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-15500 | Congruence Modulo Power of p as Linear Combination of Congruences Modulo p | Let $p$ be a prime number.
Let $S = \set {a_1, a_2, \ldots, a_p}$ be a complete residue system modulo $p$.
Then for all integers $n \in \Z$ and non-negative integer $s \in \Z_{\ge 0}$, there exists a congruence of the form:
:$n \equiv \ds \sum_{j \mathop = 0}^s b_j p^j \pmod {p^{s + 1} }$
where $b_j \in S$. | Proof by induction on $s$: | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $S = \set {a_1, a_2, \ldots, a_p}$ be a [[Definition:Complete Residue System|complete residue system modulo $p$]].
Then for all [[Definition:Integer|integers]] $n \in \Z$ and [[Definition:Non-Negative Integer|non-negative integer]] $s \in \Z_{\ge 0}$, there ... | Proof by [[Principle of Mathematical Induction|induction]] on $s$: | Congruence Modulo Power of p as Linear Combination of Congruences Modulo p | https://proofwiki.org/wiki/Congruence_Modulo_Power_of_p_as_Linear_Combination_of_Congruences_Modulo_p | https://proofwiki.org/wiki/Congruence_Modulo_Power_of_p_as_Linear_Combination_of_Congruences_Modulo_p | [
"Residue Classes",
"Proofs by Induction"
] | [
"Definition:Prime Number",
"Definition:Complete Residue System",
"Definition:Integer",
"Definition:Positive/Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-15501 | Number of Non-Dividing Primes Less than n is Less than Euler Phi Function of n | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$.
Let $\map \phi n$ denote the Euler $\phi$ function of $n$.
Then:
:$\map w n < \map \phi n$ | Let $P = \set {p < n: p \text { prime}, p \nmid n}$.
Let $Q = \set {0 < q < n: q \perp n}$, where $q \perp n$ denotes that $q$ and $n$ are coprime.
Let $p \in P$.
From Prime not Divisor implies Coprime, $p$ is coprime to $n$.
That is:
:$p \in Q$
So, by definition of subset:
:$P \subseteq Q$
From Integer is Coprime to 1... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\map w n$ denote the number of [[Definition:Prime Number|primes]] strictly less than $n$ which are not [[Definition:Divisor of Integer|divisors]] of $n$.
Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler... | Let $P = \set {p < n: p \text { prime}, p \nmid n}$.
Let $Q = \set {0 < q < n: q \perp n}$, where $q \perp n$ denotes that $q$ and $n$ are [[Definition:Coprime Integers|coprime]].
Let $p \in P$.
From [[Prime not Divisor implies Coprime]], $p$ is [[Definition:Coprime Integers|coprime]] to $n$.
That is:
:$p \in Q$
... | Number of Non-Dividing Primes Less than n is Less than Euler Phi Function of n | https://proofwiki.org/wiki/Number_of_Non-Dividing_Primes_Less_than_n_is_Less_than_Euler_Phi_Function_of_n | https://proofwiki.org/wiki/Number_of_Non-Dividing_Primes_Less_than_n_is_Less_than_Euler_Phi_Function_of_n | [
"Euler Phi Function",
"Prime Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Euler Phi Function"
] | [
"Definition:Coprime/Integers",
"Prime not Divisor implies Coprime",
"Definition:Coprime/Integers",
"Definition:Subset",
"Integer is Coprime to 1",
"One is not Prime",
"Definition:Euler Phi Function"
] |
proofwiki-15502 | Schatunowsky's Theorem | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $\map w n$ denote the number of primes strictly less than $n$ which are not divisors of $n$.
Let $\map \phi n$ denote the Euler $\phi$ function of $n$.
Then $30$ is the largest integer $n$ such that:
:$\map w n = \map \phi n - 1$ | The above equation is equivalent to the property that all numbers greater than $1$ that are coprime to it but less are prime.
For an integer to have this property:
If it is greater than $p^2$ for some prime $p$, then it must be divisible by $p$.
If not, it will be coprime to $p^2$, a composite number.
Let $p_n$ denote ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $\map w n$ denote the number of [[Definition:Prime Number|primes]] strictly less than $n$ which are not [[Definition:Divisor of Integer|divisors]] of $n$.
Let $\map \phi n$ denote the [[Definition:Euler Phi Function|Euler... | The above equation is equivalent to the property that all numbers greater than $1$ that are [[Definition:Coprime Integers|coprime]] to it but less are [[Definition:Prime Number|prime]].
For an [[Definition:Integer|integer]] to have this property:
If it is greater than $p^2$ for some [[Definition:Prime Number|prime]] ... | Schatunowsky's Theorem | https://proofwiki.org/wiki/Schatunowsky's_Theorem | https://proofwiki.org/wiki/Schatunowsky's_Theorem | [
"Euler Phi Function",
"Prime Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Euler Phi Function",
"Definition:Integer"
] | [
"Definition:Coprime/Integers",
"Definition:Prime Number",
"Definition:Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Coprime/Integers",
"Definition:Composite Number",
"Definition:Prime Number",
"Absolute Value of Integer is not less than Divisors",
"Bertra... |
proofwiki-15503 | Position of Card after n Modified Perfect Faro Shuffles | Let $D$ be a deck of cards $D$ of size $2 r$.
Let $C$ be a card in position $x$ of $D$.
Let $n$ modified perfect faro shuffles be performed on $C$.
Then $C$ will be in position $w$, where:
:$w \equiv 2^n x \pmod {2 r + 1}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:After $n$ modified perfect faro shuffles, $C$ will be in position $a_n$ where $a_n \equiv 2^n x \pmod {2 r + 1}$ | Let $D$ be a [[Definition:Deck of Cards|deck of cards]] $D$ of size $2 r$.
Let $C$ be a [[Definition:Card|card]] in position $x$ of $D$.
Let $n$ [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]] be performed on $C$.
Then $C$ will be in position $w$, where:
:$w \equiv 2^n x \pmod {2 r + 1}$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:After $n$ [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]], $C$ will be in position $a_n$ where $a_n \equiv 2^n x \pmod {2 r + ... | Position of Card after n Modified Perfect Faro Shuffles | https://proofwiki.org/wiki/Position_of_Card_after_n_Modified_Perfect_Faro_Shuffles | https://proofwiki.org/wiki/Position_of_Card_after_n_Modified_Perfect_Faro_Shuffles | [
"Cards"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Modified Perfect Faro Shuffle"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Modified Perfect Faro Shuffle",
"Principle of Mat... |
proofwiki-15504 | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order | Let $D$ be a deck of $2 m$ cards.
Let $D$ be given a sequence of modified perfect faro shuffles.
Then the cards of $D$ will return to their original order after $n$ such shuffles, where:
:$2^n \equiv 1 \pmod {2 m + 1}$ | From Position of Card after n Modified Perfect Faro Shuffles, after $n$ shuffles a card in position $x$ will be in position $2^n x \pmod {m + 1}$.
So for all $2 m$ cards in $D$, we need to find $n$ such that:
:$2^n x \equiv x \pmod {2 m + 1}$
Because $2 m + 1$ is odd, we have:
:$\gcd \set {2, 2 m + 1}$
and so from Canc... | Let $D$ be a [[Definition:Deck of Cards|deck]] of $2 m$ [[Definition:Card|cards]].
Let $D$ be given a [[Definition:Sequence|sequence]] of [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]].
Then the [[Definition:Card|cards]] of $D$ will return to their original [[Definition:Order of Cards|or... | From [[Position of Card after n Modified Perfect Faro Shuffles]], after $n$ [[Definition:Shuffle|shuffles]] a [[Definition:Card|card]] in position $x$ will be in position $2^n x \pmod {m + 1}$.
So for all $2 m$ [[Definition:Card|cards]] in $D$, we need to find $n$ such that:
:$2^n x \equiv x \pmod {2 m + 1}$
Because ... | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order | [
"Cards"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Sequence",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards",
"Definition:Shuffle"
] | [
"Position of Card after n Modified Perfect Faro Shuffles",
"Definition:Shuffle",
"Definition:Deck of Cards/Card",
"Definition:Deck of Cards/Card",
"Definition:Odd Integer",
"Cancellability of Congruences"
] |
proofwiki-15505 | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 62 Cards | Let $D$ be a deck of $62$ cards.
Let $D$ be given a sequence of modified perfect faro shuffles.
Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in. | From Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order, the cards of $D$ will return to their original order after $n$ such shuffles, where:
:$2^n \equiv 1 \pmod {63}$
We have that:
:$63 = 2^6 - 1$
and so:
:$2^6 \equiv 1 \pmod {63}$
Hence the result.
{{qed}} | Let $D$ be a [[Definition:Deck of Cards|deck]] of $62$ [[Definition:Card|cards]].
Let $D$ be given a [[Definition:Sequence|sequence]] of [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]].
Then after $6$ such [[Definition:Shuffle|shuffles]], the [[Definition:Card|cards]] of $D$ will be in the... | From [[Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order]], the [[Definition:Card|cards]] of $D$ will return to their original [[Definition:Order of Cards|order]] after $n$ such [[Definition:Shuffle|shuffles]], where:
:$2^n \equiv 1 \pmod {63}$
We have that:
:$63 = 2^6 - 1$
and so:
:$... | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 62 Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_62_Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_62_Cards | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"62"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Sequence",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Shuffle",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards"
] | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards",
"Definition:Shuffle"
] |
proofwiki-15506 | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards | Let $D$ be a deck of $8$ cards.
Let $D$ be given a sequence of modified perfect faro shuffles.
Then after $6$ such shuffles, the cards of $D$ will be in the same order they started in. | From Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order, the cards of $D$ will return to their original order after $n$ such shuffles, where:
:$2^n \equiv 1 \pmod 9$
Inspecting $2^n$ for $n$ from $1$:
{{begin-eqn}}
{{eqn | l = 2^1
| o = \equiv
| r = 2
| rr= \pmod 9
... | Let $D$ be a [[Definition:Deck of Cards|deck]] of $8$ [[Definition:Card|cards]].
Let $D$ be given a [[Definition:Sequence|sequence]] of [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]].
Then after $6$ such [[Definition:Shuffle|shuffles]], the [[Definition:Card|cards]] of $D$ will be in the ... | From [[Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order]], the [[Definition:Card|cards]] of $D$ will return to their original [[Definition:Order of Cards|order]] after $n$ such [[Definition:Shuffle|shuffles]], where:
:$2^n \equiv 1 \pmod 9$
Inspecting $2^n$ for $n$ from $1$:
{{begin-... | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 8 Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_8_Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_8_Cards | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"8"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Sequence",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Shuffle",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards"
] | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards",
"Definition:Shuffle"
] |
proofwiki-15507 | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards | Let $D$ be a deck of $12$ cards.
Let $D$ be given a sequence of modified perfect faro shuffles.
Then after $12$ such shuffles, the cards of $D$ will be in the same order they started in. | From Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order, the cards of $D$ will return to their original order after $n$ such shuffles, where:
:$2^n \equiv 1 \pmod {13}$
From Fermat's Little Theorem:
:$2^{12} \equiv 1 \pmod {13}$
so we know that $n$ is at most $12$.
But $n$ may be smaller... | Let $D$ be a [[Definition:Deck of Cards|deck]] of $12$ [[Definition:Card|cards]].
Let $D$ be given a [[Definition:Sequence|sequence]] of [[Definition:Modified Perfect Faro Shuffle|modified perfect faro shuffles]].
Then after $12$ such [[Definition:Shuffle|shuffles]], the [[Definition:Card|cards]] of $D$ will be in th... | From [[Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order]], the [[Definition:Card|cards]] of $D$ will return to their original [[Definition:Order of Cards|order]] after $n$ such [[Definition:Shuffle|shuffles]], where:
:$2^n \equiv 1 \pmod {13}$
From [[Fermat's Little Theorem]]:
:$2^{12... | Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order/Examples/Deck of 12 Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_12_Cards | https://proofwiki.org/wiki/Number_of_Modified_Perfect_Faro_Shuffles_to_return_Deck_of_Cards_to_Original_Order/Examples/Deck_of_12_Cards | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"12"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Sequence",
"Definition:Modified Perfect Faro Shuffle",
"Definition:Shuffle",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards"
] | [
"Number of Modified Perfect Faro Shuffles to return Deck of Cards to Original Order",
"Definition:Deck of Cards/Card",
"Definition:Order of Cards",
"Definition:Shuffle",
"Fermat's Little Theorem"
] |
proofwiki-15508 | Difference of Two Fourth Powers | :$x^4 - y^4 = \paren {x - y} \paren {x + y} \paren {x^2 + y^2}$ | {{begin-eqn}}
{{eqn | l = x^4 - y^4
| r = \paren {x^2}^2 - \paren {y^2}^2
| c =
}}
{{eqn | r = \paren {x^2 - y^2} \paren {x^2 + y^2}
| c = Difference of Two Squares
}}
{{eqn | r = \paren {x - y} \paren {x + y} \paren {x^2 + y^2}
| c = Difference of Two Squares
}}
{{end-eqn}}
{{qed}} | :$x^4 - y^4 = \paren {x - y} \paren {x + y} \paren {x^2 + y^2}$ | {{begin-eqn}}
{{eqn | l = x^4 - y^4
| r = \paren {x^2}^2 - \paren {y^2}^2
| c =
}}
{{eqn | r = \paren {x^2 - y^2} \paren {x^2 + y^2}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \paren {x - y} \paren {x + y} \paren {x^2 + y^2}
| c = [[Difference of Two Squares]]
}}
{{end-eqn}}
{{qed}} | Difference of Two Fourth Powers/Proof 1 | https://proofwiki.org/wiki/Difference_of_Two_Fourth_Powers | https://proofwiki.org/wiki/Difference_of_Two_Fourth_Powers/Proof_1 | [
"Difference of Two Fourth Powers",
"Fourth Powers",
"Examples of Use of Difference of Two Powers"
] | [] | [
"Difference of Two Squares",
"Difference of Two Squares"
] |
proofwiki-15509 | Sum of Two Odd Powers | Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {-1}^j a^{2 n - j} b^j
| c =
}}
{{eqn | r = \paren {a + b} \paren {a... | {{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = a^{2 n + 1} - \paren {-\paren {b^{2 n + 1} } }
| c =
}}
{{eqn | r = a^{2 n + 1} - \paren {-b}^{2 n + 1}
| c = as $n$ is odd
}}
{{eqn | r = \paren {a - \paren {-b} } \sum_{j \mathop = 0}^{2 n} a^{2 n - j} \paren {-b}^j
| c = Difference of... | Let $\F$ be one of the [[Definition:Standard Number System|standard number systems]], that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {... | {{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = a^{2 n + 1} - \paren {-\paren {b^{2 n + 1} } }
| c =
}}
{{eqn | r = a^{2 n + 1} - \paren {-b}^{2 n + 1}
| c = as $n$ is [[Definition:Odd Integer|odd]]
}}
{{eqn | r = \paren {a - \paren {-b} } \sum_{j \mathop = 0}^{2 n} a^{2 n - j} \paren {-b}^... | Sum of Two Odd Powers | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers | [
"Sum of Two Odd Powers",
"Sum of Two Powers",
"Polynomial Theory",
"Algebra"
] | [
"Definition:Number",
"Definition:Positive/Integer"
] | [
"Definition:Odd Integer",
"Difference of Two Powers"
] |
proofwiki-15510 | Sum of Two Odd Powers | Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {-1}^j a^{2 n - j} b^j
| c =
}}
{{eqn | r = \paren {a + b} \paren {a... | From Difference of Two Powers:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $x = a$ and $y = -b$.
Then:
{{begin-eqn}}
{{eqn | l = x^3 + y^3
| r = x^3 - \paren {-y^3}
| c =
}}
{{eqn | r = x^3 - \paren {-y}^3
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } \sum_{... | Let $\F$ be one of the [[Definition:Standard Number System|standard number systems]], that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {... | From [[Difference of Two Powers]]:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
Let $x = a$ and $y = -b$.
Then:
{{begin-eqn}}
{{eqn | l = x^3 + y^3
| r = x^3 - \paren {-y^3}
| c =
}}
{{eqn | r = x^3 - \paren {-y}^3
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } ... | Sum of Two Odd Powers/Examples/Sum of Two Cubes/Proof 1 | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes/Proof_1 | [
"Sum of Two Odd Powers",
"Sum of Two Powers",
"Polynomial Theory",
"Algebra"
] | [
"Definition:Number",
"Definition:Positive/Integer"
] | [
"Difference of Two Powers"
] |
proofwiki-15511 | Sum of Two Odd Powers | Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {-1}^j a^{2 n - j} b^j
| c =
}}
{{eqn | r = \paren {a + b} \paren {a... | From Sum of Two Odd Powers:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $3 = 2 \times 1 + 1$.
Hence setting $n = 1$ gives the result.
{{qed}} | Let $\F$ be one of the [[Definition:Standard Number System|standard number systems]], that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} + b^{2 n + 1}
| r = \paren {a + b} \sum_{j \mathop = 0}^{2 n} \paren {... | From [[Sum of Two Odd Powers]]:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $3 = 2 \times 1 + 1$.
Hence setting $n = 1$ gives the result.
{{qed}} | Sum of Two Odd Powers/Examples/Sum of Two Cubes/Proof 2 | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers | https://proofwiki.org/wiki/Sum_of_Two_Odd_Powers/Examples/Sum_of_Two_Cubes/Proof_2 | [
"Sum of Two Odd Powers",
"Sum of Two Powers",
"Polynomial Theory",
"Algebra"
] | [
"Definition:Number",
"Definition:Positive/Integer"
] | [
"Sum of Two Odd Powers"
] |
proofwiki-15512 | Difference of Two Fifth Powers | :$x^5 - y^5 = \paren {x - y} \paren {x^4 + x^3 y + x^2 y^2 + x y^3 + y^4}$ | From Difference of Two Powers:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
The result follows directly by setting $n = 5$.
{{qed}} | :$x^5 - y^5 = \paren {x - y} \paren {x^4 + x^3 y + x^2 y^2 + x y^3 + y^4}$ | From [[Difference of Two Powers]]:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
The result follows directly by setting $n = 5$.
{{qed}} | Difference of Two Fifth Powers | https://proofwiki.org/wiki/Difference_of_Two_Fifth_Powers | https://proofwiki.org/wiki/Difference_of_Two_Fifth_Powers | [
"Fifth Powers",
"Examples of Use of Difference of Two Powers"
] | [] | [
"Difference of Two Powers"
] |
proofwiki-15513 | Sum of Two Fifth Powers | :$x^5 + y^5 = \paren {x + y} \paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$ | From Sum of Two Odd Powers:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $5 = 2 \times 2 + 1$.
The result follows directly by setting $n = 2$.
{{qed}} | :$x^5 + y^5 = \paren {x + y} \paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$ | From [[Sum of Two Odd Powers]]:
:$a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$
We have that $5 = 2 \times 2 + 1$.
The result follows directly by setting $n = 2$.
