nyuuzyou/wikis huggingface dataset:
Ambiguous Case for Spherical Triangle/Angle-Angle-Side

== Theorem ==
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the angles $\sphericalangle A$ and $\sphericalangle B$ be known.
Let the side $b$ also be known.
Then it may not be possible to know the value of $a$.
== Proof ==
From the Spherical Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin a = \dfrac {\sin b \sin A} {\sin B}$
We find that $0 < \sin a \le 1$.
We have that:
:$\sin a = \map \sin {\pi - a}$
and so unless $\sin a = 1$ and so $a = \dfrac \pi 2$, it is not possible to tell which of $a$ or $\pi - a$ provides the correct solution.
== Also see ==
* Ambiguous Case for Triangle Side-Side-Angle Congruence
== Sources ==
* 
* 
* 
* 
Category:Ambiguous Case for Spherical Triangle
Ambiguous Case for Spherical Triangle/Side-Side-Angle

== Theorem ==
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Let the sides $a$ and $b$ be known.
Let the angle $\sphericalangle B$ also be known.
Then it may not be possible to know the value of $\sphericalangle A$.
== Proof ==
From the Spherical Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin A} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is not possible to tell which of $A$ or $\pi - A$ provides the correct solution.
== Also see ==
* Ambiguous Case for Triangle Side-Side-Angle Congruence
== Sources ==
* : Chapter $\text I$. Spherical Trigonometry: $6$. The sine-formula.
* 
* 
Category:Ambiguous Case for Spherical Triangle
Ambiguous Case for Triangle Side-Side-Angle Congruence

== Theorem ==
Let $\triangle ABC$ be a triangle.
Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively.
Let the sides $a$ and $b$ be known.
Let the angle $\angle B$ also be known.
Then it may not be possible to know the value of $\angle A$.
This is known as the ambiguous case.
== Proof 1 ==
== Proof 2 ==
== Also see ==
* Ambiguous Case for Spherical Triangle
== Sources ==
* : Chapter $\text I$. Spherical Trigonometry: $6$. The sine-formula.
* 
* 
Category:Ambiguous Case for Triangle Side-Side-Angle Congruence
Category:Ambiguous Case
Category:Triangles
Category:Named Theorems
Ambiguous Case for Triangle Side-Side-Angle Congruence/Proof 1

== Theorem ==
== Proof ==
From the Law of Sines, we have:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
from which:
:$\sin A = \dfrac {\sin a \sin B} {\sin b}$
We find that $0 < \sin A \le 1$.
We have that:
:$\sin A = \map \sin {\pi - A}$
and so unless $\sin A = 1$ and so $A = \dfrac \pi 2$, it is not possible to tell which of $A$ or $\pi - A$ provides the correct solution.
:
== Sources ==
* 
* 
* 
Category:Ambiguous Case for Triangle Side-Side-Angle Congruence
Ambiguous Times

