of Larger Triangle",
"De... |
proofwiki-12190 | Diameters of Parallelogram Bisect each other | Let $\Box ABCD$ be a parallelogram with diameters $AC$ and $BD$.
Let $AC$ and $BD$ intersect at $E$.
Then $E$ is the midpoint of both $AC$ and $BD$. | :400px
By definition of parallelogram:
By Opposite Sides and Angles of Parallelogram are Equal:
:$AB = CD$
:$AD = BC$
:$\angle ABC = \angle ADC$
:$\angle BAD = \angle BCD$
Therefore by Triangle Side-Angle-Side Congruence:
:$\triangle ABC = \triangle ADC$
:$\triangle BAD = \triangle BCD$
Thus:
:$\angle ADE = \angle CBE$... | Let $\Box ABCD$ be a [[Definition:Parallelogram|parallelogram]] with [[Definition:Diameter of Parallelogram|diameters]] $AC$ and $BD$.
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|intersect]] at $E$.
Then $E$ is the [[Definition:Midpoint of Line|midpoint]] of both $AC$ and $BD$. | :[[File:DiametersOfParallelogramBisect.png|400px]]
By definition of [[Definition:Parallelogram|parallelogram]]:
By [[Opposite Sides and Angles of Parallelogram are Equal]]:
:$AB = CD$
:$AD = BC$
:$\angle ABC = \angle ADC$
:$\angle BAD = \angle BCD$
Therefore by [[Triangle Side-Angle-Side Congruence]]:
:$\triangle A... | Diameters of Parallelogram Bisect each other/Proof 1 | https://proofwiki.org/wiki/Diameters_of_Parallelogram_Bisect_each_other | https://proofwiki.org/wiki/Diameters_of_Parallelogram_Bisect_each_other/Proof_1 | [
"Parallelograms",
"Diameters of Parallelogram Bisect each other"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Intersection (Geometry)",
"Definition:Line/Midpoint"
] | [
"File:DiametersOfParallelogramBisect.png",
"Definition:Quadrilateral/Parallelogram",
"Opposite Sides and Angles of Parallelogram are Equal",
"Triangle Side-Angle-Side Congruence",
"Triangle Angle-Side-Angle Congruence"
] |
proofwiki-12191 | Diameters of Parallelogram Bisect each other | Let $\Box ABCD$ be a parallelogram with diameters $AC$ and $BD$.
Let $AC$ and $BD$ intersect at $E$.
Then $E$ is the midpoint of both $AC$ and $BD$. | :300px
Let $\Box ABCD$ be embedded in the complex plane such that $B$ is identified with the origin $0 + 0 i$.
Let $A$ be identified with the complex number $z_1$.
Let $C$ be identified with the complex number $z_2$.
By Geometrical Interpretation of Complex Subtraction:
:$z_1 - z_2 = AC$
Then:
:$AE = m \paren {z_1 - z_... | Let $\Box ABCD$ be a [[Definition:Parallelogram|parallelogram]] with [[Definition:Diameter of Parallelogram|diameters]] $AC$ and $BD$.
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|intersect]] at $E$.
Then $E$ is the [[Definition:Midpoint of Line|midpoint]] of both $AC$ and $BD$. | :[[File:DiametersOfParallelogramBisect-Complex.png|300px]]
Let $\Box ABCD$ be embedded in the [[Definition:Complex Plane|complex plane]] such that $B$ is identified with the [[Definition:Origin|origin]] $0 + 0 i$.
Let $A$ be identified with the [[Definition:Complex Number|complex number]] $z_1$.
Let $C$ be identifi... | Diameters of Parallelogram Bisect each other/Proof 2 | https://proofwiki.org/wiki/Diameters_of_Parallelogram_Bisect_each_other | https://proofwiki.org/wiki/Diameters_of_Parallelogram_Bisect_each_other/Proof_2 | [
"Parallelograms",
"Diameters of Parallelogram Bisect each other"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Intersection (Geometry)",
"Definition:Line/Midpoint"
] | [
"File:DiametersOfParallelogramBisect-Complex.png",
"Definition:Complex Number/Complex Plane",
"Definition:Coordinate System/Origin",
"Definition:Complex Number",
"Definition:Complex Number",
"Geometrical Interpretation of Complex Subtraction",
"Geometrical Interpretation of Complex Addition",
"Linear ... |
proofwiki-12192 | Circumcenter of Triangle is Orthocenter of Medial | Let $\triangle ABC$ be a triangle.