{{qed}} | Sum of Two Fifth Powers | https://proofwiki.org/wiki/Sum_of_Two_Fifth_Powers | https://proofwiki.org/wiki/Sum_of_Two_Fifth_Powers | [
"Fifth Powers",
"Examples of Use of Sum of Two Odd Powers"
] | [] | [
"Sum of Two Odd Powers"
] |
proofwiki-15514 | Difference of Two Sixth Powers | :$x^6 - y^6 = \paren {x - y} \paren {x + y} \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$ | {{begin-eqn}}
{{eqn | l = x^6 - y^6
| r = \paren {x^3}^2 - \paren {y^3}^2
| c =
}}
{{eqn | r = \paren {x^3 - y^3} \paren {x^3 + y^3}
| c =
}}
{{eqn | r = \paren {x - y} \paren {x^2 + x y + y^2} \paren {x^3 + y^3}
| c = Difference of Two Cubes
}}
{{eqn | r = \paren {x - y} \paren {x^2 + x y + y... | :$x^6 - y^6 = \paren {x - y} \paren {x + y} \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$ | {{begin-eqn}}
{{eqn | l = x^6 - y^6
| r = \paren {x^3}^2 - \paren {y^3}^2
| c =
}}
{{eqn | r = \paren {x^3 - y^3} \paren {x^3 + y^3}
| c =
}}
{{eqn | r = \paren {x - y} \paren {x^2 + x y + y^2} \paren {x^3 + y^3}
| c = [[Difference of Two Cubes]]
}}
{{eqn | r = \paren {x - y} \paren {x^2 + x y... | Difference of Two Sixth Powers | https://proofwiki.org/wiki/Difference_of_Two_Sixth_Powers | https://proofwiki.org/wiki/Difference_of_Two_Sixth_Powers | [
"Sixth Powers",
"Examples of Use of Difference of Two Powers"
] | [] | [
"Difference of Two Powers/Examples/Difference of Two Cubes",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-15515 | Sum of Fourth Powers with Product of Squares | :$x^4 + x^2 y^2 + y^4 = \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$ | {{begin-eqn}}
{{eqn | l = x^6 - y^6
| r = \paren {x - y} \paren {x + y} \paren {x^2 + 2 x y + 2 y^2} \paren {x^2 - 2 x y + 2 y^2}
| c = Difference of Two Sixth Powers
}}
{{eqn | r = \paren {x^2 - y^2} \paren {x^2 + 2 x y + 2 y^2} \paren {x^2 - 2 x y + 2 y^2}
| c = Difference of Two Squares
}}
{{end-eq... | :$x^4 + x^2 y^2 + y^4 = \paren {x^2 + x y + y^2} \paren {x^2 - x y + y^2}$ | {{begin-eqn}}
{{eqn | l = x^6 - y^6
| r = \paren {x - y} \paren {x + y} \paren {x^2 + 2 x y + 2 y^2} \paren {x^2 - 2 x y + 2 y^2}
| c = [[Difference of Two Sixth Powers]]
}}
{{eqn | r = \paren {x^2 - y^2} \paren {x^2 + 2 x y + 2 y^2} \paren {x^2 - 2 x y + 2 y^2}
| c = [[Difference of Two Squares]]
}}
... | Sum of Fourth Powers with Product of Squares | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_with_Product_of_Squares | https://proofwiki.org/wiki/Sum_of_Fourth_Powers_with_Product_of_Squares | [
"Fourth Powers",
"Square Function"
] | [] | [
"Difference of Two Sixth Powers",
"Difference of Two Squares",
"Difference of Two Powers/Examples/Difference of Two Cubes"
] |
proofwiki-15516 | Difference of Two Odd Powers | Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} - b^{2 n + 1}
| r = \paren {a - b} \sum_{j \mathop = 0}^{2 n} a^{2 n - j} b^j
| c =
}}
{{eqn |... | A direct application of Difference of Two Powers:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }
| c =
}}
{{end-eqn}}
and ... | Let $\mathbb F$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \Z_{\ge 0}$ be a [[Definition:Positive Integer|positive integer]].
Then for all $a, b \in \mathbb F$:
{{begin-eqn}}
{{eqn | l = a^{2 n + 1} - b^{2 n + 1}
| r = \paren ... | A direct application of [[Difference of Two Powers]]:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j
| c =
}}
{{eqn | r = \paren {a - b} \paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \dotsb + a b^{n - 2} + b^{n - 1} }
| c =
}}
{{end-eqn}}... | Difference of Two Odd Powers | https://proofwiki.org/wiki/Difference_of_Two_Odd_Powers | https://proofwiki.org/wiki/Difference_of_Two_Odd_Powers | [
"Difference of Two Powers"
] | [
"Definition:Number",
"Definition:Positive/Integer"
] | [
"Difference of Two Powers"
] |
proofwiki-15517 | Difference of Two Even-Times Odd Powers | Let $\F$ be one of the standard number systems, that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{> 0}$ be a (strictly) positive odd integer.
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a - b} \paren {a + b} \paren {\sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j} \paren {\sum_{j \mathop = 0}^{n -... | {{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a^n}^2 - \paren {b^n}^2
| c =
}}
{{eqn | r = \paren {a^n - b^n} \paren {a^n + b^n}
| c =
}}
{{eqn | r = \paren {a - b} \paren {\sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j} \paren {a^n + b^n}
| c = Difference of Two Powers
}}
{{eqn | ... | Let $\F$ be one of the [[Definition:Standard Number System|standard number systems]], that is $\Z, \Q, \R$ and so on.
Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive]] [[Definition:Odd Integer|odd integer]].
Then:
{{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a... | {{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a^n}^2 - \paren {b^n}^2
| c =
}}
{{eqn | r = \paren {a^n - b^n} \paren {a^n + b^n}
| c =
}}
{{eqn | r = \paren {a - b} \paren {\sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j} \paren {a^n + b^n}
| c = [[Difference of Two Powers]]
}}
{{eq... | Difference of Two Even-Times Odd Powers | https://proofwiki.org/wiki/Difference_of_Two_Even-Times_Odd_Powers | https://proofwiki.org/wiki/Difference_of_Two_Even-Times_Odd_Powers | [
"Difference of Two Powers"
] | [
"Definition:Number",
"Definition:Strictly Positive/Integer",
"Definition:Odd Integer"
] | [
"Difference of Two Powers",
"Sum of Two Odd Powers"
] |
proofwiki-15518 | Difference of Two Even Powers | Let $\GF$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then for all $a, b \in \GF$:
{{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a - b} \paren {a + b} \sum_{j \mathop = 0}^{n - 1} a^{2 \paren {n - j - 1} } b^{2 j}
... | {{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a^2}^n - \paren {b^2}^n
| c =
}}
{{eqn | r = \paren {a^2 - b^2} \paren {\sum_{j \mathop = 0}^{n - 1} \paren {a^2} ^{n - j - 1} \paren {b^2}^j}
| c = Difference of Two Powers
}}
{{eqn | r = \paren {a - b} \paren {a + b} \sum_{j \mathop = 0}^{n ... | Let $\GF$ denote one of the [[Definition:Standard Number System|standard number systems]], that is $\Z$, $\Q$, $\R$ and $\C$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then for all $a, b \in \GF$:
{{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a ... | {{begin-eqn}}
{{eqn | l = a^{2 n} - b^{2 n}
| r = \paren {a^2}^n - \paren {b^2}^n
| c =
}}
{{eqn | r = \paren {a^2 - b^2} \paren {\sum_{j \mathop = 0}^{n - 1} \paren {a^2} ^{n - j - 1} \paren {b^2}^j}
| c = [[Difference of Two Powers]]
}}
{{eqn | r = \paren {a - b} \paren {a + b} \sum_{j \mathop = 0}... | Difference of Two Even Powers | https://proofwiki.org/wiki/Difference_of_Two_Even_Powers | https://proofwiki.org/wiki/Difference_of_Two_Even_Powers | [
"Difference of Two Powers"
] | [
"Definition:Number",
"Definition:Strictly Positive/Integer"
] | [
"Difference of Two Powers",
"Difference of Two Squares"
] |
proofwiki-15519 | Factors of Difference of Two Odd Powers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = x^{2 n + 1} - y^{2 n + 1}
| r = \paren {x - y} \prod_{k \mathop = 1}^n \paren {x^2 - 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}
| c =
}}
{{eqn | r = \paren {x - y} \paren {x^2 - 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2... | From Factorisation of $z^n - a$:
:$x^{2 n + 1} - y^{2 n + 1} = \ds \prod_{k \mathop = 0}^{2 n} \paren {x - \alpha^k y}$
where $\alpha$ is a primitive complex $2 n + 1$th roots of unity, for example:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n + 1} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = x^{2 n + 1} - y^{2 n + 1}
| r = \paren {x - y} \prod_{k \mathop = 1}^n \paren {x^2 - 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}
| c =
}}
{{eqn | r = \paren {x - y} \paren {x^2 ... | From [[Factorisation of z^n-a|Factorisation of $z^n - a$]]:
:$x^{2 n + 1} - y^{2 n + 1} = \ds \prod_{k \mathop = 0}^{2 n} \paren {x - \alpha^k y}$
where $\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $2 n + 1$th roots of unity]], for example:
{{begin-eqn}}
{{eqn | l = \alpha
| r =... | Factors of Difference of Two Odd Powers | https://proofwiki.org/wiki/Factors_of_Difference_of_Two_Odd_Powers | https://proofwiki.org/wiki/Factors_of_Difference_of_Two_Odd_Powers | [
"Difference of Two Powers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n-a",
"Definition:Root of Unity/Complex/Primitive",
"Complex Roots of Unity occur in Conjugate Pairs",
"Definition:Root of Unity/Complex",
"Definition:Multiplication/Complex Numbers",
"Complex Roots of Unity occur in Conjugate Pairs",
"Modulus in Terms of Conjugate",
"Modulus of Co... |
proofwiki-15520 | Factors of Sum of Two Odd Powers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
{{begin-eqn}}
{{eqn | l = x^{2 n + 1} + y^{2 n + 1}
| r = \paren {x + y} \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}
| c =
}}
{{eqn | r = \paren {x + y} \paren {x^2 + 2 x y \cos \dfrac {2 \pi} {2 n + 1} + y^2... | {{begin-eqn}}
{{eqn | l = x^{2 n + 1} + y^{2 n + 1}
| r = x^{2 n + 1} - \paren {-\paren {y^{2 n + 1} } }
| c =
}}
{{eqn | r = x^{2 n + 1} - \paren {-y}^{2 n + 1}
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } \prod_{k \mathop = 1}^n \paren {x^2 - 2 x \paren {-y} \cos \dfrac {2 \pi k} {2 n + 1} + \pa... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = x^{2 n + 1} + y^{2 n + 1}
| r = \paren {x + y} \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {2 \pi k} {2 n + 1} + y^2}
| c =
}}
{{eqn | r = \paren {x + y} \paren {x^2 ... | {{begin-eqn}}
{{eqn | l = x^{2 n + 1} + y^{2 n + 1}
| r = x^{2 n + 1} - \paren {-\paren {y^{2 n + 1} } }
| c =
}}
{{eqn | r = x^{2 n + 1} - \paren {-y}^{2 n + 1}
| c =
}}
{{eqn | r = \paren {x - \paren {-y} } \prod_{k \mathop = 1}^n \paren {x^2 - 2 x \paren {-y} \cos \dfrac {2 \pi k} {2 n + 1} + \pa... | Factors of Sum of Two Odd Powers | https://proofwiki.org/wiki/Factors_of_Sum_of_Two_Odd_Powers | https://proofwiki.org/wiki/Factors_of_Sum_of_Two_Odd_Powers | [
"Sum of Two Odd Powers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factors of Difference of Two Odd Powers"
] |
proofwiki-15521 | Factors of Difference of Two Even Powers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$\ds x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$ | From Factorisation of $z^n - a$:
:$\ds z^{2 n} - y^{2 n} = \prod_{k \mathop = 0}^{2 n - 1} \paren {x - \alpha^k y}$
where $\alpha$ is a primitive complex $2 n$th roots of unity, for example:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i \pi / \paren {2 n} }
| c =
}}
{{eqn | r = \cos \dfrac {2 \pi} {2 n} + ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$\ds x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$ | From [[Factorisation of z^n-a|Factorisation of $z^n - a$]]:
:$\ds z^{2 n} - y^{2 n} = \prod_{k \mathop = 0}^{2 n - 1} \paren {x - \alpha^k y}$
where $\alpha$ is a [[Definition:Primitive Complex Root of Unity|primitive complex $2 n$th roots of unity]], for example:
{{begin-eqn}}
{{eqn | l = \alpha
| r = e^{2 i ... | Factors of Difference of Two Even Powers | https://proofwiki.org/wiki/Factors_of_Difference_of_Two_Even_Powers | https://proofwiki.org/wiki/Factors_of_Difference_of_Two_Even_Powers | [
"Difference of Two Powers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n-a",
"Definition:Root of Unity/Complex/Primitive",
"Complex Roots of Unity occur in Conjugate Pairs",
"Definition:Root of Unity/Complex",
"Definition:Multiplication/Complex Numbers",
"Complex Roots of Unity occur in Conjugate Pairs",
"Modulus in Terms of Conjugate",
"Modulus of Co... |
proofwiki-15522 | Factors of Sum of Two Even Powers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$x^{2 n} + y^{2 n} = \ds \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {\paren {2 k - 1} \pi} {2 n} + y^2}$ | From Factorisation of $z^n + a$:
:$z^{2 n} + y^{2 n} = \ds \prod_{k \mathop = 0}^{2 n - 1} \paren {x - \alpha_k y}$
where $\alpha_k$ are the complex $2n$th roots of negative unity:
{{begin-eqn}}
{{eqn | l = \alpha_k
| r = e^{i \paren {2 k + 1} \pi / {2 n} }
| c = from Roots of Complex Number
}}
{{eqn | r = ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$x^{2 n} + y^{2 n} = \ds \prod_{k \mathop = 1}^n \paren {x^2 + 2 x y \cos \dfrac {\paren {2 k - 1} \pi} {2 n} + y^2}$ | From [[Factorisation of z^n+a|Factorisation of $z^n + a$]]:
:$z^{2 n} + y^{2 n} = \ds \prod_{k \mathop = 0}^{2 n - 1} \paren {x - \alpha_k y}$
where $\alpha_k$ are the complex $2n$th roots of negative unity:
{{begin-eqn}}
{{eqn | l = \alpha_k
| r = e^{i \paren {2 k + 1} \pi / {2 n} }
| c = from [[Roots o... | Factors of Sum of Two Even Powers | https://proofwiki.org/wiki/Factors_of_Sum_of_Two_Even_Powers | https://proofwiki.org/wiki/Factors_of_Sum_of_Two_Even_Powers | [
"Sum of Two Powers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Factorisation of z^n+a",
"Roots of Complex Number",
"Definition:Multiplication/Complex Numbers",
"Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs",
"Modulus in Terms of Conjugate",
"Modulus of Complex Root of Unity equals 1",
"Cosine of Angle plus Straight Angle",
"Cosine ... |
proofwiki-15523 | Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen | :The $r$-sphere of $x$, $\map {S_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$. | We have:
{{begin-eqn}}
{{eqn | l = \map {S_r} x
| r = \set {y \in R : \norm {y - x} = r}
| c = {{Defof|Sphere in Normed Division Ring}}
}}
{{eqn | r = \set {y \in R : \norm {y - x} \le r} \cap \set {y \in R : \norm{y - x} \ge r}
}}
{{eqn | r = \map { {B_r}^-} x \cap \paren {R \setminus \map {B_r} x}
|... | :The [[Definition:Sphere in Normed Division Ring|$r$-sphere of $x$]], $\map {S_r} x$, is both [[Definition:Open Set of Metric Space|open]] and [[Definition:Closed Set of Metric Space|closed]] in the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] by $\norm {\,\cdot\,}$. | We have:
{{begin-eqn}}
{{eqn | l = \map {S_r} x
| r = \set {y \in R : \norm {y - x} = r}
| c = {{Defof|Sphere in Normed Division Ring}}
}}
{{eqn | r = \set {y \in R : \norm {y - x} \le r} \cap \set {y \in R : \norm{y - x} \ge r}
}}
{{eqn | r = \map { {B_r}^-} x \cap \paren {R \setminus \map {B_r} x}
... | Topological Properties of Non-Archimedean Division Rings/Spheres are Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Spheres_are_Clopen | https://proofwiki.org/wiki/Topological_Properties_of_Non-Archimedean_Division_Rings/Spheres_are_Clopen | [
"Topological Properties of Non-Archimedean Division Rings"
] | [
"Definition:Sphere/Normed Division Ring",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Metric Induced by Norm on Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Metric Induces Topology",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric S... |
proofwiki-15524 | Existence of Real Polynomial with no Real Root | There exist polynomials in real numbers $\R$ which have no roots in $\R$. | Proof by Counterexample
Take the quadratic equation:
:$(1): \quad x^2 + 1 = 0$
From the Quadratic Formula, the solution to $(1)$ is:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac {-0 \pm \sqrt {0^2 - 4 \times 1 \times 1} } {2 \times 1}
| c =
}}
{{eqn | r = \pm \sqrt {-1}
| c =
}}
{{end-eqn}}
But there is... | There exist [[Definition:Polynomial over Real Numbers|polynomials in real numbers $\R$]] which have no [[Definition:Root of Polynomial|roots]] in $\R$. | [[Proof by Counterexample]]
Take the [[Definition:Quadratic Equation|quadratic equation]]:
:$(1): \quad x^2 + 1 = 0$
From the [[Quadratic Formula]], the solution to $(1)$ is:
{{begin-eqn}}
{{eqn | l = x
| r = \dfrac {-0 \pm \sqrt {0^2 - 4 \times 1 \times 1} } {2 \times 1}
| c =
}}
{{eqn | r = \pm \sqrt ... | Existence of Real Polynomial with no Real Root | https://proofwiki.org/wiki/Existence_of_Real_Polynomial_with_no_Real_Root | https://proofwiki.org/wiki/Existence_of_Real_Polynomial_with_no_Real_Root | [
"Polynomial Theory",
"Analysis"
] | [
"Definition:Polynomial/Real Numbers",
"Definition:Root of Polynomial"
] | [
"Proof by Counterexample",
"Definition:Quadratic Equation",
"Solution to Quadratic Equation",
"Definition:Real Number"
] |
proofwiki-15525 | Magnitude of Projection of Complex Number on Another | Let $z_1$ and $z_2$ denote complex numbers in vector form.
Let $\map {\pr_1} {z_1, z_2}$ denote the projection of $z_1$ on $z_2$.
{{explain|We really need another page to explain the concept of Definition:Projection in Plane in the context of the Definition:Complex Plane}}
Then:
:$\cmod {\map {\pr_1} {z_1, z_2} } = \df... | {{ProofWanted|Needs a proper explanation of the nature of Definition:Projection in Plane}} | Let $z_1$ and $z_2$ denote [[Definition:Complex Number as Vector|complex numbers in vector form]].
Let $\map {\pr_1} {z_1, z_2}$ denote the [[Definition:Projection in Plane|projection]] of $z_1$ on $z_2$.
{{explain|We really need another page to explain the concept of [[Definition:Projection in Plane]] in the context... | {{ProofWanted|Needs a proper explanation of the nature of [[Definition:Projection in Plane]]}} | Magnitude of Projection of Complex Number on Another | https://proofwiki.org/wiki/Magnitude_of_Projection_of_Complex_Number_on_Another | https://proofwiki.org/wiki/Magnitude_of_Projection_of_Complex_Number_on_Another | [
"Complex Dot Product",
"Complex Modulus"
] | [
"Definition:Complex Number as Vector",
"Definition:Projection (Geometry)/Plane",
"Definition:Projection (Geometry)/Plane",
"Definition:Complex Number/Complex Plane",
"Definition:Dot Product/Complex",
"Definition:Complex Modulus"
] | [
"Definition:Projection (Geometry)/Plane"
] |
proofwiki-15526 | Non-Archimedean Division Ring is Totally Disconnected | Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring.
Let $\tau$ be the topology induced by the norm $\norm{\,\cdot\,}$.
Then the topological space $\struct {R, \tau}$ is totally disconnected. | Let $S$ be a subset of $R$ such that:
:$\exists x, y \in S: x \ne y$
Let $r \in \R_{>0} : r = \norm {x - y}$
Consider the open ball $\map {B_r} x$ such that:
:$x \in \map {B_r} x$
:$y \notin \map {B_r} x$
By Open Balls are Clopen then $\map {B_r} x$ is both open and closed.
By Complement of Clopen Set is Clopen then $R... | Let $\struct {R, \norm{\,\cdot\,} }$ be a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean normed division ring]].
Let $\tau$ be the [[Definition:Topology Induced by Division Ring Norm|topology induced]] by the [[Definition:Norm on Division Ring|norm]] $\norm{\,\cdot\,}$.