== Theorem ==
Let $T$ be a time of day in $12$-hour clock form.
Then $T$ is an ambiguous time :
:$T = 12:00 + n \times 5 \tfrac 5 {143} \mathrm {min}$
where:
:$n \in \set {1, 2, \ldots, 142}$
:the hour hand and minute hand are pointing in different directions.
== Proof ==
Let $T$ be an ambiguous time.
Let $T$ be specified in hours $h$ and minutes $m$, where:
:$1 \le h \le 12$ is an integer
:$0 \le m < 60$ is a real number
whether a.m. or p.m. is immaterial.
At this time $T$:
:let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock
:let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock.
$\theta$ and $\phi$ give a valid time indication by definition.
From Condition for Valid Time Indication:
:$12 \phi \mod 360 = \theta$
By definition of ambiguous time, we have that if the hour hand and minute hand were reversed, the time of day indicated is also a valid time indication.
That is:
:$12 \theta \mod 360 = \phi$
Hence:
It remains to calculate $\dfrac {360 k} {143}$ and convert the number each time into a valid time indication:
and so forth:
These are the ambiguous times corresponding to the times of day between $12:00$ and $01:00$.
It remains to calculate:
:$(1) \quad$ the angles made by the minute hand at these times
:$(2) \quad$ work out the times of day corresponding to the hour hand at these angles.
Let us take the ambiguous time $12 : 05 \tfrac 5 {143}$.
The hour hand is at $\dfrac {360} {143} \degrees$ while the minute hand is at $\dfrac {12 \times 360} {143} \degrees = 30 \tfrac {30} {143} \degrees$.
Exchanging the hour hand and minute hand gives us a time soon after $1:00$ where the minute hand is at $\dfrac {360} {143} \degrees$.
From Speed of Minute Hand, the minute hand moves at $6 \degrees$ per minute.
This leads us to a time of $\dfrac {360} {6 \times 143}$ minutes after $1:00$, or $\dfrac {60} {143}$ minutes after $1:00$.
To check our arithmetic, we investigate the hour hand position at $01 : 00 \tfrac {60} {143}$
A time of $\dfrac {60} {143}$ minutes past the hour corresponds to an angle of $\dfrac 1 2 \times \dfrac {60} {143} \degrees$, or $\dfrac {30} {143} \degrees$ past $30 \degrees$.
This agrees with our calculation of the position of the minute hand.
The remaining corresponding times are evaluated as multiples of this time.
== Examples ==
== Historical Note ==
Category:Ambiguous Times
Category:143
Ambiguous Times/Historical Note

== Historical Note on Ambiguous Times ==
This problem appears in 's posthumous $1932$ collection , so is is presumed it first appeared before that date in one of the journals to which contributed.
It is also reported that it was independently invented by while he was on hunger strike in $1985$:
:I spent long hours gazing at the clock hanging on the wall of my room.
:At night, the dim hospital illumination made it at times difficult for me to distinguish the hour hand from the minute hand, and I thought up the brainteaser:
::An absent minder watch-maker accidentally fastens two hands of equal lengths on a clock with the usual twelve-hour dial.
::Because of this, there are moments when the time can be read in either of two ways.
::Find all the ambiguous moments.
== Sources ==
* 
* : Integer $143$: Clockhands
Category:Historical Notes
Amicable Pair/Examples

== Examples of Amicable Pairs ==
=== $220$ and $284$ ===
=== $1184$ and $1210$ ===
=== $2620$ and $2924$ ===
=== $5020$ and $5564$ ===
=== $6232$ and $6368$ ===
=== $10 \, 744$ and $10 \, 856$ ===
=== $12 \, 285$ and $14 \, 595$ ===
=== $17 \, 296$ and $18 \, 416$ ===
Category:Amicable Pairs/Examples
Amicable Pair/Examples/10,744-10,856

== Example of Amicable Pair ==
$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:
:$\map {\sigma_1} {10 \, 744} = \map {\sigma_1} {10 \, 856} = 21 \, 600 = 10 \, 744 + 10 \, 856$
== Proof ==
By definition, $m$ and $n$ form an amicable pair :
:$\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Sources ==
* 
Category:Amicable Pairs
Amicable Pair/Examples/1184-1210

== Example of Amicable Pair ==
$1184$ and $1210$ are the $2$nd amicable pair:
:$\map {\sigma_1} {1184} = \map {\sigma_1} {1210} = 2394 = 1184 + 1210$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Historical Note ==
== Sources ==
* : $1184$
* : $1184$
* 
Category:Amicable Pairs
Category:1184
Category:1210
Amicable Pair/Examples/1184-1210/Historical Note

== Historical Note on the Amicable Pair $1184$ and $1210$ ==
The amicable pair $1184$ and $1210$ was discovered by in $1866$, at the age of $16$.
It is remarkable that it had until that time escaped being found, even by 's systematic exploration.
== Sources ==
* 
* : $1184$
* : $1184$
* 
* 
Category:Historical Notes
Amicable Pair/Examples/17,296-18,416

== Example of Amicable Pair ==
$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:
:$\map {\sigma_1} {17 \, 296} = \map {\sigma_1} {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$
== Proof ==
By definition, $m$ and $n$ form an amicable pair :
:$\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Historical Note ==
== Sources ==
* : $17,296$
* : $17,296$
* 
Category:Amicable Pairs
Amicable Pair/Examples/17,296-18,416/Historical Note