Let $\triangle DEF$ be the medial triangle of $\triangle ABC$.
Let $K$ be the circumcenter of $\triangle ABC$.
Then $K$ is the orthocenter of $\triangle DEF$. | :500px
Let $FG$, $DH$ and $EJ$ be the perpendicular bisectors of the sides of $AC$, $AB$ and $BC$ respectively.
From Circumscribing Circle about Triangle, the point $K$ where they intersect is the circumcenter of $\triangle ABC$.
From Perpendicular Bisector of Triangle is Altitude of Medial Triangle, $FG$, $DH$ and $EJ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle DEF$ be the [[Definition:Medial Triangle|medial triangle]] of $\triangle ABC$.
Let $K$ be the [[Definition:Circumcenter of Triangle|circumcenter]] of $\triangle ABC$.
Then $K$ is the [[Definition:Orthocenter|orthocenter]] of $\tria... | :[[File:CircumcenterMedialOrthocenter.png|500px]]
Let $FG$, $DH$ and $EJ$ be the [[Definition:Perpendicular Bisector|perpendicular bisectors]] of the [[Definition:Side of Polygon|sides]] of $AC$, $AB$ and $BC$ respectively.
From [[Circumscribing Circle about Triangle]], the [[Definition:Point|point]] $K$ where they ... | Circumcenter of Triangle is Orthocenter of Medial | https://proofwiki.org/wiki/Circumcenter_of_Triangle_is_Orthocenter_of_Medial | https://proofwiki.org/wiki/Circumcenter_of_Triangle_is_Orthocenter_of_Medial | [
"Circumcircles of Triangles",
"Orthocenters of Triangles",
"Medial Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Medial Triangle",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Orthocenter"
] | [
"File:CircumcenterMedialOrthocenter.png",
"Definition:Perpendicular Bisector",
"Definition:Polygon/Side",
"Circumscribing Circle about Triangle",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Circumcircle of Triangle/Circumcenter",
"Perpendicular Bisector of Triangle is Altitud... |
proofwiki-12193 | Position of Centroid on Euler Line | Let $\triangle ABC$ be a triangle which is not equilateral.
Let $O$ be the circumcenter of $\triangle ABC$.
Let $G$ be the centroid of $\triangle ABC$.
Let $H$ be the orthocenter of $\triangle ABC$.
Then $G$ lies on the straight line connecting $O$ and $H$ such that:
:$OG : GH = 1 : 2$
The line $OGH$ is the '''Euler li... | First it is necessary to dispose of the case where $\triangle ABC$ is equilateral.
From Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral, in that case $O$, $G$ and $H$ are the same point.
For all other triangles, $O$, $G$ and $H$ are distinct.
:500px
Let $A'$ be the midpoint of $BC$.
Let $B'$... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]] which is not [[Definition:Equilateral Triangle|equilateral]].
Let $O$ be the [[Definition:Circumcenter of Triangle|circumcenter]] of $\triangle ABC$.
Let $G$ be the [[Definition:Centroid of Triangle|centroid]] of $\triangle ABC$.
Let $H$ be the [[D... | First it is necessary to dispose of the case where $\triangle ABC$ is [[Definition:Equilateral Triangle|equilateral]].
From [[Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral]], in that case $O$, $G$ and $H$ are the same [[Definition:Point|point]].
For all other [[Definition:Triangle (Geome... | Position of Centroid on Euler Line | https://proofwiki.org/wiki/Position_of_Centroid_on_Euler_Line | https://proofwiki.org/wiki/Position_of_Centroid_on_Euler_Line | [
"Euler Line",
"Orthocenters of Triangles",
"Circumcircles of Triangles",
"Centroids of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Centroid/Triangle",
"Definition:Orthocenter",
"Definition:Line/Straight Line",
"Definition:Euler Line"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral",
"Definition:Point",
"Definition:Triangle (Geometry)",
"Definition:Distinct",
"File:EulerLineProof.png",
"Definition:Line/Midpoint",
"Definition:Line/Midpoint",
"Definition:Cen... |
proofwiki-12194 | Perpendicular Bisector of Triangle is Altitude of Medial Triangle | Let $\triangle ABC$ be a triangle.