Then the [[Definition:Topolog... | Let $S$ be a [[Definition:Subset|subset]] of $R$ such that:
:$\exists x, y \in S: x \ne y$
Let $r \in \R_{>0} : r = \norm {x - y}$
Consider the [[Definition:Open Ball of Normed Division Ring|open ball]] $\map {B_r} x$ such that:
:$x \in \map {B_r} x$
:$y \notin \map {B_r} x$
By [[Topological Properties of Non-Archim... | Non-Archimedean Division Ring is Totally Disconnected | https://proofwiki.org/wiki/Non-Archimedean_Division_Ring_is_Totally_Disconnected | https://proofwiki.org/wiki/Non-Archimedean_Division_Ring_is_Totally_Disconnected | [
"Normed Division Rings",
"Non-Archimedean Norms",
"Totally Disconnected Spaces"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Norm/Division Ring",
"Definition:Topological Space",
"Definition:Totally Disconnected Space"
] | [
"Definition:Subset",
"Definition:Open Ball/Normed Division Ring",
"Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Complement of Clopen Set is Clopen",
"Definition:Open Set/Topology",
"Definition:Separa... |
proofwiki-15527 | Distance between Points in Complex Plane | Let $A$ and $B$ be points in the complex plane such that:
:$A = \tuple {x_1, y_1}$
:$B = \tuple {x_2, y_2}$
Then the distance between $A$ and $B$ is given by:
{{begin-eqn}}
{{eqn | l = \size {AB}
| r = \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}
| c =
}}
{{eqn | r = \cmod {z_1 - z_2}
| c =
}... | We have:
{{begin-eqn}}
{{eqn | l = AB
| r = z_2 - z_1
| c = Geometrical Interpretation of Complex Subtraction
}}
{{eqn | r = \paren {x_2 + i y_2} - \paren {x_1 + i y_1}
| c =
}}
{{eqn | r = \paren {x_2 - x_1} + \paren {y_2 - y_1} i
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | ll= \leadsto
... | Let $A$ and $B$ be [[Definition:Point|points]] in the [[Definition:Complex Plane|complex plane]] such that:
:$A = \tuple {x_1, y_1}$
:$B = \tuple {x_2, y_2}$
Then the [[Definition:Distance between Points|distance]] between $A$ and $B$ is given by:
{{begin-eqn}}
{{eqn | l = \size {AB}
| r = \sqrt {\paren {x_2 ... | We have:
{{begin-eqn}}
{{eqn | l = AB
| r = z_2 - z_1
| c = [[Geometrical Interpretation of Complex Subtraction]]
}}
{{eqn | r = \paren {x_2 + i y_2} - \paren {x_1 + i y_1}
| c =
}}
{{eqn | r = \paren {x_2 - x_1} + \paren {y_2 - y_1} i
| c = {{Defof|Complex Subtraction}}
}}
{{eqn | ll= \leadst... | Distance between Points in Complex Plane | https://proofwiki.org/wiki/Distance_between_Points_in_Complex_Plane | https://proofwiki.org/wiki/Distance_between_Points_in_Complex_Plane | [
"Complex Addition",
"Geometry of Complex Plane"
] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Distance between Points",
"Definition:Complex Number as Vector"
] | [
"Geometrical Interpretation of Complex Subtraction"
] |
proofwiki-15528 | Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero | Let $z_1$ and $z_2$ be complex numbers expressed as vectors such taht $z_1$ is not parallel to $z_2$.
Let $a, b \in \R$ be real numbers such that:
:$a z_1 + b z_2 = 0$
Then $a = 0$ and $b = 0$. | Suppose it is not the case that $a = b = 0$.
Then:
{{begin-eqn}}
{{eqn | l = a z_1 + b z_2
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = a \paren {x_1 + i y_1} + b \paren {x_2 + i y_2}
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = a x_1 + b x_2
| r = 0
| c =
... | Let $z_1$ and $z_2$ be [[Definition:Complex Number as Vector|complex numbers expressed as vectors]] such taht $z_1$ is not [[Definition:Parallel Lines|parallel]] to $z_2$.
Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] such that:
:$a z_1 + b z_2 = 0$
Then $a = 0$ and $b = 0$. | Suppose it is not the case that $a = b = 0$.
Then:
{{begin-eqn}}
{{eqn | l = a z_1 + b z_2
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = a \paren {x_1 + i y_1} + b \paren {x_2 + i y_2}
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = a x_1 + b x_2
| r = 0
| c =... | Linear Combination of Non-Parallel Complex Numbers is Zero if Factors are Both Zero | https://proofwiki.org/wiki/Linear_Combination_of_Non-Parallel_Complex_Numbers_is_Zero_if_Factors_are_Both_Zero | https://proofwiki.org/wiki/Linear_Combination_of_Non-Parallel_Complex_Numbers_is_Zero_if_Factors_are_Both_Zero | [
"Geometry of Complex Plane"
] | [
"Definition:Complex Number as Vector",
"Definition:Parallel (Geometry)/Lines",
"Definition:Real Number"
] | [
"Definition:Parallel (Geometry)/Lines"
] |
proofwiki-15529 | Equation for Line through Two Points in Complex Plane/Formulation 1 | $L$ can be expressed by the equation:
:$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$ | Let $z$ be a point on the $L$.
Then:
:$z - z_1 = b \paren {z - z_2}$
where $b$ is some real number.
Then:
{{begin-eqn}}
{{eqn | l = b
| r = \frac {z - z_1} {z - z_2}
| c =
}}
{{eqn | ll= \leadsto
| l = \map \arg {\frac {z - z_1} {z_2 - z_1} }
| r = \arg b
| c =
}}
{{eqn | r = 0
| c... | $L$ can be expressed by the equation:
:$\map \arg {\dfrac {z - z_1} {z_2 - z_1} } = 0$ | Let $z$ be a point on the $L$.
Then:
:$z - z_1 = b \paren {z - z_2}$
where $b$ is some [[Definition:Real Number|real number]].
Then:
{{begin-eqn}}
{{eqn | l = b
| r = \frac {z - z_1} {z - z_2}
| c =
}}
{{eqn | ll= \leadsto
| l = \map \arg {\frac {z - z_1} {z_2 - z_1} }
| r = \arg b
| ... | Equation for Line through Two Points in Complex Plane/Formulation 1 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Formulation_1 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Formulation_1 | [
"Equation for Line through Two Points in Complex Plane"
] | [] | [
"Definition:Real Number",
"Definition:Real Number"
] |
proofwiki-15530 | Equation for Line through Two Points in Complex Plane/Parametric Form 1 | $L$ can be expressed by the equation:
:$z = z_1 + t \paren {z_2 - z_1}$
or:
:$z = \paren {1 - t} z_1 + t z_2$
This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''. | Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.
Let $z$ be an arbitrary point on $L$ represented by the point $P$.
:500px
From Geometrical Interpretation of Complex Addition:
{{begin-eqn}}
{{eqn | l = OA + AP
| r = OP
| ... | $L$ can be expressed by the equation:
:$z = z_1 + t \paren {z_2 - z_1}$
or:
:$z = \paren {1 - t} z_1 + t z_2$
This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''. | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the [[Definition:Complex Plane|complex plane]].
Let $z$ be an arbitrary [[Definition:Point|point]] on $L$ represented by the [[Definition:Point|point]] $P$.
:[[File:Line-in-Comple... | Equation for Line through Two Points in Complex Plane/Parametric Form 1 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Parametric_Form_1 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Parametric_Form_1 | [
"Equation for Line through Two Points in Complex Plane"
] | [] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Point",
"Definition:Point",
"File:Line-in-Complex-Plane-through-Two-Points.png",
"Geometrical Interpretation of Complex Addition"
] |
proofwiki-15531 | Equation for Line through Two Points in Complex Plane/Symmetric Form | $L$ can be expressed by the equation:
:$z = \dfrac {m z_1 + n z_2} {m + n}$
This form of $L$ is known as the '''symmetric form'''. | Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.
Let $z$ be an arbitrary point on $L$ represented by the point $P$.
:500px
As $AP$ and $AB$ are collinear:
:$m AP = n PB$
and so:
:$m \paren {z - z_1} = n \paren {z_2 - z_1}$
The result... | $L$ can be expressed by the equation:
:$z = \dfrac {m z_1 + n z_2} {m + n}$
This form of $L$ is known as the '''symmetric form'''. | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the [[Definition:Complex Plane|complex plane]].
Let $z$ be an arbitrary [[Definition:Point|point]] on $L$ represented by the [[Definition:Point|point]] $P$.
:[[File:Line-in-Comple... | Equation for Line through Two Points in Complex Plane/Symmetric Form | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Symmetric_Form | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Symmetric_Form | [
"Equation for Line through Two Points in Complex Plane"
] | [] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Point",
"Definition:Point",
"File:Line-in-Complex-Plane-through-Two-Points.png"
] |
proofwiki-15532 | Valuation Ring of Non-Archimedean Division Ring is Subring | Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.
Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is:
:$\OO = \set {x \in R : \norm{x} \le 1}$
Then $\OO$ is a subring of $R$:
:with a unity: $1_R$
:in which there a... | To show that $\OO$ is a subring the Subring Test is used by showing:
:$(1): \quad \OO \ne \O$
:$(2): \quad \forall x, y \in \OO: x + \paren {-y} \in \OO$
:$(3): \quad \forall x, y \in \OO: x y \in \OO$
'''(1)'''
By Norm of Unity,
:$\norm{1_R} = 1$
Hence:
:$1_R \in \OO \ne \O$
{{qed|lemma}}
'''(2)'''
Let $x, y \in \OO$.... | Let $\struct {R, \norm{\,\cdot\,}}$ be a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean normed division ring]] with [[Definition:Ring Zero|zero]] $0_R$ and [[Definition:Unity of Ring|unity]] $1_R$.
Let $\OO$ be the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]] b... | To show that $\OO$ is a [[Definition:Subring|subring]] the [[Subring Test]] is used by showing:
:$(1): \quad \OO \ne \O$
:$(2): \quad \forall x, y \in \OO: x + \paren {-y} \in \OO$
:$(3): \quad \forall x, y \in \OO: x y \in \OO$
'''(1)'''
By [[Properties of Norm on Division Ring/Norm of Unity|Norm of Unity]],
:$\nor... | Valuation Ring of Non-Archimedean Division Ring is Subring | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Subring | https://proofwiki.org/wiki/Valuation_Ring_of_Non-Archimedean_Division_Ring_is_Subring | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Subring",
"Definition:Ring with Unity",
"Definition:Proper ... | [
"Definition:Subring",
"Subring Test",
"Properties of Norm on Division Ring/Norm of Unity",
"Properties of Norm on Division Ring/Norm of Negative",
"Subring Test",
"Definition:Subring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Division Ring has No Prope... |
proofwiki-15533 | Valuation Ideal is Maximal Ideal of Induced Valuation Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$ and unity $1_R$.
Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$, that is:
:$\OO = \set{x \in R : \norm x \le 1}$
Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $... | First it is shown that $\PP$ is an ideal of $\OO$ by applying Test for Ideal.
That is, it is shown that:
:$(1): \quad \PP \ne \O$
:$(2): \quad \forall x, y \in \PP: x + \paren {-y} \in \PP$
:$(3): \quad \forall x \in \PP, y \in \OO: x y \in \PP$
By Maximal Left and Right Ideal iff Quotient Ring is Division Ring the sta... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean normed division ring]] with [[Definition:Ring Zero|zero]] $0_R$ and [[Definition:Unity of Ring|unity]] $1_R$.
Let $\OO$ be the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|valuation ring induced]]... | First it is shown that $\PP$ is an [[Definition:Ideal of Ring|ideal]] of $\OO$ by applying [[Test for Ideal]].
That is, it is shown that:
:$(1): \quad \PP \ne \O$
:$(2): \quad \forall x, y \in \PP: x + \paren {-y} \in \PP$
:$(3): \quad \forall x \in \PP, y \in \OO: x y \in \PP$
By [[Maximal Left and Right Ideal iff ... | Valuation Ideal is Maximal Ideal of Induced Valuation Ring | https://proofwiki.org/wiki/Valuation_Ideal_is_Maximal_Ideal_of_Induced_Valuation_Ring | https://proofwiki.org/wiki/Valuation_Ideal_is_Maximal_Ideal_of_Induced_Valuation_Ring | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Ring Zero",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Valuation Ideal Induced by Non-Archimedean Norm",
"Definition... | [
"Definition:Ideal of Ring",
"Test for Ideal",
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring",
"Definition:Equivalence",
"Definition:Maximal Ideal of Ring/Left",
"Test for Ideal",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring... |
proofwiki-15534 | Sequence of Imaginary Reciprocals/Boundedness | The set $S$ is bounded in $\C$. | Let $z \in S$.
Then, for example:
:$\cmod z \le 2$
That is, $S$ is contained entirely within a circle of radius $2$ whose center is at the origin.
{{qed}} | The [[Definition:Set|set]] $S$ is [[Definition:Bounded Subset of Complex Plane|bounded in $\C$]]. | Let $z \in S$.
Then, for example:
:$\cmod z \le 2$
That is, $S$ is contained entirely within a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $2$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]].
{{qed}} | Sequence of Imaginary Reciprocals/Boundedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Boundedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Boundedness | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Set",
"Definition:Bounded Metric Space/Complex"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin"
] |
proofwiki-15535 | Sequence of Imaginary Reciprocals/Limit Points | The set $S$ has exactly one limit point, and that is $z = 0$. | We have that $S$ is countably infinite and also bounded.
Hence by the Bolzano-Weierstrass Theorem $S$ has at least one limit point.
Let $\epsilon \in \R_{>0}$.
Let $\map {N_\epsilon} z$ be the $\epsilon$-neighborhood of $z$.
Let $n \in \N$ such that $n > \dfrac 1 \epsilon$.
Then $\cmod {\dfrac i n} < \epsilon$ and so:
... | The [[Definition:Set|set]] $S$ has exactly one [[Definition:Limit Point (Complex Analysis)|limit point]], and that is $z = 0$. | We have that $S$ is [[Definition:Countably Infinite Set|countably infinite]] and also [[Sequence of Imaginary Reciprocals/Boundedness|bounded]].
Hence by the [[Bolzano-Weierstrass Theorem]] $S$ has at least one [[Definition:Limit Point (Complex Analysis)|limit point]].
Let $\epsilon \in \R_{>0}$.
Let $\map {N_\epsi... | Sequence of Imaginary Reciprocals/Limit Points | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Limit_Points | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Limit_Points | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Set",
"Definition:Limit Point/Complex Analysis"
] | [
"Definition:Countably Infinite/Set",
"Sequence of Imaginary Reciprocals/Boundedness",
"Bolzano-Weierstrass Theorem",
"Definition:Limit Point/Complex Analysis",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Limit Point/Complex Analysis",
"Defini... |
proofwiki-15536 | Sequence of Imaginary Reciprocals/Closedness | The set $S$ is not closed. | From Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.
But:
:$\nexists n \in \N: \dfrac i n = 0$
so $0 \notin S$.
As $S$ does not contain (all) its limit point(s), it follows by definition that $S$ is not closed.
{{qed}} | The set $S$ is not [[Definition:Closed Set (Complex Analysis)|closed]]. | From [[Sequence of Imaginary Reciprocals/Limit Points|Sequence of Imaginary Reciprocals: Limit Points]], $S$ has one [[Definition:Limit Point (Complex Analysis)|limit point]] $z = 0$.
But:
:$\nexists n \in \N: \dfrac i n = 0$
so $0 \notin S$.
As $S$ does not contain (all) its [[Definition:Limit Point (Complex Analys... | Sequence of Imaginary Reciprocals/Closedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closedness | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Closed Set/Complex Analysis"
] | [
"Sequence of Imaginary Reciprocals/Limit Points",
"Definition:Limit Point/Complex Analysis",
"Definition:Limit Point/Complex Analysis",
"Definition:Closed Set/Complex Analysis"
] |
proofwiki-15537 | Valuation Ring of P-adic Norm on Rationals | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
The induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set:
:$\OO = \Z_{\paren p} = \set {\dfrac a b \in \Q : p \nmid b}$ | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.
Then:
{{begin-eqn}}
{{eqn | l = \OO
| r = \set {\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}
| c = {{Defof|Valuation Ring Induced by Non-Archimedean Norm}}
}}
{{eqn | o = }}
{{eqn | r = \set{\dfrac a b \in \Q : \map {\nu_p} {\d... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime]] $p$.
The [[Definition:Valuation Ring Induced by Non-Archimedean Norm|induced valuation ring]] on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the [[Defini... | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the [[Definition:P-adic Valuation|$p$-adic valuation]] on $\Q$.
Then:
{{begin-eqn}}
{{eqn | l = \OO
| r = \set {\dfrac a b \in \Q : \norm {\dfrac a b}_p \le 1}
| c = {{Defof|Valuation Ring Induced by Non-Archimedean Norm}}
}}
{{eqn | o = }}
{{eqn | r = \set{\df... | Valuation Ring of P-adic Norm on Rationals | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_on_Rationals | https://proofwiki.org/wiki/Valuation_Ring_of_P-adic_Norm_on_Rationals | [
"P-adic Number Theory",
"Valuation Ring of P-adic Norm on Rationals"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Definition:Set"
] | [
"Definition:P-adic Valuation",
"Definition:Rational Number/Canonical Form"
] |
proofwiki-15538 | Valuation Ideal of P-adic Norm on Rationals | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
The induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set:
:$\PP = p \Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b, p \divides a}$
where $\Z_{\ideal p}$ is the induced valuation ring on $\struct {\Q,\norm {... | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.
Then:
{{begin-eqn}}
{{eqn | l = \PP
| r = \set {\dfrac a b \in \Q : \norm{\dfrac a b}_p < 1}
| c = {{Defof|Valuation Ideal Induced by Non-Archimedean Norm}}
}}
{{eqn | o = }}
{{eqn | r = \set {\dfrac a b \in \Q : \map {\nu_p} {\df... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime]] $p$.
The [[Definition:Valuation Ideal Induced by Non-Archimedean Norm|induced valuation ideal]] on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the [[Defi... | Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the [[Definition:P-adic Valuation|$p$-adic valuation]] on $\Q$.
Then:
{{begin-eqn}}
{{eqn | l = \PP
| r = \set {\dfrac a b \in \Q : \norm{\dfrac a b}_p < 1}
| c = {{Defof|Valuation Ideal Induced by Non-Archimedean Norm}}
}}
{{eqn | o = }}
{{eqn | r = \set {\dfr... | Valuation Ideal of P-adic Norm on Rationals | https://proofwiki.org/wiki/Valuation_Ideal_of_P-adic_Norm_on_Rationals | https://proofwiki.org/wiki/Valuation_Ideal_of_P-adic_Norm_on_Rationals | [
"P-adic Number Theory"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Valuation Ideal Induced by Non-Archimedean Norm",
"Definition:Set",
"Definition:Valuation Ring Induced by Non-Archimedean Norm"
] | [
"Definition:P-adic Valuation",
"Definition:Rational Number/Canonical Form",
"Valuation Ring of P-adic Norm on Rationals"
] |
proofwiki-15539 | Residue Field of P-adic Norm on Rationals | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
The induced residue field on $\struct {\Q,\norm {\,\cdot\,}_p}$ is isomorphic to the field $\F_p$ of integers modulo $p$. | By Valuation Ring of P-adic Norm on Rationals:
:$\Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b}$
is the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.
By Valuation Ideal of P-adic Norm on Rationals:
:$p \Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b, p \divides a}$
is the induced valuation ide... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime]] $p$.
The [[Definition:Residue Division Ring Induced by Non-Archimedean Norm|induced residue field]] on $\struct {\Q,\norm {\,\cdot\,}_p}$ is [[Defi... | By [[Valuation Ring of P-adic Norm on Rationals|Valuation Ring of P-adic Norm on Rationals]]:
:$\Z_{\ideal p} = \set {\dfrac a b \in \Q : p \nmid b}$
is the [[Definition:Valuation Ring Induced by Non-Archimedean Norm|induced valuation ring]] on $\struct {\Q,\norm {\,\cdot\,}_p}$.
By [[Valuation Ideal of P-adic Norm on... | Residue Field of P-adic Norm on Rationals | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals | [
"P-adic Number Theory",
"Residue Field of P-adic Norm on Rationals"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Residue Division Ring Induced by Non-Archimedean Norm",
"Definition:Isomorphism (Abstract Algebra)",
"Ring of Integers Modulo Prime is Field"
] | [
"Valuation Ring of P-adic Norm on Rationals",
"Definition:Valuation Ring Induced by Non-Archimedean Norm",
"Valuation Ideal of P-adic Norm on Rationals",
"Definition:Valuation Ideal Induced by Non-Archimedean Norm",
"Definition:Residue Division Ring Induced by Non-Archimedean Norm",
"Definition:Quotient R... |
proofwiki-15540 | Sequence of Imaginary Reciprocals/Boundary Points | Every point of $S$, along with the point $z = 0$, is a boundary point of $S$. | Consider the point $z = \dfrac i n \in S$.
Let $\delta \in \R_{>0}$.
Let $\map {N_\delta} z$ be the $\delta$-neighborhood of $z$.
Then $\map {N_\delta} z$ contains at least one point of $S$ ($i / n$ itself) as well as points which are not in $S$.
Hence, by definition, $z$ is a boundary point of $S$.