== Historical Note on the Amicable Pair $17 \, 296$ and $18 \, 416$ ==
The amicable pair $17 \, 296$ and $18 \, 416$ were discovered by in $1636$.
As such, it appears that he re-discovered what had previously been discovered by the medieval Arab school.
It was, however, the second amicable pair to be known of by the Western mathematical world after $220$ and $284$, known of old.
== Sources ==
* : $17,296$
* : $17,296$
* 
* 
Category:Historical Notes
Amicable Pair/Examples/220-284

== Example of Amicable Pair ==
$220$ and $284$ are the smallest amicable pair:
:$\map {\sigma_1} {220} = \map {\sigma_1} {284} = 504 = 220 + 284$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
It can be determined by inspection of the aliquot sums of all smaller integers that there is no smaller amicable pair.
== Historical Note ==
== Sources ==
* : Preface
* : $220$
* 
* : $220$
* 
* 
* 
* 
Category:Amicable Pairs
Amicable Pair/Examples/220-284/Historical Note

== Historical Note on the Amicable Pair $220$ and $284$ ==
The amicable pair $220$ and $284$ were, according to , known to .
However, it is strongly supposed by some commentators that they were known even further back than that.
== Sources ==
* : Preface
* : $220$
* : $220$
* 
Category:Historical Notes
Amicable Pair/Examples/2620-2924

== Example of Amicable Pair ==
$2620$ and $2924$ are the $3$rd amicable pair:
:$\map {\sigma_1} {2620} = \map {\sigma_1} {2924} = 5544 = 2620 + 2924$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\map {\sigma_1} n$ denotes the divisor sum function.
Thus:
== Sources ==
* : $2620$
* : $2620$
* 
Category:Amicable Pairs
Amicable Pair/Examples/3^4 x 5 x 11 x 5281^19 x 29 x 89 (2 x 1291 x 5281^19 - 1)-3^4 x 5 x 11 x 5281^19 (2^3 x 3^3 x 5^2 x 1291 x 5281^19 - 1)

== Example of Amicable Pair ==
These integers:
:$3^4 \times 5 \times 11 \times 5281^{19} \times 29 \times 89 \paren {2 \times 1291 \times 5281^{19} - 1}$
:$3^4 \times 5 \times 11 \times 5281^{19} \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1}$
form an amicable pair.
== Proof ==
By definition, $m$ and $n$ form an amicable pair :
:$\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\map {\sigma_1} n$ denotes the divisor sum of $n$.
First it is established (by means of an online big integer calculator and integer factorisation calculator):
:$2 \times 1291 \times 5281^{19} - 1$ is prime
:$2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1$ is prime
Thus from Divisor Sum of Integer: Corollary:
Then we calculate the sum, by means of the same online big integer calculator and integer factorisation calculator:
== Sources ==
* 
* : $220$
Category:Amicable Pairs/Examples
Amicable Pair/Examples/5020-5564

== Example of Amicable Pair ==
$5020$ and $5564$ are the $4$th amicable pair:
:$\map {\sigma_1} {5020} = \map {\sigma_1} {5564} = 10 \, 584 = 5020 + 5564$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Sources ==
* : $5020$
* : $5020$
* 
Category:Amicable Pairs
Amicable Pair/Examples/59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448 - 59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472

== Example of Amicable Pair ==
:$59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448$
and
:$59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472$
are an amicable pair:
== Proof ==
By definition, $m$ and $n$ form an amicable pair :
:$\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\map {\sigma_1} n$ denotes the divisor sum function.
From :
From :
while:
== Sources ==
* 
* : $2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
Category:Amicable Pairs
Amicable Pair/Examples/59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448 - 59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472/Mistake