Let $\triangle DEF$ be the medial triangle of $\triangle ABC$.
Let a perpendicular bisector be constructed on $AC$ at $F$ to intersect $DE$ at $P$.
Then $FP$ is an altitude of $\triangle DEF$. | :500px
Consider the triangles $\triangle ABC$ and $\triangle DBE$.
By the Midline Theorem:
:$BA : BD = 2 : 1$
:$BC : BE = 2 : 1$
and:
:$AC \parallel DE$
From Parallelism implies Equal Alternate Angles:
:$\angle AFP = \angle FPE$
By construction, $\angle AFP$ is a right angle.
Thus $\angle FPE$ is also a right angle.
Th... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle DEF$ be the [[Definition:Medial Triangle|medial triangle]] of $\triangle ABC$.
Let a [[Definition:Perpendicular Bisector|perpendicular bisector]] be constructed on $AC$ at $F$ to [[Definition:Intersection (Geometry)|intersect]] $DE$ ... | :[[File:PerpendicularBisectorAltitudeOfMedial.png|500px]]
Consider the [[Definition:Triangle (Geometry)|triangles]] $\triangle ABC$ and $\triangle DBE$.
By the [[Midline Theorem]]:
:$BA : BD = 2 : 1$
:$BC : BE = 2 : 1$
and:
:$AC \parallel DE$
From [[Parallelism implies Equal Alternate Angles]]:
:$\angle AFP = \angl... | Perpendicular Bisector of Triangle is Altitude of Medial Triangle | https://proofwiki.org/wiki/Perpendicular_Bisector_of_Triangle_is_Altitude_of_Medial_Triangle | https://proofwiki.org/wiki/Perpendicular_Bisector_of_Triangle_is_Altitude_of_Medial_Triangle | [
"Perpendicular Bisectors",
"Medial Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Medial Triangle",
"Definition:Perpendicular Bisector",
"Definition:Intersection (Geometry)",
"Definition:Altitude of Triangle"
] | [
"File:PerpendicularBisectorAltitudeOfMedial.png",
"Definition:Triangle (Geometry)",
"Midline Theorem",
"Parallelism implies Equal Alternate Angles",
"Definition:Right Angle",
"Definition:Right Angle",
"Definition:Right Angle/Perpendicular",
"Definition:Polygon/Vertex",
"Definition:Altitude of Triang... |
proofwiki-12195 | Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral | Let $\triangle ABC$ be a triangle.
Let $O$ be the circumcenter of $\triangle ABC$.
Let $G$ be the centroid of $\triangle ABC$.
Let $H$ be the orthocenter of $\triangle ABC$.
Then $O$, $G$ and $H$ are the same points {{iff}} $\triangle ABC$ is equilateral.
If $\triangle ABC$ is not equilateral, then $O$, $G$ and $H$ are... | === Necessary Condition ===
Let $\triangle ABC$ be an equilateral triangle.
By definition, each side of $\triangle ABC$ is the base of an isosceles triangle.
:400px
Let $AE$, $BF$ and $CD$ be the altitudes of $\triangle ABC$ through $A$, $B$ and $C$ respectively.
From Altitudes of Triangle Meet at Point, let them inter... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $O$ be the [[Definition:Circumcenter of Triangle|circumcenter]] of $\triangle ABC$.
Let $G$ be the [[Definition:Centroid of Triangle|centroid]] of $\triangle ABC$.
Let $H$ be the [[Definition:Orthocenter|orthocenter]] of $\triangle ABC$.
The... | === Necessary Condition ===
Let $\triangle ABC$ be an [[Definition:Equilateral Triangle|equilateral triangle]].
By definition, each [[Definition:Side of Polygon|side]] of $\triangle ABC$ is the [[Definition:Base of Isosceles Triangle|base]] of an [[Definition:Isosceles Triangle|isosceles triangle]].