Let $z = 0$.
Simila... | Every point of $S$, along with the point $z = 0$, is a [[Definition:Boundary Point (Complex Analysis)|boundary point]] of $S$. | Consider the point $z = \dfrac i n \in S$.
Let $\delta \in \R_{>0}$.
Let $\map {N_\delta} z$ be the [[Definition:Neighborhood (Complex Analysis)|$\delta$-neighborhood]] of $z$.
Then $\map {N_\delta} z$ contains at least one point of $S$ ($i / n$ itself) as well as points which are not in $S$.
Hence, by definition, ... | Sequence of Imaginary Reciprocals/Boundary Points | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Boundary_Points | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Boundary_Points | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Boundary Point (Complex Analysis)"
] | [
"Definition:Neighborhood (Complex Analysis)",
"Definition:Boundary Point (Complex Analysis)",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Neighborhood (Complex Analysis)",
"Definition:Boundary Point (Complex Analysis)"
] |
proofwiki-15541 | Sequence of Imaginary Reciprocals/Interior | No point of $S$ is an interior point. | From Sequence of Imaginary Reciprocals: Boundary Points, every $z \in S$ is a boundary point of $S$.
Thus no $z \in S$ is an interior point.
{{qed}} | No point of $S$ is an [[Definition:Interior Point (Complex Analysis)|interior point]]. | From [[Sequence of Imaginary Reciprocals/Boundary Points|Sequence of Imaginary Reciprocals: Boundary Points]], every $z \in S$ is a [[Definition:Boundary Point (Complex Analysis)|boundary point]] of $S$.
Thus no $z \in S$ is an [[Definition:Interior Point (Complex Analysis)|interior point]].
{{qed}} | Sequence of Imaginary Reciprocals/Interior | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Interior | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Interior | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Interior Point (Complex Analysis)"
] | [
"Sequence of Imaginary Reciprocals/Boundary Points",
"Definition:Boundary Point (Complex Analysis)",
"Definition:Interior Point (Complex Analysis)"
] |
proofwiki-15542 | Sequence of Imaginary Reciprocals/Openness | $S$ is not an open set. | From Sequence of Imaginary Reciprocals: Interior, no $z \in S$ is an interior point.
Hence $S$ cannot be open.
{{qed}} | $S$ is not an [[Definition:Open Set (Complex Analysis)|open set]]. | From [[Sequence of Imaginary Reciprocals/Interior|Sequence of Imaginary Reciprocals: Interior]], no $z \in S$ is an [[Definition:Interior Point (Complex Analysis)|interior point]].
Hence $S$ cannot be [[Definition:Open Set (Complex Analysis)|open]].
{{qed}} | Sequence of Imaginary Reciprocals/Openness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Openness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Openness | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Open Set/Complex Analysis"
] | [
"Sequence of Imaginary Reciprocals/Interior",
"Definition:Interior Point (Complex Analysis)",
"Definition:Open Set/Complex Analysis"
] |
proofwiki-15543 | Sequence of Imaginary Reciprocals/Connectedness | $S$ is not connected. | Let $z_1 \in S$ and $z_2 \in S$ be joined by a polygonal path $P$.
Then there are points of $P$ which are not in $S$.
Hence, by definition, $S$ is not connected.
{{qed}} | $S$ is not [[Definition:Connected Set (Complex Analysis)|connected]]. | Let $z_1 \in S$ and $z_2 \in S$ be joined by a [[Definition:Polygonal Path|polygonal path]] $P$.
Then there are points of $P$ which are not in $S$.
Hence, by definition, $S$ is not [[Definition:Connected Set (Complex Analysis)|connected]].
{{qed}} | Sequence of Imaginary Reciprocals/Connectedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Connectedness | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Connectedness | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Connected Set (Complex Analysis)"
] | [
"Definition:Polygonal Path",
"Definition:Connected Set (Complex Analysis)"
] |
proofwiki-15544 | Sequence of Imaginary Reciprocals/Not an Open Region | $S$ is not an open region. | From Sequence of Imaginary Reciprocals: Openness, $S$ is not an open set.
From Sequence of Imaginary Reciprocals: Connectedness, $S$ is not connected.
Hence, by definition, $S$ is not an open region of $\C$.
{{qed}} | $S$ is not an [[Definition:Open Region of Complex Plane|open region]]. | From [[Sequence of Imaginary Reciprocals/Openness|Sequence of Imaginary Reciprocals: Openness]], $S$ is not an [[Definition:Open Set (Complex Analysis)|open set]].
From [[Sequence of Imaginary Reciprocals/Connectedness|Sequence of Imaginary Reciprocals: Connectedness]], $S$ is not [[Definition:Connected Set (Complex A... | Sequence of Imaginary Reciprocals/Not an Open Region | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Not_an_Open_Region | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Not_an_Open_Region | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Open Region/Complex"
] | [
"Sequence of Imaginary Reciprocals/Openness",
"Definition:Open Set/Complex Analysis",
"Sequence of Imaginary Reciprocals/Connectedness",
"Definition:Connected Set (Complex Analysis)",
"Definition:Open Region/Complex"
] |
proofwiki-15545 | Sequence of Imaginary Reciprocals/Closure | The closure of $S$ is the set:
:$\set {0, i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$ | By definition, the closure of $S$ is the set $S$ along with all its limit points.
Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.
Hence the result.
{{qed}} | The [[Definition:Closure (Topology)|closure]] of $S$ is the [[Definition:Set|set]]:
:$\set {0, i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$ | By definition, the [[Definition:Closure (Topology)|closure]] of $S$ is the [[Definition:Set|set]] $S$ along with all its [[Definition:Limit Point (Complex Analysis)|limit points]].
[[Sequence of Imaginary Reciprocals/Limit Points|Sequence of Imaginary Reciprocals: Limit Points]], $S$ has one [[Definition:Limit Point (... | Sequence of Imaginary Reciprocals/Closure | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closure | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closure | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Closure (Topology)",
"Definition:Set"
] | [
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:Limit Point/Complex Analysis",
"Sequence of Imaginary Reciprocals/Limit Points",
"Definition:Limit Point/Complex Analysis"
] |
proofwiki-15546 | Sequence of Imaginary Reciprocals/Countability | The set $S$ is countably infinite. | Let $\phi: \N \to S$ be the mapping defined as:
:$\forall n \in \N: \map \phi n = \dfrac i n$
$\phi$ is a bijection.
Hence the result by definition of countably infinite.
{{qed}} | The set $S$ is [[Definition:Countably Infinite Set|countably infinite]]. | Let $\phi: \N \to S$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall n \in \N: \map \phi n = \dfrac i n$
$\phi$ is a [[Definition:Bijection|bijection]].
Hence the result by definition of [[Definition:Countably Infinite Set|countably infinite]].
{{qed}} | Sequence of Imaginary Reciprocals/Countability | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Countability | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Countability | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Countably Infinite/Set"
] | [
"Definition:Mapping",
"Definition:Bijection",
"Definition:Countably Infinite/Set"
] |
proofwiki-15547 | Sequence of Imaginary Reciprocals/Not Compact | The set $S$ is not compact. | From Sequence of Imaginary Reciprocals: Boundedness, $S$ is bounded in $\C$.
But from Sequence of Imaginary Reciprocals: Closedness, $S$ is not closed.
Hence the result by definition of compact.
{{qed}} | The set $S$ is not [[Definition:Compact Subset of Complex Plane|compact]]. | From [[Sequence of Imaginary Reciprocals/Boundedness|Sequence of Imaginary Reciprocals: Boundedness]], $S$ is [[Definition:Bounded Subset of Complex Plane|bounded in $\C$]].
But from [[Sequence of Imaginary Reciprocals/Closedness|Sequence of Imaginary Reciprocals: Closedness]], $S$ is not [[Definition:Closed Set (Comp... | Sequence of Imaginary Reciprocals/Not Compact | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Not_Compact | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Not_Compact | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Compact Space/Metric Space/Complex"
] | [
"Sequence of Imaginary Reciprocals/Boundedness",
"Definition:Bounded Metric Space/Complex",
"Sequence of Imaginary Reciprocals/Closedness",
"Definition:Closed Set/Complex Analysis",
"Definition:Compact Space/Metric Space/Complex"
] |
proofwiki-15548 | Sequence of Imaginary Reciprocals/Closure is Compact | The closure $S^-$ of the set $S$ is compact. | From Topological Closure is Closed, $S^-$ is closed.
From Sequence of Imaginary Reciprocals: Boundedness, $S$ is bounded in $\C$.
It follows trivially that $S^-$ is also bounded in $\C$.
Hence the result by definition of compact.
{{qed}} | The [[Definition:Closure (Topology)|closure]] $S^-$ of the set $S$ is [[Definition:Compact Subset of Complex Plane|compact]]. | From [[Topological Closure is Closed]], $S^-$ is [[Definition:Closed Set (Complex Analysis)|closed]].
From [[Sequence of Imaginary Reciprocals/Boundedness|Sequence of Imaginary Reciprocals: Boundedness]], $S$ is [[Definition:Bounded Subset of Complex Plane|bounded in $\C$]].
It follows trivially that $S^-$ is also [[... | Sequence of Imaginary Reciprocals/Closure is Compact | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closure_is_Compact | https://proofwiki.org/wiki/Sequence_of_Imaginary_Reciprocals/Closure_is_Compact | [
"Sequence of Imaginary Reciprocals"
] | [
"Definition:Closure (Topology)",
"Definition:Compact Space/Metric Space/Complex"
] | [
"Topological Closure is Closed",
"Definition:Closed Set/Complex Analysis",
"Sequence of Imaginary Reciprocals/Boundedness",
"Definition:Bounded Metric Space/Complex",
"Definition:Bounded Metric Space/Complex",
"Definition:Compact Space/Metric Space/Complex"
] |
proofwiki-15549 | Product of Complex Conjugates/Examples/3 Arguments | Let $z_1, z_2, z_3 \in \C$ be complex numbers.
Let $\overline z$ denote the complex conjugate of the complex number $z$.
Then:
:$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$ | From Product of Complex Conjugates: General Result:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
The result follows by setting $n = 3$. | Let $z_1, z_2, z_3 \in \C$ be [[Definition:Complex Number|complex numbers]].
Let $\overline z$ denote the [[Definition:Complex Conjugate|complex conjugate]] of the [[Definition:Complex Number|complex number]] $z$.
Then:
:$\overline {z_1 z_2 z_3} = \overline {z_1} \cdot \overline {z_2} \cdot \overline {z_3}$ | From [[Product of Complex Conjugates/General Result|Product of Complex Conjugates: General Result]]:
:$\ds \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$
The result follows by setting $n = 3$. | Product of Complex Conjugates/Examples/3 Arguments/Proof 2 | https://proofwiki.org/wiki/Product_of_Complex_Conjugates/Examples/3_Arguments | https://proofwiki.org/wiki/Product_of_Complex_Conjugates/Examples/3_Arguments/Proof_2 | [
"Product of Complex Conjugates"
] | [
"Definition:Complex Number",
"Definition:Complex Conjugate",
"Definition:Complex Number"
] | [
"Product of Complex Conjugates/General Result"
] |
proofwiki-15550 | Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1 | :$(a): \quad \OO$ is a local ring.
:$(b): \quad \PP$ is the unique maximal left ideal of $\OO$
:$(c): \quad \PP$ is the unique maximal right ideal of $\OO$ | By Valuation Ideal is Maximal Ideal of Induced Valuation Ring then:
:$\PP$ is a maximal left ideal of $\OO$.
Let $J \subsetneq \OO$ be any maximal left ideal of $\OO$.
Let $x \in \OO \setminus \PP$.
{{AimForCont}} $x \in J$.
By Norm of Inverse then:
:$\norm {x^{-1}} = 1 / \norm x = 1 / 1 = 1$
Hence:
:$x^{-1} \in \OO$
... | :$(a): \quad \OO$ is a [[Definition:Noncommutative Local Ring|local ring]].
:$(b): \quad \PP$ is the [[Definition:Unique|unique]] [[Definition:Maximal Left Ideal of Ring|maximal left ideal]] of $\OO$
:$(c): \quad \PP$ is the [[Definition:Unique|unique]] [[Definition:Maximal Right Ideal of Ring|maximal right ideal]] of ... | By [[Valuation Ideal is Maximal Ideal of Induced Valuation Ring|Valuation Ideal is Maximal Ideal of Induced Valuation Ring]] then:
:$\PP$ is a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]] of $\OO$.
Let $J \subsetneq \OO$ be any [[Definition:Maximal Left Ideal of Ring|maximal left ideal]] of $\OO$.
Le... | Valuation Ideal is Maximal Ideal of Induced Valuation Ring/Corollary 1 | https://proofwiki.org/wiki/Valuation_Ideal_is_Maximal_Ideal_of_Induced_Valuation_Ring/Corollary_1 | https://proofwiki.org/wiki/Valuation_Ideal_is_Maximal_Ideal_of_Induced_Valuation_Ring/Corollary_1 | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [
"Definition:Local Ring/Noncommutative",
"Definition:Unique",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Unique",
"Definition:Maximal Ideal of Ring/Right"
] | [
"Valuation Ideal is Maximal Ideal of Induced Valuation Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring/Left",
"Properties of Norm on Division Ring/Norm of Inverse",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Contradiction",
"Definition:Maximal Ideal of Ring/Left... |
proofwiki-15551 | Residue Field of P-adic Norm on Rationals/Lemma 1 | :$\phi$ is a homomorphism. | Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \Z_{\paren p}$.
Hence $\Z \subset \Z_{\paren p}$ is a subring of $\Z_{\paren p}$.
Let $i: \Z \to \Z_{\paren p}$ be the inclusion mapping defined by:
:$\map i a = a$
By Inclusion Mapping is Monomorphism then $i$ is a ring monomorphism.
Let $q: \Z_{\paren p} ... | :$\phi$ is a [[Definition:Ring Homomorphism|homomorphism]]. | Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \Z_{\paren p}$.
Hence $\Z \subset \Z_{\paren p}$ is a [[Definition:Subring|subring]] of $\Z_{\paren p}$.
Let $i: \Z \to \Z_{\paren p}$ be the [[Definition:Inclusion Mapping|inclusion mapping]] defined by:
:$\map i a = a$
By [[Inclusion Mapping is Monomo... | Residue Field of P-adic Norm on Rationals/Lemma 1 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_1 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_1 | [
"Residue Field of P-adic Norm on Rationals"
] | [
"Definition:Ring Homomorphism"
] | [
"Definition:Subring",
"Definition:Inclusion Mapping",
"Inclusion Mapping is Monomorphism",
"Definition:Ring Monomorphism",
"Definition:Quotient Epimorphism/Ring",
"Quotient Epimorphism is Epimorphism/Ring",
"Definition:Ring Epimorphism",
"Definition:Composition of Mappings",
"Composition of Ring Hom... |
proofwiki-15552 | Residue Field of P-adic Norm on Rationals/Lemma 2 | :$p \Z = \map \ker \phi$ | Let $\map \ker \phi$ denote the kernel of $\phi$.
Then:
{{begin-eqn}}
{{eqn | l = a
| o = \in
| r = \map \ker \phi
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map \phi a
| r = p \Z_{\ideal p}
| c = {{Defof|Kernel of Ring Homomorphism}}
}}
{{eqn | ll= \leadstoandfrom
| l = a + p... | :$p \Z = \map \ker \phi$ | Let $\map \ker \phi$ denote the [[Definition:Kernel of Ring Homomorphism|kernel]] of $\phi$.
Then:
{{begin-eqn}}
{{eqn | l = a
| o = \in
| r = \map \ker \phi
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \map \phi a
| r = p \Z_{\ideal p}
| c = {{Defof|Kernel of Ring Homomorphism}}
}}
{... | Residue Field of P-adic Norm on Rationals/Lemma 2 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_2 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_2 | [
"Residue Field of P-adic Norm on Rationals"
] | [] | [
"Definition:Kernel of Ring Homomorphism",
"Element in Right Coset iff Product with Inverse in Subgroup"
] |
proofwiki-15553 | Residue Field of P-adic Norm on Rationals/Lemma 3 | :$\phi : \Z \to \Z_{\paren p} / p \Z_{\paren p}$ is a surjection. | Let $a / b \in \Z_{\paren p}$, where $a / b$ are in canonical form.
Then $p \nmid b$
Let $\F_p$ be the field of integers modulo $p$.
By the definition of a field:
:$\exists b' \in \Z: b b' \equiv 1 \pmod p$
By the definition of congruence modulo $p$:
:$p \divides b b' - 1$
By Divisor Divides Multiple:
:$\forall a \in \... | :$\phi : \Z \to \Z_{\paren p} / p \Z_{\paren p}$ is a [[Definition:Surjective|surjection]]. | Let $a / b \in \Z_{\paren p}$, where $a / b$ are in [[Definition:Canonical Form of Rational Number|canonical form]].
Then $p \nmid b$
Let $\F_p$ be the [[Ring of Integers Modulo Prime is Field|field of integers modulo $p$]].
By the definition of a [[Definition:Field (Abstract Algebra)|field]]:
:$\exists b' \in \Z: ... | Residue Field of P-adic Norm on Rationals/Lemma 3 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_3 | https://proofwiki.org/wiki/Residue_Field_of_P-adic_Norm_on_Rationals/Lemma_3 | [
"Residue Field of P-adic Norm on Rationals"
] | [
"Definition:Surjection"
] | [
"Definition:Rational Number/Canonical Form",
"Ring of Integers Modulo Prime is Field",
"Definition:Field (Abstract Algebra)",
"Definition:Congruence (Number Theory)/Integers",
"Divisor Divides Multiple",
"Valuation Ideal of P-adic Norm on Rationals",
"Element in Left Coset iff Product with Inverse in Su... |
proofwiki-15554 | Equation for Line through Two Points in Complex Plane/Parametric Form 2 | $L$ can be expressed by the equations:
{{begin-eqn}}
{{eqn | l = x - x_1
| r = t \paren {x_2 - x_1}
}}
{{eqn | l = y - y_1
| r = t \paren {y_2 - y_1}
}}
{{end-eqn}}
These are the '''parametric equations of $L$''', where $t$ is the parameter. | From Equation for Line through Two Points in Complex Plane: Parametric Form 1:
:$z = z_1 + t \paren {z_2 - z_1}$
Letting:
{{begin-eqn}}
{{eqn | l = z
| r = x + i y
}}
{{eqn | l = z_1
| r = x_1 + i y_1
}}
{{eqn | l = z_2
| r = x_2 + i y_2
}}
{{end-eqn}}
the parametric equations follow by equating real ... | $L$ can be expressed by the equations:
{{begin-eqn}}
{{eqn | l = x - x_1
| r = t \paren {x_2 - x_1}
}}
{{eqn | l = y - y_1
| r = t \paren {y_2 - y_1}
}}
{{end-eqn}}
These are the '''parametric equations of $L$''', where $t$ is the [[Definition:Parameter|parameter]]. | From [[Equation for Line through Two Points in Complex Plane/Parametric Form 1|Equation for Line through Two Points in Complex Plane: Parametric Form 1]]:
:$z = z_1 + t \paren {z_2 - z_1}$
Letting:
{{begin-eqn}}
{{eqn | l = z
| r = x + i y
}}
{{eqn | l = z_1
| r = x_1 + i y_1
}}
{{eqn | l = z_2
| ... | Equation for Line through Two Points in Complex Plane/Parametric Form 2 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Parametric_Form_2 | https://proofwiki.org/wiki/Equation_for_Line_through_Two_Points_in_Complex_Plane/Parametric_Form_2 | [
"Equation for Line through Two Points in Complex Plane"
] | [
"Definition:Parameter"
] | [
"Equation for Line through Two Points in Complex Plane/Parametric Form 1",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part"
] |
proofwiki-15555 | Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1 | $L$ can be expressed by the equation:
:$z - \dfrac {z_1 + z_2} 2 = i t\paren {z_2 - z_1}$
or:
:$z = \dfrac {z_1 + z_2} 2 + i t\paren {z_2 - z_1}$
This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''. | Let $z_1$ and $z_2$ be represented by the points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the complex plane.
Let $z$ be an arbitrary point on $L$ represented by the point $P$.