== Source Work ==
:
:The Dictionary
::$2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
== Mistake ==
:Together with its amicable friend, $2^4 \times 7 \times 9,288,811,670,405,087 \times 45,556,233,678,753,109,045,286,896,851,222,527$, the largest known pair of amicable numbers. [ and , Computer Mathematics and Applications v27]
The factors are incorrect.
The correct numbers are:
:$2^{47} \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
and:
:$2^{47} \times 9,288,811,670,405,087 \times 45,556,233,678,753,109,045,286,896,851,222,527$
Also, the correct name of the journal the article was found in is .
It also needs to be pointed out that when the article was written, plenty of larger amicable pairs had been found, so this was far from being the largest pair known.
== Sources ==
* : $2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$
Category:Mistakes/Curious and Interesting Numbers
Amicable Pair/Examples/6232-6368

== Example of Amicable Pair ==
$6232$ and $6368$ are the $5$th amicable pair:
:$\map {\sigma_1} {6232} = \map {\sigma_1} {6368} = 12 \, 600 = 6232 + 6368$
== Proof ==
By definition, $m$ and $n$ form an amicable pair :
:$\map {\sigma_1} m = \map {\sigma_1} n = m + n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Sources ==
* 
Category:Amicable Pairs
Amicable Pair/Sequence

== Sequence ==
The sequence of amicable pairs begins:
:$\tuple {220, 284}, \tuple {1184, 1210}, \tuple {2620, 2924}, \tuple {5020, 5564}, \tuple {6232, 6368}, \tuple {10744, 10856}, \tuple {12285, 14595}, \tuple {17296, 18416}, \tuple {63020, 76084}$
== Sources ==
* 
Category:Amicable Pairs
Amicable Pair with Smallest Common Prime Factor 5

== Theorem ==
An amicable pair whose smallest common prime factor is greater than $3$ has the elements:
:$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8693 \times 19 \, 479 \, 382 \, 229$
and:
:$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 307 \, 071 \, 129$
This was at one point the smallest known counterexample to the observation that:
:most amicable pairs consist of even integers
:most of the rest, whose elements are odd, have both elements divisible by $3$.
== Proof ==
It is to be demonstrated that these numbers are amicable.
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
is the prime decomposition of $n$.
When $k_i = 1$ the individual factor becomes $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1} = p_i + 1$.
We establish the contributions to the divisor sum values of $m_1$ and $m_2$ by taking the prime factors in turn, and extracting the prime factors of each result.
First, the elements common to both:
This gives a common factor of both $\sigma_1$ values of:
The remaining prime factors of $m_1$:
The remaining prime factors of $m_2$:
and so:
It remains to be shown that:
:$\map {\sigma_1} {m_1} = \map {\sigma_1} {m_2} = m_1 + m_2$
So:
== Historical Note ==
== Sources ==
* : $220$
Category:Amicable Pairs
Amicable Pair with Smallest Common Prime Factor 5/Historical Note

== Historical Note on Amicable Pair with Smallest Common Prime Factor 5 ==
In $1986$, reported in his of $1986$ that "numerous mathematicians" had conjectured that all amicable pairs both of whose elements are odd are multiples of $3$.
This amicable pair, a counterexample, was communicated by letter dated $15$th May $1987$ from to .
The latter published it, with mistakes, in his of $1994$:
::$5 \cdot 7^2 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19^3 \cdot 23 \cdot 37 \cdot 181 \begin{cases} 101 \cdot 8643 \cdot 1947938229 \\ 365147 \cdot 47303071129 \end{cases}$
 repeated the mistakes in his of $1997$.
 finally published the corrected version in his of $2004$.
It was for some time believed to be the smallest such amicable pair, but smaller ones have since been discovered.
== Sources ==
* : $220$
* : $220$
Category:Historical Notes
Amicable Pair with Smallest Common Prime Factor 5/Mistake