:[[File:Equilate... | Orthocenter, Centroid and Circumcenter Coincide iff Triangle is Equilateral | https://proofwiki.org/wiki/Orthocenter,_Centroid_and_Circumcenter_Coincide_iff_Triangle_is_Equilateral | https://proofwiki.org/wiki/Orthocenter,_Centroid_and_Circumcenter_Coincide_iff_Triangle_is_Equilateral | [
"Equilateral Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Circumcircle of Triangle/Circumcenter",
"Definition:Centroid/Triangle",
"Definition:Orthocenter",
"Definition:Point",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Distinct"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Triangle (Geometry)/Isosceles",
"File:EquilateralWithAltitudes.png",
"Definition:Altitude of Triangle",
"Altitudes of Triangle Meet at Point",
"Definition:Intersection ... |
proofwiki-12196 | Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles | Let $\triangle ABC$ be a triangle.
Then:
::the altitude from $AB$ to $C$
::the median from $AB$ to $C$
::the perpendicular bisector of $AB$
:are all the same straight line
{{iff}}:
:$\triangle ABC$ is isosceles where $AB$ is the base. | === Necessary Condition ===
Let $\triangle ABC$ be an isosceles triangle whose base is $AB$.
Let $D$ be the midpoint of $AB$.
:300px
By definition of isosceles triangle, $AC = BC$.
We have $AD = DB$ by construction, and $CD$ is common.
So by Triangle Side-Side-Side Congruence:
:$\triangle ACD = \triangle BCD$
From Two ... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Then:
::the [[Definition:Altitude of Triangle|altitude]] from $AB$ to $C$
::the [[Definition:Median of Triangle|median]] from $AB$ to $C$
::the [[Definition:Perpendicular Bisector|perpendicular bisector]] of $AB$
:are all the same [[Definition:Stra... | === Necessary Condition ===
Let $\triangle ABC$ be an [[Definition:Isosceles Triangle|isosceles triangle]] whose [[Definition:Base of Isosceles Triangle|base]] is $AB$.
Let $D$ be the [[Definition:Midpoint of Line|midpoint]] of $AB$.
:[[File:IsoscelesAltitudeMedianPerpBis.png|300px]]
By definition of [[Definition... | Altitude, Median and Perpendicular Bisector Coincide iff Triangle is Isosceles | https://proofwiki.org/wiki/Altitude,_Median_and_Perpendicular_Bisector_Coincide_iff_Triangle_is_Isosceles | https://proofwiki.org/wiki/Altitude,_Median_and_Perpendicular_Bisector_Coincide_iff_Triangle_is_Isosceles | [
"Isosceles Triangles",
"Perpendicular Bisectors",
"Medians of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Altitude of Triangle",
"Definition:Median of Triangle",
"Definition:Perpendicular Bisector",
"Definition:Line/Straight Line",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Base"
] | [
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Triangle (Geometry)/Isosceles/Base",
"Definition:Line/Midpoint",
"File:IsoscelesAltitudeMedianPerpBis.png",
"Definition:Triangle (Geometry)/Isosceles",
"Triangle Side-Side-Side Congruence",
"Two Angles on Straight Line make Two Right Angles",
"De... |
proofwiki-12197 | Three Regular Tessellations | There exist exactly $3$ regular tessellations of the plane. | Let $m$ be the number of sides of each of the regular polygons that form the regular tessellation.
Let $n$ be the number of those regular polygons which meet at each vertex.
From Internal Angles of Regular Polygon, the internal angles of each polygon measure $\dfrac {\paren {m - 2} 180 \degrees} m$.
The sum of the inte... | There exist exactly $3$ [[Definition:Regular Tessellation|regular tessellations]] of [[Definition:The Plane|the plane]]. | Let $m$ be the number of [[Definition:Side of Polygon|sides]] of each of the [[Definition:Regular Polygon|regular polygons]] that form the [[Definition:Regular Tessellation|regular tessellation]].