:400px
We have that $L$ passes through the point:
:$\dfrac {z_1 + z_2} 2$
and is perpendicular to the straight line:
... | $L$ can be expressed by the equation:
:$z - \dfrac {z_1 + z_2} 2 = i t\paren {z_2 - z_1}$
or:
:$z = \dfrac {z_1 + z_2} 2 + i t\paren {z_2 - z_1}$
This form of $L$ is known as the '''parametric form''', where $t$ is the '''parameter'''. | Let $z_1$ and $z_2$ be represented by the [[Definition:Point|points]] $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ respectively in the [[Definition:Complex Plane|complex plane]].
Let $z$ be an arbitrary [[Definition:Point|point]] on $L$ represented by the [[Definition:Point|point]] $P$.
:[[File:Perpendicular ... | Equation for Perpendicular Bisector of Two Points in Complex Plane/Parametric Form 1 | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points_in_Complex_Plane/Parametric_Form_1 | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points_in_Complex_Plane/Parametric_Form_1 | [
"Equation for Perpendicular Bisector of Two Points in Complex Plane"
] | [] | [
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Point",
"Definition:Point",
"File:Perpendicular Bisector of Two Points in Complex Plane.png",
"Definition:Point",
"Definition:Right Angle/Perpendicular",
"Definition:Line/Straight Line",
"Multiplication by Imaginary Unit is E... |
proofwiki-15556 | Equation of Hyperbola in Complex Plane | Let $\C$ be the complex plane.
Let $H$ be a hyperbola in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alpha} - \cmod {z - \beta} = d$
where $\cmod {\, \cdot \,}$ denotes complex modulus. | By definition of complex modulus:
:$\cmod {z - \alpha}$ is the distance from $z$ to $\alpha$
:$\cmod {z - \beta}$ is the distance from $z$ to $\beta$.
Thus $\cmod {z - \alpha} - \cmod {z - \beta}$ is the difference of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.
This is precisely the equidistance property... | Let $\C$ be the [[Definition:Complex Plane|complex plane]].
Let $H$ be a [[Definition:Hyperbola|hyperbola]] in $\C$ whose [[Definition:Major Axis of Hyperbola|major axis]] is $d \in \R_{>0}$ and whose [[Definition:Focus of Hyperbola|foci]] are at $\alpha, \beta \in \C$.
Then $C$ may be written as:
:$\cmod {z - \alph... | By definition of [[Definition:Complex Modulus|complex modulus]]:
:$\cmod {z - \alpha}$ is the [[Definition:Distance between Points|distance]] from $z$ to $\alpha$
:$\cmod {z - \beta}$ is the [[Definition:Distance between Points|distance]] from $z$ to $\beta$.
Thus $\cmod {z - \alpha} - \cmod {z - \beta}$ is the [[Defi... | Equation of Hyperbola in Complex Plane | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Complex_Plane | https://proofwiki.org/wiki/Equation_of_Hyperbola_in_Complex_Plane | [
"Hyperbolas",
"Geometry of Complex Plane",
"Equation of Hyperbola in Complex Plane"
] | [
"Definition:Complex Number/Complex Plane",
"Definition:Hyperbola",
"Definition:Hyperbola/Major Axis",
"Definition:Hyperbola/Focus",
"Definition:Complex Modulus"
] | [
"Definition:Complex Modulus",
"Definition:Distance between Points",
"Definition:Distance between Points",
"Definition:Subtraction/Real Numbers",
"Definition:Distance between Points",
"Definition:Hyperbola/Equidistance",
"Equidistance of Hyperbola equals Transverse Axis",
"Definition:Constant",
"Defi... |
proofwiki-15557 | Geodesic Equation/2d Surface Embedded in 3d Euclidean Space | Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function:
:$\mathbf r = \map {\mathbf r} {u, v}$
Then a geodesic of $\sigma$ satisfies the following system of differential equations:
:$\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \df... | A curve on the surface $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$
The arc length between the points corresponding to $t_0$ and $t_1$ equals:
:$\ds J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' + G v'^2} \rd t$
The following derivatives will appear in Euler's Equations:
{{begin-eqn}}
{{... | Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a [[Definition:Smooth Real Function|smooth]] [[Definition:Surface|surface]] specified by a [[Definition:Vector-Valued Function|vector-valued function]]:
:$\mathbf r = \map {\mathbf r} {u, v}$
Then a [[Definition:Geodesic Curve|geodesic]] of $\sigma$ satisfies the fo... | A [[Definition:Curve|curve]] on the [[Definition:Surface|surface]] $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$
The [[Definition:Arc Length|arc length]] between the [[Definition:Point|points]] corresponding to $t_0$ and $t_1$ equals:
:$\ds J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' ... | Geodesic Equation/2d Surface Embedded in 3d Euclidean Space | https://proofwiki.org/wiki/Geodesic_Equation/2d_Surface_Embedded_in_3d_Euclidean_Space | https://proofwiki.org/wiki/Geodesic_Equation/2d_Surface_Embedded_in_3d_Euclidean_Space | [
"Calculus of Variations"
] | [
"Definition:Smooth Real Function",
"Definition:Surface",
"Definition:Vector-Valued Function",
"Definition:Geodesic",
"Definition:Differential Equation/System",
"Definition:Real Function",
"Definition:First Fundamental Form"
] | [
"Definition:Line/Curve",
"Definition:Surface",
"Definition:Arc Length",
"Definition:Point",
"Definition:Derivative",
"Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions",
"Definition:Derivative",
"Necessary Condition for Integral Functional to have... |
proofwiki-15558 | Magnitude and Direction of Equilibrant | Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space.
The '''equilibrant''' $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ is:
:$\mathbf E = -\ds \sum_{k \mathop = 1}^n \mathbf F_k$
That is, the magnitude and direction of $\mathbf E$... | From Newton's First Law of Motion, the total force on $B$ must equal zero in order for $B$ to remain stationary.
That is, $\mathbf E$ must be such that:
:$\mathbf E + \ds \sum_{k \mathop = 1}^n \mathbf F_k = \bszero$
That is:
:$\mathbf E = -\ds \sum_{k \mathop = 1}^n \mathbf F_k$
{{qed}}
Category:Force
Category:Equilib... | Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a [[Definition:Set|set]] of $n$ [[Definition:Force|forces]] acting on a [[Definition:Particle|particle]] $B$ at a [[Definition:Point|point]] $P$ in [[Definition:Ordinary Space|space]].
The '''equilibrant''' $\mathbf E$ of $\mathbf F_1, \mathbf F_2, \ldots, \mathb... | From [[Newton's First Law of Motion]], the total [[Definition:Force|force]] on $B$ must equal [[Definition:Zero Vector|zero]] in order for $B$ to remain [[Definition:Stationary|stationary]].
That is, $\mathbf E$ must be such that:
:$\mathbf E + \ds \sum_{k \mathop = 1}^n \mathbf F_k = \bszero$
That is:
:$\mathbf E... | Magnitude and Direction of Equilibrant | https://proofwiki.org/wiki/Magnitude_and_Direction_of_Equilibrant | https://proofwiki.org/wiki/Magnitude_and_Direction_of_Equilibrant | [
"Force",
"Equilibrants"
] | [
"Definition:Set",
"Definition:Force",
"Definition:Particle",
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Magnitude",
"Definition:Direction"
] | [
"Newton's Laws of Motion/First Law",
"Definition:Force",
"Definition:Zero Vector",
"Definition:Stationary",
"Category:Force",
"Category:Equilibrants"
] |
proofwiki-15559 | Modulus of Exponential of i z where z is on Circle | Let $C$ be the circle embedded in the complex plane given by the equation:
:$z = R e^{i \theta}$
Then:
:$\cmod {e^{i z} } = e^{-R \sin \theta}$ | {{begin-eqn}}
{{eqn | l = \cmod {e^{i z} }
| r = \cmod {\map \exp {i R \, \map \exp {i \theta} } }
| c =
}}
{{eqn | r = \cmod {\map \exp {i R \paren {\cos \theta + i \sin \theta} } }
| c =
}}
{{eqn | r = \cmod {\map \exp {R \paren {-\sin \theta + i \cos \theta} } }
| c =
}}
{{eqn | r = \cmod ... | Let $C$ be the [[Definition:Circle|circle]] embedded in the [[Definition:Complex Plane|complex plane]] given by the equation:
:$z = R e^{i \theta}$
Then:
:$\cmod {e^{i z} } = e^{-R \sin \theta}$ | {{begin-eqn}}
{{eqn | l = \cmod {e^{i z} }
| r = \cmod {\map \exp {i R \, \map \exp {i \theta} } }
| c =
}}
{{eqn | r = \cmod {\map \exp {i R \paren {\cos \theta + i \sin \theta} } }
| c =
}}
{{eqn | r = \cmod {\map \exp {R \paren {-\sin \theta + i \cos \theta} } }
| c =
}}
{{eqn | r = \cmod ... | Modulus of Exponential of i z where z is on Circle | https://proofwiki.org/wiki/Modulus_of_Exponential_of_i_z_where_z_is_on_Circle | https://proofwiki.org/wiki/Modulus_of_Exponential_of_i_z_where_z_is_on_Circle | [
"Complex Modulus",
"Exponential Function",
"Circles",
"Modulus of Exponential of i z where z is on Circle"
] | [
"Definition:Circle",
"Definition:Complex Number/Complex Plane"
] | [
"Modulus and Argument of Complex Exponential"
] |
proofwiki-15560 | Sum of Complex Numbers in Exponential Form/General Result | Let $n \in \Z_{>0}$ be a positive integer.
For all $k \in \set {1, 2, \dotsc, n}$, let:
:$z_k = r_k e^{i \theta_k}$
be non-zero complex numbers in exponential form.
Let:
:$r e^{i \theta} = \ds \sum_{k \mathop = 1}^n z_k = z_1 + z_2 + \dotsb + z_k$
Then:
{{begin-eqn}}
{{eqn | l = r
| r = \sqrt {\sum_{k \mathop = 1... | Let:
{{begin-eqn}}
{{eqn | l = r e^{i \theta}
| r = \sum_{k \mathop = 1}^n z_k
| c =
}}
{{eqn | r = z_1 + z_2 + \dotsb + z_k
| c =
}}
{{eqn | r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2} + \dotsb + r_n \paren {\cos \theta_n + i \sin \theta_n}
| ... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
For all $k \in \set {1, 2, \dotsc, n}$, let:
:$z_k = r_k e^{i \theta_k}$
be [[Definition:Complex Zero|non-zero]] [[Definition:Complex Number|complex numbers]] in [[Definition:Exponential Form of Complex Number|exponential form]].
Let:
:$r e^... | Let:
{{begin-eqn}}
{{eqn | l = r e^{i \theta}
| r = \sum_{k \mathop = 1}^n z_k
| c =
}}
{{eqn | r = z_1 + z_2 + \dotsb + z_k
| c =
}}
{{eqn | r = r_1 \paren {\cos \theta_1 + i \sin \theta_1} + r_2 \paren {\cos \theta_2 + i \sin \theta_2} + \dotsb + r_n \paren {\cos \theta_n + i \sin \theta_n}
| ... | Sum of Complex Numbers in Exponential Form/General Result | https://proofwiki.org/wiki/Sum_of_Complex_Numbers_in_Exponential_Form/General_Result | https://proofwiki.org/wiki/Sum_of_Complex_Numbers_in_Exponential_Form/General_Result | [
"Complex Addition"
] | [
"Definition:Positive/Integer",
"Definition:Zero (Number)/Complex",
"Definition:Complex Number",
"Definition:Complex Number/Polar Form/Exponential Form"
] | [
"Definition:Complex Modulus",
"Sum of Squares of Sine and Cosine",
"Cosine of Difference",
"Definition:Argument of Complex Number",
"Definition:Complex Modulus",
"Definition:Complex Modulus",
"Definition:Division/Field/Complex Numbers"
] |
proofwiki-15561 | Bias of Sample Variance | Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\ds {S_n}^2 = \frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$
is a biased estimator of $\sigma^2$, with:
:$\map {\operatorname {bias}... | By definition, ${S_n}^2$ is a biased estimator of $\sigma^2$ {{iff}}:
:$\expect { {S_n}^2} \ne \sigma^2$
We have:
{{begin-eqn}}
{{eqn | l = \expect { {S_n}^2}
| r = \expect {\frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2}
}}
{{eqn | r = \expect {\frac 1 n \sum_{i \mathop = 1}^n \paren {\paren {X_i - \... | Let $X_1, X_2, \ldots, X_n$ form a [[Definition:Random Sample (Statistics)|random sample]] from a [[Definition:Population|population ]] with [[Definition:Expectation|mean]] $\mu$ and [[Definition:Variance|variance]] $\sigma^2$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\ds {S_n}^2 = \frac 1... | By definition, ${S_n}^2$ is a [[Definition:Bias in Estimator|biased]] [[Definition:Estimator|estimator]] of $\sigma^2$ {{iff}}:
:$\expect { {S_n}^2} \ne \sigma^2$
We have:
{{begin-eqn}}
{{eqn | l = \expect { {S_n}^2}
| r = \expect {\frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2}
}}
{{eqn | r = \ex... | Bias of Sample Variance | https://proofwiki.org/wiki/Bias_of_Sample_Variance | https://proofwiki.org/wiki/Bias_of_Sample_Variance | [
"Inductive Statistics"
] | [
"Definition:Random Sample (Statistics)",
"Definition:Population",
"Definition:Expectation",
"Definition:Variance",
"Definition:Bias/Estimator",
"Definition:Estimator"
] | [
"Definition:Bias/Estimator",
"Definition:Estimator",
"Square of Difference",
"Summation is Linear",
"Definition:Independent Random Variables/Discrete/General Definition",
"Expectation is Linear",
"Expectation is Linear",
"Variance of Sample Mean",
"Definition:Bias/Estimator",
"Definition:Estimator... |
proofwiki-15562 | Bessel's Correction | Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\ds \hat {\sigma^2} = \frac 1 {n - 1} \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2$
is an unbiased estimator of $\sigma^2$. | We know that $\hat {\sigma^2}$ is an unbiased estimator of $\sigma^2$ {{iff}}:
:$\expect {\hat {\sigma^2} } = \sigma^2$
In Bias of Sample Variance, it is shown that:
:$\ds \expect {\frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2} = \paren {1 - \frac 1 n} \sigma^2$
By Expectation is Linear:
{{begin-eqn}}
{{e... | Let $X_1, X_2, \ldots, X_n$ form a [[Definition:Random Sample (Statistics)|random sample]] from a [[Definition:Population|population]] with [[Definition:Expectation|mean]] $\mu$ and [[Definition:Variance|variance]] $\sigma^2$.
Let:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
Then:
:$\ds \hat {\sigma^2} = ... | We know that $\hat {\sigma^2}$ is an [[Definition:Unbiased Estimator|unbiased estimator]] of $\sigma^2$ {{iff}}:
:$\expect {\hat {\sigma^2} } = \sigma^2$
In [[Bias of Sample Variance]], it is shown that:
:$\ds \expect {\frac 1 n \sum_{i \mathop = 1}^n \paren {X_i - \bar X}^2} = \paren {1 - \frac 1 n} \sigma^2$
B... | Bessel's Correction | https://proofwiki.org/wiki/Bessel's_Correction | https://proofwiki.org/wiki/Bessel's_Correction | [
"Inductive Statistics"
] | [
"Definition:Random Sample (Statistics)",
"Definition:Population",
"Definition:Expectation",
"Definition:Variance",
"Definition:Unbiased Estimator"
] | [
"Definition:Unbiased Estimator",
"Bias of Sample Variance",
"Expectation is Linear",
"Expectation is Linear",
"Definition:Unbiased Estimator",
"Category:Inductive Statistics"
] |
proofwiki-15563 | Sample Mean is Unbiased Estimator of Population Mean | Let $X_1, X_2, \ldots, X_n$ form a random sample from a population with mean $\mu$ and variance $\sigma^2$.
Then:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
is an unbiased estimator of $\mu$. | If $\bar X$ be an unbiased estimator of $\mu$, then:
:$\expect {\bar X} = \mu$
{{Explain|The structure of this proof is confusing. It starts with what looks like an assertion of what we are trying to prove.}}
We have:
{{begin-eqn}}
{{eqn | l = \expect {\bar X}
| r = \expect {\frac 1 n \sum_{i \mathop = 1}^n X_i... | Let $X_1, X_2, \ldots, X_n$ form a [[Definition:Random Sample (Statistics)|random sample]] from a population with [[Definition:Expectation|mean]] $\mu$ and [[Definition:Variance|variance]] $\sigma^2$.
Then:
:$\ds \bar X = \frac 1 n \sum_{i \mathop = 1}^n X_i$
is an [[Definition:Unbiased Estimator|unbiased estimator... | If $\bar X$ be an [[Definition:Unbiased Estimator|unbiased estimator]] of $\mu$, then:
:$\expect {\bar X} = \mu$
{{Explain|The structure of this proof is confusing. It starts with what looks like an assertion of what we are trying to prove.}}
We have:
{{begin-eqn}}
{{eqn | l = \expect {\bar X}
| r = \expec... | Sample Mean is Unbiased Estimator of Population Mean | https://proofwiki.org/wiki/Sample_Mean_is_Unbiased_Estimator_of_Population_Mean | https://proofwiki.org/wiki/Sample_Mean_is_Unbiased_Estimator_of_Population_Mean | [
"Inductive Statistics"
] | [
"Definition:Random Sample (Statistics)",
"Definition:Expectation",
"Definition:Variance",
"Definition:Unbiased Estimator"
] | [
"Definition:Unbiased Estimator",
"Expectation is Linear",
"Definition:Unbiased Estimator"
] |
proofwiki-15564 | Ostrowski's Theorem/Archimedean Norm | Let $\norm {\, \cdot \,}$ be a non-trivial Archimedean norm on the rational numbers $\Q$.
Then $\norm {\, \cdot \,}$ is equivalent to the absolute value $\size {\, \cdot \,}$. | By Characterisation of Non-Archimedean Division Ring Norms then:
:$\exists n \in \N$ such that $\norm n > 1$
Let $n_0 = \min \set {n \in \N : \norm n > 1}$
By Norm of Unity then:
:$n_0 > 1$
Let $\alpha = \dfrac {\log \norm {n_0} } {\log n_0}$
Since $n_0, \norm n_0 > 1$ then:
:$\alpha > 0$
==== Lemma 1.1 ====
{{:Ostrows... | Let $\norm {\, \cdot \,}$ be a [[Definition:Nontrivial Division Ring Norm|non-trivial]] [[Definition:Archimedean Division Ring Norm|Archimedean]] [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$.
Then $\norm {\, \cdot \,}$ is [[Definition:Equivalent Division Ring ... | By [[Characterisation of Non-Archimedean Division Ring Norms]] then:
:$\exists n \in \N$ such that $\norm n > 1$
Let $n_0 = \min \set {n \in \N : \norm n > 1}$
By [[Properties of Norm on Division Ring/Norm of Unity|Norm of Unity]] then:
:$n_0 > 1$
Let $\alpha = \dfrac {\log \norm {n_0} } {\log n_0}$
Since $n_0, \... | Ostrowski's Theorem/Archimedean Norm | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm | [
"Ostrowski's Theorem"
] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Definition:Non-Archimedean/Norm (Division Ring)/Archimedean",
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:Absolute Value"
] | [
"Characterisation of Non-Archimedean Division Ring Norms",
"Properties of Norm on Division Ring/Norm of Unity",
"Ostrowski's Theorem/Archimedean Norm/Lemma 1.1",
"Ostrowski's Theorem/Archimedean Norm/Lemma 1.2",
"Equivalent Norms on Rational Numbers",
"Definition:Equivalent Division Ring Norms",
"Defini... |
proofwiki-15565 | Ostrowski's Theorem/Non-Archimedean Norm | Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.
Then $\norm {\, \cdot \,}$ is equivalent to the $p$-adic norm $\norm {\, \cdot \,}_p$ for some prime $p$. | From Characterisation of Non-Archimedean Division Ring Norms:
:$\forall n \in \N: \norm n \le 1$
==== Lemma 2.1 ====
{{:Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1}}{{qed|lemma}}
Let $n_0 = \min \set {n \in N : \norm n < 1}$.
==== Lemma 2.2 ====
{{:Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2}}{{qed|lemma}... | Let $\norm {\, \cdot \,}$ be a [[Definition:Nontrivial Division Ring Norm|non-trivial]] [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$.
Then $\norm {\, \cdot \,}$ is [[Definition:Equivalent Divisi... | From [[Characterisation of Non-Archimedean Division Ring Norms]]:
:$\forall n \in \N: \norm n \le 1$
==== [[Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1|Lemma 2.1]] ====
{{:Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1}}{{qed|lemma}}
Let $n_0 = \min \set {n \in N : \norm n < 1}$.
==== [[Ostrowski's The... | Ostrowski's Theorem/Non-Archimedean Norm | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm | [
"Ostrowski's Theorem"
] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:P-adic Norm",
"Definition:Prime Number"
] | [
"Characterisation of Non-Archimedean Division Ring Norms",
"Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1",
"Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2",
"Prime not Divisor implies Coprime",
"Definition:Coprime/Integers",
"Definition:P-adic Norm",
"Definition:P-adic Valuation/Integers",
"... |
proofwiki-15566 | Ostrowski's Theorem/Archimedean Norm/Lemma 1.1 | :$\forall n \in N: \norm n \le n^\alpha$ | By the definition of $\alpha$ then:
:$\norm {n_0} = {n_0}^\alpha$
By the definition of $n_0$ then:
:${n_0}^\alpha > 1$
Let $n \in \N$.
By Basis Representation Theorem then $n$ can be written:
:$n = a_0 + a_1 n_0 + a_2 {n_0}^2 + \cdots + a_s {n_0}^s$
where $0 \le a_i < n_0$ and $a_s \ne 0$
Since all of the $a_i < n_0$, ... | :$\forall n \in N: \norm n \le n^\alpha$ | By the definition of $\alpha$ then:
:$\norm {n_0} = {n_0}^\alpha$
By the definition of $n_0$ then:
:${n_0}^\alpha > 1$
Let $n \in \N$.
By [[Basis Representation Theorem]] then $n$ can be written:
:$n = a_0 + a_1 n_0 + a_2 {n_0}^2 + \cdots + a_s {n_0}^s$
where $0 \le a_i < n_0$ and $a_s \ne 0$
Since all of the $a... | Ostrowski's Theorem/Archimedean Norm/Lemma 1.1 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm/Lemma_1.1 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm/Lemma_1.1 | [
"Ostrowski's Theorem"
] | [] | [
"Basis Representation Theorem",
"Sum of Infinite Geometric Sequence",
"Limit of Root of Positive Real Number",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Inequality Rule for Real Sequences"
] |
proofwiki-15567 | Ostrowski's Theorem/Archimedean Norm/Lemma 1.2 | :$\forall n \in N: \norm n \ge n^\alpha$ | By the definition of $\alpha$:
:$\norm {n_0} = {n_0}^\alpha$
By the definition of $n_0$:
:${n_0}^\alpha > 1$
Let $n \in \N$.
By Basis Representation Theorem then $n$ can be written:
:$n = a_0 + a_1 n_0 + a_2 {n_0}^2 + \cdots + a_s {n_0}^s$
where $0 \le a_i < n_0$ and $a_s \ne 0$
By Bounds for Integer Expressed in Base ... | :$\forall n \in N: \norm n \ge n^\alpha$ | By the definition of $\alpha$:
:$\norm {n_0} = {n_0}^\alpha$
By the definition of $n_0$:
:${n_0}^\alpha > 1$
Let $n \in \N$.
By [[Basis Representation Theorem]] then $n$ can be written:
:$n = a_0 + a_1 n_0 + a_2 {n_0}^2 + \cdots + a_s {n_0}^s$
where $0 \le a_i < n_0$ and $a_s \ne 0$
By [[Bounds for Integer Expre... | Ostrowski's Theorem/Archimedean Norm/Lemma 1.2 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm/Lemma_1.2 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Archimedean_Norm/Lemma_1.2 | [
"Ostrowski's Theorem"
] | [] | [
"Basis Representation Theorem",
"Bounds for Integer Expressed in Base k",
"Ostrowski's Theorem/Archimedean Norm/Lemma 1.1",
"Reverse Triangle Inequality/Normed Division Ring",
"Limit of Root of Positive Real Number",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Inequality Rule for Real Seque... |
proofwiki-15568 | Roots of Complex Number/Examples/z^4 + 81 = 0 | The roots of the polynomial:
:$z^4 + 81$
are:
:$\set {3 \cis 45 \degrees, 3 \cis 135 \degrees, 3 \cis 225 \degrees, 3 \cis 315 \degrees}$ | From Factorisation of $z^n + 1$:
:$z^4 + 1 = \ds \prod_{k \mathop = 0}^3 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 4}$
Thus:
:$z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 4}$
{{begin-eqn}}
{{eqn | n = k = 0
| l = z
| r = \cos \dfrac \pi 4 + i \sin \dfrac \pi 4
| c =
}}
{{eqn | n = k = 1
... | The [[Definition:Root of Polynomial|roots]] of the [[Definition:Polynomial over Complex Numbers|polynomial]]:
:$z^4 + 81$
are:
:$\set {3 \cis 45 \degrees, 3 \cis 135 \degrees, 3 \cis 225 \degrees, 3 \cis 315 \degrees}$ | From [[Factorisation of z^n+1|Factorisation of $z^n + 1$]]:
:$z^4 + 1 = \ds \prod_{k \mathop = 0}^3 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 4}$
Thus:
:$z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 4}$
{{begin-eqn}}
{{eqn | n = k = 0
| l = z
| r = \cos \dfrac \pi 4 + i \sin \dfrac \pi 4
... | Roots of Complex Number/Examples/z^4 + 81 = 0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^4_+_81_=_0 | https://proofwiki.org/wiki/Roots_of_Complex_Number/Examples/z^4_+_81_=_0 | [
"Examples of Complex Roots"
] | [
"Definition:Root of Polynomial",
"Definition:Polynomial/Complex Numbers"
] | [
"Factorisation of z^n+1"
] |
proofwiki-15569 | Equivalent Norms on Rational Numbers | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$.
Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent {{iff}}:
:$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$ | === Necessary Condition ===
{{:Equivalent Norms on Rational Numbers/Necessary Condition}}{{qed|lemma}} | Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be [[Definition:Norm on Division Ring|norms]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$.
Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are [[Definition:Equivalent Division Ring Norms|equivalent]] {{iff}}:
:$\exists \alpha \in \R_{\g... | === [[Equivalent Norms on Rational Numbers/Necessary Condition|Necessary Condition]] ===
{{:Equivalent Norms on Rational Numbers/Necessary Condition}}{{qed|lemma}} | Equivalent Norms on Rational Numbers | https://proofwiki.org/wiki/Equivalent_Norms_on_Rational_Numbers | https://proofwiki.org/wiki/Equivalent_Norms_on_Rational_Numbers | [
"Normed Division Rings"
] | [
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms"
] | [
"Equivalent Norms on Rational Numbers/Necessary Condition"
] |
proofwiki-15570 | Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1 | :$\exists n \in \N: 0 < \norm n < 1$ | Because $\norm {\, \cdot \,}$ is non-trivial:
:$\exists \dfrac a b \in \Q : 0 < \norm {\dfrac a b} \mbox { and } \norm {\dfrac a b} \ne 1$
By Norm of Inverse:
:$\norm {\dfrac a b} > 1 \implies \norm {\dfrac b a} < 1$
Hence either $\norm {\dfrac a b} < 1$ or $\norm {\dfrac b a} < 1$.
{{WLOG}}, assume $\norm {\dfrac a b}... | :$\exists n \in \N: 0 < \norm n < 1$ | Because $\norm {\, \cdot \,}$ is [[Definition:Nontrivial Division Ring Norm|non-trivial]]:
:$\exists \dfrac a b \in \Q : 0 < \norm {\dfrac a b} \mbox { and } \norm {\dfrac a b} \ne 1$
By [[Properties of Norm on Division Ring/Norm of Inverse|Norm of Inverse]]:
:$\norm {\dfrac a b} > 1 \implies \norm {\dfrac b a} < 1$
... | Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.1 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm/Lemma_2.1 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm/Lemma_2.1 | [
"Ostrowski's Theorem"
] | [] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Properties of Norm on Division Ring/Norm of Inverse",
"Properties of Norm on Division Ring/Norm of Quotient",
"Definition:Absolute Value",
"Properties of Norm on Division Ring/Norm of Negative",
"Definition:Non-Archimedean/Norm (Division Ring)"
] |
proofwiki-15571 | Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2 | :$n_0$ is a prime number. | {{AimForCont}} $n_0$ is a composite number.
Let $n_1, n_2 \in \N$ such that $n_1, n_2 < n_0$ and $n_0 = n_1 n_2$.
By the definition of $n_0$ then:
:$\norm {n_1} = 1$
:$\norm {n_2} = 1$
By {{NormAxiomNonArch|2}}:
:$\norm {n_0} = \norm {n_1 n_2} = \norm {n_1} \norm {n_2} = 1$
This contradicts the assumption that $\no... | :$n_0$ is a [[Definition:Prime Number|prime number]]. | {{AimForCont}} $n_0$ is a [[Definition:Composite Number|composite number]].
Let $n_1, n_2 \in \N$ such that $n_1, n_2 < n_0$ and $n_0 = n_1 n_2$.
By the definition of $n_0$ then:
:$\norm {n_1} = 1$
:$\norm {n_2} = 1$
By {{NormAxiomNonArch|2}}:
:$\norm {n_0} = \norm {n_1 n_2} = \norm {n_1} \norm {n_2} = 1$
This ... | Ostrowski's Theorem/Non-Archimedean Norm/Lemma 2.2 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm/Lemma_2.2 | https://proofwiki.org/wiki/Ostrowski's_Theorem/Non-Archimedean_Norm/Lemma_2.2 | [
"Ostrowski's Theorem"
] | [
"Definition:Prime Number"
] | [
"Definition:Composite Number",
"Definition:Contradiction",
"Definition:Prime Number"
] |
proofwiki-15572 | Sum of Cosines of k pi over 5 | :$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$ | We have:
{{begin-eqn}}
{{eqn | l = 144 \degrees
| r = 180 \degrees - 36 \degrees
| c =
}}
{{eqn | ll= \leadsto
| l = \cos 36 \degrees
| r = -\cos 144 \degrees
| c = Cosine of Supplementary Angle
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = 108 \degrees
| r = 180 \degrees - 72 \degr... | :$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$ | We have:
{{begin-eqn}}
{{eqn | l = 144 \degrees
| r = 180 \degrees - 36 \degrees
| c =
}}
{{eqn | ll= \leadsto
| l = \cos 36 \degrees
| r = -\cos 144 \degrees
| c = [[Cosine of Supplementary Angle]]
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = 108 \degrees
| r = 180 \degrees - 7... | Sum of Cosines of k pi over 5 | https://proofwiki.org/wiki/Sum_of_Cosines_of_k_pi_over_5 | https://proofwiki.org/wiki/Sum_of_Cosines_of_k_pi_over_5 | [
"Complex 5th Roots of Unity"
] | [] | [
"Cosine of Supplementary Angle",
"Cosine of Supplementary Angle",
"File:Sum of Cosines of k pi over 5.png"
] |
proofwiki-15573 | Sum of Products of nth Roots of Unity taken up to n-1 at a Time is Zero | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $U_n = \set {z \in \C: z^n = 1}$ be the complex $n$th roots of unity.
Then the sum of the products of the elements of $U_n$ taken $2, 3, 4, \dotsc n - 1$ at a time is zero. | The elements of $U_n = \set {z \in \C: z^n = 1}$ are the solutions to the equation:
:$z^n - 1 = 0$
Thus by definition the coefficients of the powers of $z$:
:$z^2, z^3, \ldots, z^{n - 1}$
are all zero.
The result follows directly from Viète's Formulas.
{{qed}} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $U_n = \set {z \in \C: z^n = 1}$ be the [[Definition:Complex Roots of Unity|complex $n$th roots of unity]].
Then the [[Definition:Sum (Addition)|sum]] of the [[Definition:Product (Algebra)|products]] of the [[Definitio... | The [[Definition:Element|elements]] of $U_n = \set {z \in \C: z^n = 1}$ are the solutions to the equation:
:$z^n - 1 = 0$
Thus by definition the [[Definition:Coefficient|coefficients]] of the [[Definition:Power (Algebra)|powers]] of $z$:
:$z^2, z^3, \ldots, z^{n - 1}$
are all [[Definition:Zero (Number)|zero]].
The r... | Sum of Products of nth Roots of Unity taken up to n-1 at a Time is Zero | https://proofwiki.org/wiki/Sum_of_Products_of_nth_Roots_of_Unity_taken_up_to_n-1_at_a_Time_is_Zero | https://proofwiki.org/wiki/Sum_of_Products_of_nth_Roots_of_Unity_taken_up_to_n-1_at_a_Time_is_Zero | [
"Viète's Formulas",
"Complex Roots of Unity"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Root of Unity/Complex",
"Definition:Addition/Sum",
"Definition:Multiplication/Product",
"Definition:Element",
"Definition:Zero (Number)"
] | [
"Definition:Element",
"Definition:Coefficient",
"Definition:Power (Algebra)",
"Definition:Zero (Number)",
"Viète's Formulas"
] |
proofwiki-15574 | Absolute Value of Complex Dot Product is Commutative | Let $z_1$ and $z_2$ be complex numbers.
Let $z_1 \circ z_2$ denote the (complex) dot product of $z_1$ and $z_2$.
Then:
:$\size {z_1 \circ z_2} = \size {z_2 \circ z_1}$
where $\size {\, \cdot \,}$ denotes the absolute value function. | From Dot Product Operator is Commutative:
:$z_1 \circ z_2 = z_2 \circ z_1$
The result follows trivially.
{{qed}} | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Let $z_1 \circ z_2$ denote the [[Definition:Complex Dot Product|(complex) dot product]] of $z_1$ and $z_2$.
Then:
:$\size {z_1 \circ z_2} = \size {z_2 \circ z_1}$
where $\size {\, \cdot \,}$ denotes the [[Definition:Absolute Value|absolute value f... | From [[Dot Product Operator is Commutative]]:
:$z_1 \circ z_2 = z_2 \circ z_1$
The result follows trivially.
{{qed}} | Absolute Value of Complex Dot Product is Commutative | https://proofwiki.org/wiki/Absolute_Value_of_Complex_Dot_Product_is_Commutative | https://proofwiki.org/wiki/Absolute_Value_of_Complex_Dot_Product_is_Commutative | [
"Absolute Value Function",
"Complex Dot Product"
] | [
"Definition:Complex Number",
"Definition:Dot Product/Complex",
"Definition:Absolute Value"
] | [
"Dot Product Operator is Commutative"
] |
proofwiki-15575 | Absolute Value of Complex Cross Product is Commutative | Let $z_1$ and $z_2$ be complex numbers.
Let $z_1 \times z_2$ denote the (complex) cross product of $z_1$ and $z_2$.
Then:
:$\size {z_1 \times z_2} = \size {z_2 \times z_1}$
where $\size {\, \cdot \,}$ denotes the absolute value function. | {{begin-eqn}}
{{eqn | l = \size {z_2 \times z_1}
| r = \size {-z_1 \times z_2}
| c = Complex Cross Product is Anticommutative
}}
{{eqn | r = \size {z_1 \times z_2}
| c = {{Defof|Absolute Value}}
}}
{{end-eqn}}
Hence the result.
{{qed}} | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Let $z_1 \times z_2$ denote the [[Definition:Complex Cross Product|(complex) cross product]] of $z_1$ and $z_2$.
Then:
:$\size {z_1 \times z_2} = \size {z_2 \times z_1}$
where $\size {\, \cdot \,}$ denotes the [[Definition:Absolute Value|absolute ... | {{begin-eqn}}
{{eqn | l = \size {z_2 \times z_1}
| r = \size {-z_1 \times z_2}
| c = [[Complex Cross Product is Anticommutative]]
}}
{{eqn | r = \size {z_1 \times z_2}
| c = {{Defof|Absolute Value}}
}}
{{end-eqn}}
Hence the result.
{{qed}} | Absolute Value of Complex Cross Product is Commutative | https://proofwiki.org/wiki/Absolute_Value_of_Complex_Cross_Product_is_Commutative | https://proofwiki.org/wiki/Absolute_Value_of_Complex_Cross_Product_is_Commutative | [
"Absolute Value Function",
"Complex Cross Product"
] | [
"Definition:Complex Number",
"Definition:Vector Cross Product/Complex",
"Definition:Absolute Value"
] | [
"Vector Cross Product is Anticommutative/Complex"
] |
proofwiki-15576 | Area of Quadrilateral in Determinant Form | Let $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$, $C = \tuple {x_3, y_3}$ and $D = \tuple {x_4, y_4}$ be points in the Cartesian plane.
Let $A$, $B$, $C$ and $D$ form the vertices of a quadrilateral.
The area $\AA$ of $\Box ABCD$ is given by:
:<nowiki>$\AA = \dfrac 1 2 \paren {\size {\paren {\begin{vmatrix}
x_1 & y... | $\Box ABCD$ can be divided into $2$ triangles: $\triangle ABC$ and $\triangle ADC$.
Hence $\AA$ is the sum of the areas of $\triangle ABC$ and $\triangle ADC$.
From Area of Triangle in Determinant Form:
{{begin-eqn}}
{{eqn | l = \map \Area {\triangle ABC}
| r = <nowiki>\dfrac 1 2 \size {\paren {\begin{vmatrix}
x_... | Let $A = \tuple {x_1, y_1}$, $B = \tuple {x_2, y_2}$, $C = \tuple {x_3, y_3}$ and $D = \tuple {x_4, y_4}$ be [[Definition:Point|points]] in the [[Definition:Cartesian Plane|Cartesian plane]].
Let $A$, $B$, $C$ and $D$ form the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Quadrilateral|quadrilateral]].
... | $\Box ABCD$ can be divided into $2$ [[Definition:Triangle (Geometry)|triangles]]: $\triangle ABC$ and $\triangle ADC$.
Hence $\AA$ is the [[Definition:Sum (Addition)|sum]] of the [[Definition:Area|areas]] of $\triangle ABC$ and $\triangle ADC$.
From [[Area of Triangle in Determinant Form]]:
{{begin-eqn}}
{{eqn | l =... | Area of Quadrilateral in Determinant Form | https://proofwiki.org/wiki/Area_of_Quadrilateral_in_Determinant_Form | https://proofwiki.org/wiki/Area_of_Quadrilateral_in_Determinant_Form | [
"Area of Quadrilateral in Determinant Form",
"Areas of Quadrilaterals"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Polygon/Vertex",
"Definition:Quadrilateral",
"Definition:Area"
] | [
"Definition:Triangle (Geometry)",
"Definition:Addition/Sum",
"Definition:Area",
"Area of Triangle in Determinant Form"
] |
proofwiki-15577 | Three Points in Ultrametric Space have Two Equal Distances/Corollary 5 | Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.
Let $a, b \in \Z_{\ne 0}$ be coprime:
:$a \perp b$
Then:
:$\norm a = 1$ or $\norm b = 1$ | By Bézout's Identity then:
:$\exists n, m \in \Z : m a + n b = 1$
By Norm of Unity:
:$\norm {m a + n b} = 1$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|5}}:
:$\norm a, \norm b, \norm n, \norm m \le 1$
Let $\norm a < 1$.
By {{Norm-axiom-mult|2}}:
:$\norm {m a} = \norm m \norm a < 1$
Hence:
... | Let $\norm {\, \cdot \,}$ be a [[Definition:Nontrivial Division Ring Norm|non-trivial]] [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] [[Definition:Norm on Division Ring|norm]] on the [[Definition:Rational Numbers|rational numbers]] $\Q$.
Let $a, b \in \Z_{\ne 0}$ be [[Definition:Coprime Integers|co... | By [[Bézout's Identity]] then:
:$\exists n, m \in \Z : m a + n b = 1$
By [[Properties of Norm on Division Ring/Norm of Unity|Norm of Unity]]:
:$\norm {m a + n b} = 1$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|5}}:
:$\norm a, \norm b, \norm n, \norm m \le 1$
Let $\norm a < 1$.
By {{N... | Three Points in Ultrametric Space have Two Equal Distances/Corollary 5 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_5 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_5 | [
"Three Points in Ultrametric Space have Two Equal Distances"
] | [
"Definition:Trivial Norm/Division Ring/Nontrivial",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Norm/Division Ring",
"Definition:Rational Number",
"Definition:Coprime/Integers"
] | [
"Bézout's Identity",
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15578 | Exponential of 2 m i Arccotangent of p | :$\map \exp {2 m i \arccot p} \paren {\dfrac {p i + 1} {p i - 1} }^m = 1$ | Let $z = \arccot p$.