Let $n$ be the number of those [[Definition:Regular Polygon|regular polygons]] which meet at each [[Definition:Vertex of P... | Three Regular Tessellations | https://proofwiki.org/wiki/Three_Regular_Tessellations | https://proofwiki.org/wiki/Three_Regular_Tessellations | [
"Three Regular Tessellations",
"Regular Tessellations",
"3"
] | [
"Definition:Regular Tessellation",
"Definition:Plane Surface/The Plane"
] | [
"Definition:Polygon/Side",
"Definition:Polygon/Regular",
"Definition:Regular Tessellation",
"Definition:Polygon/Regular",
"Definition:Polygon/Vertex",
"Internal Angles of Regular Polygon",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Regular",
"Definition:Polygon/Internal Angle",
"Sum ... |
proofwiki-12198 | Integer is Sum of Three Triangular Numbers | Let $n$ be a positive integer.
Then $n$ is the sum of $3$ triangular numbers. | From Integer as Sum of Three Odd Squares, $8 n + 3$ is the sum of $3$ odd squares.
So:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 8 n + 3
| r = \paren {2 x + 1}^2 + \paren {2 y + 1}^2 + \paren {2 z + 1}^2
| c = for some $x, y, z \in \Z_{\ge 0}$
}}
{{eqn | r = 4 x^2 + 4 x + 4 y^2 + 4 y + ... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Then $n$ is the sum of $3$ [[Definition:Triangular Number|triangular numbers]]. | From [[Integer as Sum of Three Odd Squares]], $8 n + 3$ is the sum of $3$ [[Definition:Odd Integer|odd]] [[Definition:Square Number|squares]].
So:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{\ge 0}
| l = 8 n + 3
| r = \paren {2 x + 1}^2 + \paren {2 y + 1}^2 + \paren {2 z + 1}^2
| c = for some $x, y,... | Integer is Sum of Three Triangular Numbers | https://proofwiki.org/wiki/Integer_is_Sum_of_Three_Triangular_Numbers | https://proofwiki.org/wiki/Integer_is_Sum_of_Three_Triangular_Numbers | [
"Triangular Numbers"
] | [
"Definition:Positive/Integer",
"Definition:Triangular Number"
] | [
"Integer as Sum of Three Odd Squares",
"Definition:Odd Integer",
"Definition:Square Number",
"Closed Form for Triangular Numbers",
"Definition:Triangular Number"
] |
proofwiki-12199 | Smallest Magic Square is of Order 3 | Apart from the trivial order $1$ magic square:
{{:Magic Square/Examples/Order 1}}
the smallest magic square is the order $3$ magic square:
{{:Magic Square/Examples/Order 3}} | Suppose there were an order $2$ magic square.
The row and column total is $\dfrac {1 + 2 + 3 + 4} 2 = 5$.
Any row or column with a $1$ in it must therefore also have a $4$ in it.
But there are:
:one row
:one column
:one diagonal
all of which have a $1$ in them.
Therefore the $4$ would need to go in all $3$ cells.
But $... | Apart from the trivial [[Magic Square/Examples/Order 1|order $1$ magic square]]:
{{:Magic Square/Examples/Order 1}}
the smallest [[Definition:Magic Square|magic square]] is the [[Magic Square/Examples/Order 3|order $3$ magic square]]:
{{:Magic Square/Examples/Order 3}} | Suppose there were an [[Definition:Order of Magic Square|order $2$]] [[Definition:Magic Square|magic square]].
The row and column total is $\dfrac {1 + 2 + 3 + 4} 2 = 5$.
Any row or column with a $1$ in it must therefore also have a $4$ in it.
But there are:
:one row
:one column
:one diagonal
all of which have a $1$... | Smallest Magic Square is of Order 3 | https://proofwiki.org/wiki/Smallest_Magic_Square_is_of_Order_3 | https://proofwiki.org/wiki/Smallest_Magic_Square_is_of_Order_3 | [
"Magic Squares"
] | [
"Magic Square/Examples/Order 1",
"Definition:Magic Square",
"Magic Square/Examples/Order 3"
] | [
"Definition:Magic Square/Order",
"Definition:Magic Square",
"Definition:Magic Square",
"Definition:Magic Square/Order",
"Definition:Magic Square"
] |
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