Then:
{{begin-eqn}}
{{eqn | l = p
| r = \frac {\cos z} {\sin z}
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadsto
| l = p
| r = i \dfrac {\map \exp {i z} + \map \exp {-i z} } {\map \exp {i z} - \map \exp {-i z} }
| c = Euler's Cotangent Identity
}}
{{eqn | r = ... | :$\map \exp {2 m i \arccot p} \paren {\dfrac {p i + 1} {p i - 1} }^m = 1$ | Let $z = \arccot p$.
Then:
{{begin-eqn}}
{{eqn | l = p
| r = \frac {\cos z} {\sin z}
| c = {{Defof|Real Arccotangent}}
}}
{{eqn | ll= \leadsto
| l = p
| r = i \dfrac {\map \exp {i z} + \map \exp {-i z} } {\map \exp {i z} - \map \exp {-i z} }
| c = [[Euler's Cotangent Identity]]
}}
{{eqn ... | Exponential of 2 m i Arccotangent of p | https://proofwiki.org/wiki/Exponential_of_2_m_i_Arccotangent_of_p | https://proofwiki.org/wiki/Exponential_of_2_m_i_Arccotangent_of_p | [
"Arccotangent Function"
] | [] | [
"Euler's Cotangent Identity"
] |
proofwiki-15579 | Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero | Let $z \in \C$ be a complex number.
Then:
:$\cmod {z - 1} < \cmod {z + 1} \iff \map \Re z > 0$ | {{begin-eqn}}
{{eqn | l = \cmod {z - 1}
| o = <
| r = \cmod {z + 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \cmod {z + 1}
| o = >
| r = \cmod {z - 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {x + 1}^2 + y^2
| o = >
| r = \paren {x - 1}^2 + y^2
... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Then:
:$\cmod {z - 1} < \cmod {z + 1} \iff \map \Re z > 0$ | {{begin-eqn}}
{{eqn | l = \cmod {z - 1}
| o = <
| r = \cmod {z + 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \cmod {z + 1}
| o = >
| r = \cmod {z - 1}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {x + 1}^2 + y^2
| o = >
| r = \paren {x - 1}^2 + y^2
... | Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero | https://proofwiki.org/wiki/Modulus_z_-_1_Less_than_Modulus_z_+_1_iff_Real_z_Greater_than_Zero | https://proofwiki.org/wiki/Modulus_z_-_1_Less_than_Modulus_z_+_1_iff_Real_z_Greater_than_Zero | [
"Complex Modulus"
] | [
"Definition:Complex Number"
] | [] |
proofwiki-15580 | Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition | Let $z_1$, $z_2$ and $z_3$ be complex numbers.
Let $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle.
Then:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$ | Let $T$ be the equilateral triangle whose vertices are $z_1$, $z_2$ and $z_3$.
We have that $z_2 - z_1$ and $z_3 - z_1$ are two sides of $T$ which meet at $z_1$.
From the geometry of $T$ it follows that $z_2 - z_1$ is at an angle of $\pi/3$ to $z_3 - z_1$.
Similarly, $z_1 - z_3$ and $z_2 - z_3$ are two sides of $T$ whi... | Let $z_1$, $z_2$ and $z_3$ be [[Definition:Complex Number|complex numbers]].
Let $z_1$, $z_2$ and $z_3$ represent on the [[Definition:Complex Plane|complex plane]] the [[Definition:Vertex of Polygon|vertices]] of an [[Definition:Equilateral Triangle|equilateral triangle]].
Then:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1... | Let $T$ be the [[Definition:Equilateral Triangle|equilateral triangle]] whose [[Definition:Vertex of Polygon|vertices]] are $z_1$, $z_2$ and $z_3$.
We have that $z_2 - z_1$ and $z_3 - z_1$ are two sides of $T$ which meet at $z_1$.
From the geometry of $T$ it follows that $z_2 - z_1$ is at an angle of $\pi/3$ to $z_3 ... | Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane/Sufficient_Condition | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane/Sufficient_Condition | [
"Vertices of Equilateral Triangle in Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Equilateral"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Vertex"
] |
proofwiki-15581 | Vertices of Equilateral Triangle in Complex Plane/Necessary Condition | Let $z_1$, $z_2$ and $z_3$ be complex numbers.
Let $z_1$, $z_2$ and $z_3$ fulfil the condition:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Then $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle. | Let:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Then:
{{begin-eqn}}
{{eqn | l = {z_1}^2 + {z_2}^2 + {z_3}^2
| r = z_1 z_2 + z_2 z_3 + z_3 z_1
| c =
}}
{{eqn | ll= \leadsto
| l = {z_2}^2 - z_1 z_2 - z_2 z_3 + z_3 z_1
| r = - {z_1}^2 - {z_3}^2 + 2 z_3 z_1
| c =
}}
{{eqn |... | Let $z_1$, $z_2$ and $z_3$ be [[Definition:Complex Number|complex numbers]].
Let $z_1$, $z_2$ and $z_3$ fulfil the condition:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Then $z_1$, $z_2$ and $z_3$ represent on the [[Definition:Complex Plane|complex plane]] the [[Definition:Vertex of Polygon|vertice... | Let:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
Then:
{{begin-eqn}}
{{eqn | l = {z_1}^2 + {z_2}^2 + {z_3}^2
| r = z_1 z_2 + z_2 z_3 + z_3 z_1
| c =
}}
{{eqn | ll= \leadsto
| l = {z_2}^2 - z_1 z_2 - z_2 z_3 + z_3 z_1
| r = - {z_1}^2 - {z_3}^2 + 2 z_3 z_1
| c =
}}
{{eqn ... | Vertices of Equilateral Triangle in Complex Plane/Necessary Condition | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane/Necessary_Condition | https://proofwiki.org/wiki/Vertices_of_Equilateral_Triangle_in_Complex_Plane/Necessary_Condition | [
"Vertices of Equilateral Triangle in Complex Plane"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Complex Plane",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)/Equilateral"
] | [
"Definition:Angle",
"Definition:Angle",
"Definition:Angle",
"Definition:Triangle (Geometry)/Equilateral"
] |
proofwiki-15582 | Squares of Diagonals of Parallelogram | Let $ABCD$ be a parallelogram.
:400px
Then:
:$AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$ | {{begin-eqn}}
{{eqn | l = AC^2
| r = AB^2 + BC^2 - 2 (AB) (BC) \cos \angle B
| c = Cosine Rule
}}
{{eqn | l = BD^2
| r = BC^2 + CD^2 - 2 (CD) (BC) \cos \angle C
| c = Cosine Rule
}}
{{eqn | r = DA^2 + CD^2 - 2 (AB) (CD) \cos \angle C
| c = as $AB = CD$ and $BC = DA$
}}
{{eqn | ll= \leadsto... | Let $ABCD$ be a [[Definition:Parallelogram|parallelogram]].
:[[File:DiameterOfParallelogram.png|400px]]
Then:
:$AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$ | {{begin-eqn}}
{{eqn | l = AC^2
| r = AB^2 + BC^2 - 2 (AB) (BC) \cos \angle B
| c = [[Cosine Rule]]
}}
{{eqn | l = BD^2
| r = BC^2 + CD^2 - 2 (CD) (BC) \cos \angle C
| c = [[Cosine Rule]]
}}
{{eqn | r = DA^2 + CD^2 - 2 (AB) (CD) \cos \angle C
| c = as $AB = CD$ and $BC = DA$
}}
{{eqn | ll= ... | Squares of Diagonals of Parallelogram | https://proofwiki.org/wiki/Squares_of_Diagonals_of_Parallelogram | https://proofwiki.org/wiki/Squares_of_Diagonals_of_Parallelogram | [
"Parallelograms"
] | [
"Definition:Quadrilateral/Parallelogram",
"File:DiameterOfParallelogram.png"
] | [
"Law of Cosines",
"Law of Cosines",
"Definition:Supplementary Angles",
"Cosine of Supplementary Angle"
] |
proofwiki-15583 | Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder | Let $\sigma$ be the surface of a cylinder.
Let $\sigma$ be embedded in 3-dimensional Euclidean space.
Let $\sigma$ be parameterised by $\tuple {\phi, z}$ as
:$\mathbf r = \tuple {a \cos \phi, a \sin \phi, z}$
where
:$a > 0$
and
:$z, \phi \in \R$
Then geodesics on $\sigma$ are of the following form:
:$z = C_1 \phi + C_2... | From the given parametrization it follows that:
{{begin-eqn}}
{{eqn | l = E
| r = \mathbf r_\phi \cdot \mathbf r_\phi
}}
{{eqn | r = \tuple {-a \sin \phi, a \cos \phi, 0} \cdot \tuple {-a \sin \phi, a \cos \phi, 0}
}}
{{eqn | r = a^2
}}
{{eqn | l = F
| r = \mathbf r_\phi \cdot \mathbf r_z
}}
{{eqn | r = \tu... | Let $\sigma$ be the [[Definition:Surface|surface]] of a [[Definition:Right Circular Cylinder|cylinder]].
Let $\sigma$ be embedded in [[Definition:Real Euclidean Space|3-dimensional Euclidean space]].
Let $\sigma$ be [[Definition:Parametric Equation|parameterised]] by $\tuple {\phi, z}$ as
:$\mathbf r = \tuple {a \co... | From the given [[Definition:Parametric Equation|parametrization]] it follows that:
{{begin-eqn}}
{{eqn | l = E
| r = \mathbf r_\phi \cdot \mathbf r_\phi
}}
{{eqn | r = \tuple {-a \sin \phi, a \cos \phi, 0} \cdot \tuple {-a \sin \phi, a \cos \phi, 0}
}}
{{eqn | r = a^2
}}
{{eqn | l = F
| r = \mathbf r_\phi ... | Geodesic Equation/2d Surface Embedded in 3d Euclidean Space/Cylinder | https://proofwiki.org/wiki/Geodesic_Equation/2d_Surface_Embedded_in_3d_Euclidean_Space/Cylinder | https://proofwiki.org/wiki/Geodesic_Equation/2d_Surface_Embedded_in_3d_Euclidean_Space/Cylinder | [
"Calculus of Variations"
] | [
"Definition:Surface",
"Definition:Right Circular Cylinder",
"Definition:Euclidean Space/Real",
"Definition:Parametric Equation",
"Definition:Geodesic",
"Definition:Real Number",
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Definition:Parametric Equation",
"Definition:Real Function",
"Definition:First Fundamental Form",
"Definition:Differentiable Mapping",
"Geodesic Equation/2d Surface Embedded in 3d Euclidean Space",
"Definition:Differential Equation",
"Definition:Real Number",
"Definition:Primitive (Calculus)/Constant... |
proofwiki-15584 | Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary | Let $z_1$ and $z_2$ be complex numbers such that:
:$\cmod {z_1 + z_2} = \cmod {z_1 - z_2}$
Then $\dfrac {z_2} {z_1}$ is wholly imaginary. | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}
| r = \cmod {z_1 - z_2}
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x_1 + x_2}^2 + \paren {y_1 + y_2}^2
| r = \paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2
| c = {{Defof|Complex Modulus}}
}}
{... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]] such that:
:$\cmod {z_1 + z_2} = \cmod {z_1 - z_2}$
Then $\dfrac {z_2} {z_1}$ is [[Definition:Wholly Imaginary|wholly imaginary]]. | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
Then:
{{begin-eqn}}
{{eqn | l = \cmod {z_1 + z_2}
| r = \cmod {z_1 - z_2}
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {x_1 + x_2}^2 + \paren {y_1 + y_2}^2
| r = \paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2
| c = {{Defof|Complex Modulus}}
}... | Modulus of Sum equals Modulus of Distance implies Quotient is Imaginary | https://proofwiki.org/wiki/Modulus_of_Sum_equals_Modulus_of_Distance_implies_Quotient_is_Imaginary | https://proofwiki.org/wiki/Modulus_of_Sum_equals_Modulus_of_Distance_implies_Quotient_is_Imaginary | [
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Wholly Imaginary"
] | [
"Square of Sum",
"Square of Difference",
"Definition:Complex Number/Wholly Imaginary"
] |
proofwiki-15585 | Difference of Even Powers of z + a and z - a | Let $m \in \Z$ be an integer such that $m > 1$.
Then for all complex number $z$:
:$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \ds \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$ | From Factors of Difference of Two Even Powers:
:$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \ds \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$
Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get:
{{begin-eqn}}
{{eqn | o =
| r = \paren {z + a}^{2 m} - \par... | Let $m \in \Z$ be an [[Definition:Integer|integer]] such that $m > 1$.
Then for all [[Definition:Complex Number|complex number]] $z$:
:$\paren {z + a}^{2 m} - \paren {z - a}^{2 m} = 4 m a z \ds \prod_{k \mathop = 1}^{m - 1} \paren {z^2 + a^2 \cot^2 \dfrac {k \pi} {2 m} }$ | From [[Factors of Difference of Two Even Powers]]:
:$x^{2 n} - y^{2 n} = \paren {x - y} \paren {x + y} \ds \prod_{k \mathop = 1}^{n - 1} \paren {x^2 - 2 x y \cos \dfrac {k \pi} n + y^2}$
Substituting $z + a$ for $x$, $z - a$ for $y$, and $m$ for $n$ we get:
{{begin-eqn}}
{{eqn | o =
| r = \paren {z + a}^{2 m}... | Difference of Even Powers of z + a and z - a | https://proofwiki.org/wiki/Difference_of_Even_Powers_of_z_+_a_and_z_-_a | https://proofwiki.org/wiki/Difference_of_Even_Powers_of_z_+_a_and_z_-_a | [
"Difference of Two Powers"
] | [
"Definition:Integer",
"Definition:Complex Number"
] | [
"Factors of Difference of Two Even Powers",
"Sine of Supplementary Angle",
"Sine of Right Angle",
"Translation of Index Variable of Product",
"Product of Sines of Fractions of Pi"
] |
proofwiki-15586 | Real Part of Complex Product | Let $z_1$ and $z_2$ be complex numbers.
Then:
:$\map \Re {z_1 z_2} = \map \Re {z_1} \map \Re {z_2} - \map \Im {z_1} \map \Im {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
By definition of complex multiplication:
:$z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Re {z_1 z_2}
| r = x_1 x_2 - y_1 y_2
| c = {{Defof|Real Part}}
}}
{{eqn | r = \map \Re {z_1} \map \Re {z_2} - \map \Im... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\map \Re {z_1 z_2} = \map \Re {z_1} \map \Re {z_2} - \map \Im {z_1} \map \Im {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
By definition of [[Definition:Complex Multiplication|complex multiplication]]:
:$z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Re {z_1 z_2}
| r = x_1 x_2 - y_1 y_2
| c = {{Defof|Real Part}}
}}
{{eqn | r... | Real Part of Complex Product | https://proofwiki.org/wiki/Real_Part_of_Complex_Product | https://proofwiki.org/wiki/Real_Part_of_Complex_Product | [
"Complex Multiplication"
] | [
"Definition:Complex Number"
] | [
"Definition:Multiplication/Complex Numbers"
] |
proofwiki-15587 | Imaginary Part of Complex Product | Let $z_1$ and $z_2$ be complex numbers.
Then:
:$\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
By definition of complex multiplication:
:$z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$
Then
{{begin-eqn}}
{{eqn | l = \map \Im {z_1 z_2}
| r = x_1 y_2 + x_2 y_1
| c = {{Defof|Imaginary Part}}
}}
{{eqn | r = \map \Re {z_1} \, \map \Im {z_2} + \... | Let $z_1$ and $z_2$ be [[Definition:Complex Number|complex numbers]].
Then:
:$\map \Im {z_1 z_2} = \map \Re {z_1} \, \map \Im {z_2} + \map \Im {z_1} \, \map \Re {z_2}$ | Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$.
By definition of [[Definition:Complex Multiplication|complex multiplication]]:
:$z_1 z_2 = x_1 x_2 - y_1 y_2 + i \paren {x_1 y_2 + x_2 y_1}$
Then
{{begin-eqn}}
{{eqn | l = \map \Im {z_1 z_2}
| r = x_1 y_2 + x_2 y_1
| c = {{Defof|Imaginary Part}}
}}
{{eqn... | Imaginary Part of Complex Product | https://proofwiki.org/wiki/Imaginary_Part_of_Complex_Product | https://proofwiki.org/wiki/Imaginary_Part_of_Complex_Product | [
"Complex Multiplication"
] | [
"Definition:Complex Number"
] | [
"Definition:Multiplication/Complex Numbers"
] |
proofwiki-15588 | Equation for Perpendicular Bisector of Two Points | Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two points in the cartesian plane.
Let $L$ be the perpendicular bisector of the straight line through $z_1$ and $z_2$ in the complex plane.
$L$ can be expressed by the equation:
:$y - \dfrac {y_1 + y_2} 2 = \dfrac {x_1 - x_2} {y_2 - y_1} \paren {x - \dfrac {x_1 + x_2} ... | Let $M$ be the straight line passing through $z_1$ and $z_2$.
Let $Q$ be the midpoint of $M$.
We have that:
:$Q = \tuple {\dfrac {x_1 + x_2} 2, \dfrac {y_1 + y_2} 2}$
The slope of $M$ is $\dfrac {y_2 - y_1} {x_2 - x_1}$.
As $L$ is perpendicular to the $M$, its slope is $\dfrac {x_1 - x_2} {y_2 - y_1}$.
Thus by Equation... | Let $\tuple {x_1, y_1}$ and $\tuple {y_1, y_2}$ be two [[Definition:Point|points]] in the [[Definition:Cartesian Plane|cartesian plane]].
Let $L$ be the [[Definition:Perpendicular Bisector|perpendicular bisector]] of the [[Definition:Straight Line|straight line]] through $z_1$ and $z_2$ in the [[Definition:Complex Pla... | Let $M$ be the [[Definition:Straight Line|straight line]] passing through $z_1$ and $z_2$.
Let $Q$ be the [[Definition:Midpoint|midpoint]] of $M$.
We have that:
:$Q = \tuple {\dfrac {x_1 + x_2} 2, \dfrac {y_1 + y_2} 2}$
The [[Definition:Slope of Straight Line|slope]] of $M$ is $\dfrac {y_2 - y_1} {x_2 - x_1}$.
As ... | Equation for Perpendicular Bisector of Two Points | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points | https://proofwiki.org/wiki/Equation_for_Perpendicular_Bisector_of_Two_Points | [
"Straight Lines",
"Perpendiculars"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Perpendicular Bisector",
"Definition:Line/Straight Line",
"Definition:Complex Number/Complex Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Midpoint",
"Definition:Slope/Straight Line",
"Definition:Right Angle/Perpendicular",
"Definition:Slope/Straight Line",
"Equation of Straight Line in Plane/Point-Slope Form",
"Category:Straight Lines",
"Category:Perpendiculars"
] |
proofwiki-15589 | Element of Center in Group whose Order is Power of 2 | Let $n \in \Z$ be an integer such that $n \ge 2$.
Let $G$ be a group whose order is $2^n$.
Let $x \in G$ be of order $2^{n - 1}$ in $G$.
Then $x^{2^{n - 2} }$ is an element of the center of $G$. | Let $H = \gen x$ be the subgroup of $G$ generated by $x$.
We have that the index of $H$ in $G$ is $2$.
Then by Subgroup of Index 2 is Normal, $H$ is normal in $G$.
Let $y = x^{2^{n - 2} }$.
We have that $y \in H$, by definition of the construction of $H$.
Then the order of $y$ in $G$ is $2$.
As $H$ is normal, we have:
... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 2$.
Let $G$ be a [[Definition:Group|group]] whose [[Definition:Order of Group|order]] is $2^n$.
Let $x \in G$ be of [[Definition:Order of Group Element|order]] $2^{n - 1}$ in $G$.
Then $x^{2^{n - 2} }$ is an [[Definition:Element|element]] of the [... | Let $H = \gen x$ be the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by $x$.
We have that the [[Definition:Index of Subgroup|index]] of $H$ in $G$ is $2$.
Then by [[Subgroup of Index 2 is Normal]], $H$ is [[Definition:Normal Subgroup|normal]] in $G$.
Let $y = x^{2^{n - 2} }$.
We have that $y \in H$,... | Element of Center in Group whose Order is Power of 2 | https://proofwiki.org/wiki/Element_of_Center_in_Group_whose_Order_is_Power_of_2 | https://proofwiki.org/wiki/Element_of_Center_in_Group_whose_Order_is_Power_of_2 | [
"Centers of Groups"
] | [
"Definition:Integer",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Order of Group Element",
"Definition:Element",
"Definition:Center (Abstract Algebra)/Group"
] | [
"Definition:Generated Subgroup",
"Definition:Index of Subgroup",
"Subgroup of Index 2 is Normal",
"Definition:Normal Subgroup",
"Definition:Order of Group Element",
"Definition:Normal Subgroup",
"Order of Conjugate Element equals Order of Element",
"Definition:Order of Group Element",
"Order of Powe... |
proofwiki-15590 | Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups | Let $G$ be a group.
Let $L$ and $M$ be abelian subgroups of $G$.
Let $H = \gen {L, M}$ be the subgroup of $G$ generated by $L$ and $M$.
Then $L \cap M$ is a normal subgroup of $H$. | From
We have that $L$ and $M$ are abelian.
From Intersection of Subgroups is Subgroup, we have that $L \cap M$ is a subgroup of both $L$ and $M$.
From Subgroup of Abelian Group is Normal, we also have that $L \cap M$ is a normal subgroup of both $L$ and $M$.
Thus from Normalizer of Subgroup is Largest Subgroup contain... | Let $G$ be a [[Definition:Group|group]].
Let $L$ and $M$ be [[Definition:Abelian Group|abelian]] [[Definition:Subgroup|subgroups]] of $G$.
Let $H = \gen {L, M}$ be the [[Definition:Generated Subgroup|subgroup of $G$ generated]] by $L$ and $M$.
Then $L \cap M$ is a [[Definition:Normal Subgroup|normal subgroup]] of $... | From
We have that $L$ and $M$ are [[Definition:Abelian Group|abelian]].
From [[Intersection of Subgroups is Subgroup]], we have that $L \cap M$ is a [[Definition:Subgroup|subgroup]] of both $L$ and $M$.
From [[Subgroup of Abelian Group is Normal]], we also have that $L \cap M$ is a [[Definition:Normal Subgroup|norm... | Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups | https://proofwiki.org/wiki/Intersection_of_Abelian_Subgroups_is_Normal_Subgroup_of_Subgroup_Generated_by_those_Subgroups | https://proofwiki.org/wiki/Intersection_of_Abelian_Subgroups_is_Normal_Subgroup_of_Subgroup_Generated_by_those_Subgroups | [
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Abelian Group",
"Definition:Subgroup",
"Definition:Generated Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Abelian Group",
"Intersection of Subgroups is Subgroup",
"Definition:Subgroup",
"Subgroup of Abelian Group is Normal",
"Definition:Normal Subgroup",
"Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup",
"Normalizer is Subgroup",
"Definition:Subgroup",
... |
proofwiki-15591 | Characterisation of Non-Archimedean Division Ring Norms/Corollary 5 | If $\norm {\, \cdot \,}$ is non-Archimedean then:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$
where $n \cdot 1_R =
\begin{cases}
\underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\
0 & : n = 0 \\
\\
-\underbrace {\paren {1_R + 1_R + \dots + 1_R} }_{\text {$-n$ times} } & : n < 0 \\
\end{cases}... | By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|1}}:
:$\sup \set {\norm{n \cdot 1_R}: n \in \N_{> 0}} = 1$
By {{NormAxiomNonArch|1}}:
:$\norm {0 \cdot 1_R} = 0 \le 1$
Let $n < 0$ then:
{{begin-eqn}}
{{eqn | l = \norm {n \cdot 1_R}
| r = \norm {-\underbrace {\paren {1_R + 1_R + \dots + 1_R} ... | If $\norm {\, \cdot \,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] then:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = 1$
where $n \cdot 1_R =
\begin{cases}
\underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} } & : n > 0 \\
0 & : n = 0 \\
\\
-\underbrace {\paren {1_R + 1_R + \dots + ... | By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|1}}:
:$\sup \set {\norm{n \cdot 1_R}: n \in \N_{> 0}} = 1$
By {{NormAxiomNonArch|1}}:
:$\norm {0 \cdot 1_R} = 0 \le 1$
Let $n < 0$ then:
{{begin-eqn}}
{{eqn | l = \norm {n \cdot 1_R}
| r = \norm {-\underbrace {\paren {1_R + 1_R + \dots + 1_... | Characterisation of Non-Archimedean Division Ring Norms/Corollary 5 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_5 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_5 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Properties of Norm on Division Ring/Norm of Negative",
"Characterisation of Non-Archimedean Division Ring Norms"
] |
proofwiki-15592 | Subgroup Containing all Squares of Group Elements is Normal | Let $G$ be a group.
Let $H$ be a subgroup of $G$ with the property that:
:$\forall x \in G: x^2 \in H$
Then $H$ is normal in $G$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {x h}^2 h^{-1} \paren {x^{-1} }^2
| r = x h x h h^{-1} x^{-1} x^{-1}
| c = {{Group-axiom|1}}
}}
{{eqn | r = x h x x^{-1} x^{-1}
| c = {{Group-axiom|3}}
}}
{{eqn | r = x h x^{-1}
| c = {{Group-axiom|3}}
}}
{{end-eqn}}
Because $\paren {x h}^2$ and $\paren ... | Let $G$ be a [[Definition:Group|group]].
Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$ with the property that:
:$\forall x \in G: x^2 \in H$
Then $H$ is [[Definition:Normal Subgroup|normal]] in $G$. | We have:
{{begin-eqn}}
{{eqn | l = \paren {x h}^2 h^{-1} \paren {x^{-1} }^2
| r = x h x h h^{-1} x^{-1} x^{-1}
| c = {{Group-axiom|1}}
}}
{{eqn | r = x h x x^{-1} x^{-1}
| c = {{Group-axiom|3}}
}}
{{eqn | r = x h x^{-1}
| c = {{Group-axiom|3}}
}}
{{end-eqn}}
Because $\paren {x h}^2$ and $\pare... | Subgroup Containing all Squares of Group Elements is Normal | https://proofwiki.org/wiki/Subgroup_Containing_all_Squares_of_Group_Elements_is_Normal | https://proofwiki.org/wiki/Subgroup_Containing_all_Squares_of_Group_Elements_is_Normal | [
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Subgroup",
"Definition:Normal Subgroup"
] | [
"Definition:Element",
"Definition:Normal Subgroup"
] |
proofwiki-15593 | Finite Order Elements of Infinite Abelian Group form Normal Subgroup | Let $G$ be an infinite abelian group.
Let $H \subseteq G$ be the subset of $G$ defined as:
:$H := \set {x \in G: x \text { is of finite order in } G}$
Then $H$ forms a normal subgroup of $G$. | Let $e$ be the identity element of $G$.
From Identity is Only Group Element of Order 1, $\order e = 1$ and so $H \ne \O$.
Let $a \in H$.
Then by Order of Group Element equals Order of Inverse:
:$\order a = \order {a^{-1} }$
and so $a \in H$.
Let $a, b \in H$.
From Order of Product of Abelian Group Elements Divides LCM ... | Let $G$ be an [[Definition:Infinite Group|infinite]] [[Definition:Abelian Group| abelian group]].
Let $H \subseteq G$ be the [[Definition:Subset|subset]] of $G$ defined as:
:$H := \set {x \in G: x \text { is of finite order in } G}$
Then $H$ forms a [[Definition:Normal Subgroup|normal subgroup]] of $G$. | Let $e$ be the [[Definition:Identity Element|identity element]] of $G$.
From [[Identity is Only Group Element of Order 1]], $\order e = 1$ and so $H \ne \O$.
Let $a \in H$.
Then by [[Order of Group Element equals Order of Inverse]]:
:$\order a = \order {a^{-1} }$
and so $a \in H$.
Let $a, b \in H$.
From [[Order... | Finite Order Elements of Infinite Abelian Group form Normal Subgroup | https://proofwiki.org/wiki/Finite_Order_Elements_of_Infinite_Abelian_Group_form_Normal_Subgroup | https://proofwiki.org/wiki/Finite_Order_Elements_of_Infinite_Abelian_Group_form_Normal_Subgroup | [
"Normal Subgroups",
"Infinite Groups",
"Abelian Groups"
] | [
"Definition:Infinite Group",
"Definition:Abelian Group",
"Definition:Subset",
"Definition:Normal Subgroup"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Identity is Only Group Element of Order 1",
"Order of Group Element equals Order of Inverse",
"Order of Product of Abelian Group Elements Divides LCM of Orders of Elements",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Int... |
proofwiki-15594 | Commutator is Identity iff Elements Commute | Let $G$ be a group whose identity is $e$.
Let $x, y \in G$.
Let $\sqbrk {x, y}$ denote the commutator of $x$ and $y$.
Then $\sqbrk {x, y} = e$ {{iff}} $x$ and $y$ commute. | As $G$ is a group, it is {{afortiori}} a monoid.
Hence Product of Commuting Elements with Inverses applies:
:$x y x^{-1} y^{-1} = e = x^{-1} y^{-1} x y$
{{qed}} | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let $x, y \in G$.
Let $\sqbrk {x, y}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $x$ and $y$.
Then $\sqbrk {x, y} = e$ {{iff}} $x$ and $y$ [[Definition:Commuting Elements|commute]]. | As $G$ is a [[Definition:Group|group]], it is {{afortiori}} a [[Definition:Monoid|monoid]].
Hence [[Product of Commuting Elements with Inverses]] applies:
:$x y x^{-1} y^{-1} = e = x^{-1} y^{-1} x y$
{{qed}} | Commutator is Identity iff Elements Commute | https://proofwiki.org/wiki/Commutator_is_Identity_iff_Elements_Commute | https://proofwiki.org/wiki/Commutator_is_Identity_iff_Elements_Commute | [
"Group Commutators"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutator/Group",
"Definition:Commutative/Elements"
] | [
"Definition:Group",
"Definition:Monoid",
"Product of Commuting Elements with Inverses"
] |
proofwiki-15595 | Commutator of Quotient Group Elements | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $\sqbrk {x, y}$ denote the commutator of $x, y \in G$:
:$\sqbrk {x, y} = x^{-1} y^{-1} x y$
Then:
:$\forall x, y \in G: \sqbrk {x N, y N} = \sqbrk {x, y} N$
where $x N$ and $y N$ are left cosets of $N$, and so elements of the quotient group $G / N$ of $G$ by ... | {{begin-eqn}}
{{eqn | l = \sqbrk {x N, y N}
| r = \paren {x N}^{-1} \paren {y N}^{-1} \paren {x N} \paren {y N}
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = \paren {x^{-1} N} \paren {y^{-1} N} \paren {x N} \paren {y N}
| c = Quotient Group is Group: inverse of $x N$ is $x^{-1} N$
}}
{{e... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $\sqbrk {x, y}$ denote the [[Definition:Commutator of Group Elements|commutator]] of $x, y \in G$:
:$\sqbrk {x, y} = x^{-1} y^{-1} x y$
Then:
:$\forall x, y \in G: \sqbrk {x N, y N} = \sqbrk {x, y} N$
w... | {{begin-eqn}}
{{eqn | l = \sqbrk {x N, y N}
| r = \paren {x N}^{-1} \paren {y N}^{-1} \paren {x N} \paren {y N}
| c = {{Defof|Commutator of Group Elements}}
}}
{{eqn | r = \paren {x^{-1} N} \paren {y^{-1} N} \paren {x N} \paren {y N}
| c = [[Quotient Group is Group]]: [[Definition:Inverse Element|inve... | Commutator of Quotient Group Elements | https://proofwiki.org/wiki/Commutator_of_Quotient_Group_Elements | https://proofwiki.org/wiki/Commutator_of_Quotient_Group_Elements | [
"Group Commutators",
"Quotient Groups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Commutator/Group",
"Definition:Coset/Left Coset",
"Definition:Element",
"Definition:Quotient Group"
] | [
"Quotient Group is Group",
"Definition:Inverse (Abstract Algebra)/Inverse"
] |
proofwiki-15596 | Quotient Group is Abelian iff All Commutators in Divisor | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $G / N$ be the quotient group of $G$ by $N$.
Then the quotient group $G / N$ is abelian {{iff}}:
:$\forall x, y \in G: \sqbrk {x, y} \in N$
where $\sqbrk {x, y}$ denotes the commutator of $x$ and $y$.
That is, {{iff}} $x y x^{-1} y^{-1} \in N$ for all $x, y \... | Let $x, y \in G$.
First we establish the following:
{{begin-eqn}}
{{eqn | l = \paren {x N} \paren {y N}
| r = \paren {y N} \paren {x N}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {x N} \paren {y N}
| r = \paren {y N} \paren {x N} N
| c = Quotient Group is Group: $N$ is the identity... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $G / N$ be the [[Definition:Quotient Group|quotient group]] of $G$ by $N$.
Then the [[Definition:Quotient Group|quotient group]] $G / N$ is [[Definition:Abelian Group|abelian]] {{iff}}:
:$\forall x, y \... | Let $x, y \in G$.
First we establish the following:
{{begin-eqn}}
{{eqn | l = \paren {x N} \paren {y N}
| r = \paren {y N} \paren {x N}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {x N} \paren {y N}
| r = \paren {y N} \paren {x N} N
| c = [[Quotient Group is Group]]: $N$ is the [[... | Quotient Group is Abelian iff All Commutators in Divisor/Proof 1 | https://proofwiki.org/wiki/Quotient_Group_is_Abelian_iff_All_Commutators_in_Divisor | https://proofwiki.org/wiki/Quotient_Group_is_Abelian_iff_All_Commutators_in_Divisor/Proof_1 | [
"Group Commutators",
"Quotient Groups",
"Abelian Groups",
"Quotient Group is Abelian iff All Commutators in Divisor"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Quotient Group",
"Definition:Abelian Group",
"Definition:Commutator/Group"
] | [
"Quotient Group is Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Commutator of Quotient Group Elements",
"Commutator of Quotient Group Elements",
"Definition:Abelian Group",
"Definition:Abelian Group"
] |
proofwiki-15597 | Quotient Group is Abelian iff All Commutators in Divisor | Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.
Let $G / N$ be the quotient group of $G$ by $N$.
Then the quotient group $G / N$ is abelian {{iff}}:
:$\forall x, y \in G: \sqbrk {x, y} \in N$
where $\sqbrk {x, y}$ denotes the commutator of $x$ and $y$.
That is, {{iff}} $x y x^{-1} y^{-1} \in N$ for all $x, y \... | Let $G / N$ be abelian.
Then:
{{begin-eqn}}
{{eqn | q = \forall a N, b N \in G / N
| l = a N b N
| r = b N a N
| c =
}}
{{eqn | ll= \leadsto
| l = a b N
| r = b a N
| c =
}}
{{eqn | ll= \leadsto
| l = a b \paren {b a}^{-1}
| o = \in
| r = N
| c =
}}
{{eqn |... | Let $G$ be a [[Definition:Group|group]].
Let $N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $G / N$ be the [[Definition:Quotient Group|quotient group]] of $G$ by $N$.
Then the [[Definition:Quotient Group|quotient group]] $G / N$ is [[Definition:Abelian Group|abelian]] {{iff}}:
:$\forall x, y \... | Let $G / N$ be [[Definition:Abelian Group|abelian]].
Then:
{{begin-eqn}}
{{eqn | q = \forall a N, b N \in G / N
| l = a N b N
| r = b N a N
| c =
}}
{{eqn | ll= \leadsto
| l = a b N
| r = b a N
| c =
}}
{{eqn | ll= \leadsto
| l = a b \paren {b a}^{-1}
| o = \in
... | Quotient Group is Abelian iff All Commutators in Divisor/Proof 2 | https://proofwiki.org/wiki/Quotient_Group_is_Abelian_iff_All_Commutators_in_Divisor | https://proofwiki.org/wiki/Quotient_Group_is_Abelian_iff_All_Commutators_in_Divisor/Proof_2 | [
"Group Commutators",
"Quotient Groups",
"Abelian Groups",
"Quotient Group is Abelian iff All Commutators in Divisor"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Quotient Group",
"Definition:Abelian Group",
"Definition:Commutator/Group"
] | [
"Definition:Abelian Group"
] |
proofwiki-15598 | Sufficient Condition for Quotient Group by Intersection to be Abelian | Let $G$ be a group.
Let $N$ and $K$ be normal subgroups of $G$.
Let the quotient groups $G / N$ and $G / K$ be abelian.
Then the quotient group $G / \paren {N \cap K}$ is also abelian. | From Intersection of Normal Subgroups is Normal, we have that $N \cap K$ is normal in $G$.
We are given that $G / N$ and $G / K$ are abelian.
Hence:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \sqbrk {x, y}
| o = \in
| r = N
| c = Quotient Group is Abelian iff All Commutators in Divisor
}... | Let $G$ be a [[Definition:Group|group]].
Let $N$ and $K$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$.
Let the [[Definition:Quotient Group|quotient groups]] $G / N$ and $G / K$ be [[Definition:Abelian Group|abelian]].
Then the [[Definition:Quotient Group|quotient group]] $G / \paren {N \cap K}$ is also... | From [[Intersection of Normal Subgroups is Normal]], we have that $N \cap K$ is [[Definition:Normal Subgroup|normal]] in $G$.
We are given that $G / N$ and $G / K$ are [[Definition:Abelian Group|abelian]].
Hence:
{{begin-eqn}}
{{eqn | q = \forall x, y \in G
| l = \sqbrk {x, y}
| o = \in
| r = N
... | Sufficient Condition for Quotient Group by Intersection to be Abelian | https://proofwiki.org/wiki/Sufficient_Condition_for_Quotient_Group_by_Intersection_to_be_Abelian | https://proofwiki.org/wiki/Sufficient_Condition_for_Quotient_Group_by_Intersection_to_be_Abelian | [
"Quotient Groups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Abelian Group",
"Definition:Quotient Group",
"Definition:Abelian Group"
] | [
"Intersection of Normal Subgroups is Normal",
"Definition:Normal Subgroup",
"Definition:Abelian Group",
"Quotient Group is Abelian iff All Commutators in Divisor",
"Quotient Group is Abelian iff All Commutators in Divisor"
] |
proofwiki-15599 | Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are | Let $G$ be a group.
Let $N$ and $K$ be normal subgroups of $G$.
Let the quotient groups $G / N$ and $G / K$ be cyclic.
Then the quotient group $G / \paren {N \cap K}$ is not necessarily cyclic. | Proof by Counterexample:
Let $G = \set {e, a, b, c}$ be the Klein $4$-group whose identity element is $e$.
Let $N = \set {e, a}$ and $K = \set {e, b}$.
By Subgroups of Klein Four-Group, both $N$ and $K$ are subgroups of $G$.
By Prime Group is Cyclic, both $N$ and $K$ are cyclic.
By Subgroup of Abelian Group is Normal, ... | Let $G$ be a [[Definition:Group|group]].
Let $N$ and $K$ be [[Definition:Normal Subgroup|normal subgroups]] of $G$.
Let the [[Definition:Quotient Group|quotient groups]] $G / N$ and $G / K$ be [[Definition:Cyclic Group|cyclic]].
Then the [[Definition:Quotient Group|quotient group]] $G / \paren {N \cap K}$ is not ne... | [[Proof by Counterexample]]:
Let $G = \set {e, a, b, c}$ be the [[Definition:Klein Four-Group|Klein $4$-group]] whose [[Definition:Identity Element|identity element]] is $e$.
Let $N = \set {e, a}$ and $K = \set {e, b}$.
By [[Subgroups of Klein Four-Group]], both $N$ and $K$ are [[Definition:Subgroup|subgroups]] of $... | Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are | https://proofwiki.org/wiki/Quotient_Group_by_Intersection_of_Normal_Subgroups_not_necessarily_Cyclic_if_Quotient_Groups_are | https://proofwiki.org/wiki/Quotient_Group_by_Intersection_of_Normal_Subgroups_not_necessarily_Cyclic_if_Quotient_Groups_are | [
"Quotient Groups"
] | [
"Definition:Group",
"Definition:Normal Subgroup",
"Definition:Quotient Group",
"Definition:Cyclic Group",
"Definition:Quotient Group",
"Definition:Cyclic Group"
] | [
"Proof by Counterexample",
"Definition:Klein Four-Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Subgroups of Klein Four-Group",
"Definition:Subgroup",
"Prime Group is Cyclic",
"Definition:Cyclic Group",
"Subgroup of Abelian Group is Normal",
"Definition:Normal Subgroup",
"T... |
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