== Source Work ==
:
:The Dictionary
::$220$
== Mistake ==
:Most known amicable pairs have both numbers in the pair divisible by $3$. However, this is not a general rule: this counterexample by te Riele may be the smallest such: $5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181$ multiplied by either $101 \times 8643 \times 1 \, 947 \, 938 \, 229$ or by $365 \, 147 \times 47 \, 303 \, 071 \, 129$.
== Correction ==
Those numbers are incorrect.
This should read:
:... multiplied by either $101 \times 8693 \times 19 \, 479 \, 382 \, 229$ or by $365 \, 147 \times 47 \, 307 \, 071 \, 129$.
Analysing the divisor sum values of the numbers given reveals the incorrectitude.
Let:
:$m_1 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 101 \times 8643 \times 1 \, 947 \, 938 \, 229$
and:
:$m_2 = 5 \times 7^2 \times 11^2 \times 13 \times 17 \times 19^3 \times 23 \times 37 \times 181 \times 365 \, 147 \times 47 \, 303 \, 071 \, 129$
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
is the prime decomposition of $n$.
When $k_i = 1$ the individual factor becomes $\dfrac {p_i^2 - 1} {p_i - 1} = \dfrac {\paren {p_i + 1} \paren {p_i - 1} } {p_i - 1} = p_i + 1$.
First we make sure we have all the prime factors:
All other factors given are indeed prime.
We establish the contributions to the divisor sum values of $m_1$ and $m_2$ by taking the prime factors in turn, and extracting the prime factors of each result.
First, the elements common to both:
This gives a common factor of both $\sigma_1$ values of:
The remaining prime factors of $m_1$:
Thus:
This gives us the prime decomposition of the rest of $\map {\sigma_1} {m_1}$:
The remaining prime factors of $m_2$:
Thus:
This gives us the prime decomposition of the rest of $\map {\sigma_1} {m_2}$:
Thus it is seen that $\map {\sigma_1} {m_1} \ne \map {\sigma_1} {m_2}$, and so $m_1$ and $m_2$ are not amicable.
The error originates in .
He corrects this mistake in :
:In an $87-05-15$ letter announced a $33$-digit specimen (misquoted in UPINT2)
::$5 \cdot 7^2 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19^3 \cdot 23 \cdot 37 \cdot 181 \begin{cases} 101 \cdot 8693 \cdot 19479382229 \\ 365147 \cdot 47307071129 \end{cases}$
Smaller counterexamples have since been found.
== Sources ==
* : $220$
Category:Mistakes/Curious and Interesting Numbers
Amicable Pairs with Common Factor 3/Mistake

== Source Work ==
:
:The Dictionary
::$220$
== Mistake ==
:Most known amicable pairs have both numbers in the pair divisible by $3$.
There is a crucial word missing out here. This should read:
:Most known odd amicable pairs have both numbers in the pair divisible by $3$.
It is known that amicable pairs whose elements are both even cannot have either element divisible by $3$.
== Sources ==
* : $220$
Category:Mistakes/Curious and Interesting Numbers
Amicable Triplet/Examples/103,340,640-123,228,768-124,015,008

== Example of Amicable Triplet ==
The following numbers form an amicable triplet:
:$103 \, 340 \, 640$
:$123 \, 228 \, 768$
:$124 \, 015 \, 008$
== Proof ==
Let $\map s n$ denote the aliquot sum of $n$.
By definition:
:$\map s n = \map {\sigma_1} n - n$
where $\sigma_1$ denotes the divisor sum function.
Thus:
== Sources ==
* 
* 
* : $220$
* : $220$
* 
Category:Amicable Triplets
Category:103,340,640
Category:123,228,768
Category:124,015,008
Amicable Triplet/Examples/1980-2016-2556

== Example of Amicable Triplet ==
:$\tuple {1980, 2016, 2556}$ form an amicable triplet.
== Proof ==
We have
Hence the result, by definition of amicable triplet.
== Sources ==
* 
* : $1980$
* 
Category:Amicable Triplets
Category:1980
Category:2016
Category:2556
Amplitude of Beats

== Theorem ==
Let $W_1$ and $W_2$ be harmonic waves whose frequencies are $f_1$ and $f_2$.
Let the amplitude of $W_1$ and $W_2$ both be $a$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of beats.
The amplitude $A_b$ of those beats at time $t$ is:
:$A_b = 2 a \map \cos {\pi \size {f_1 - f_2} t - \dfrac \epsilon 2}$
== Proof ==
Let $\omega_1$ and $\omega_2$ denote the angular frequency of $W_1$ and $W_2$ respectively.
Let us consider the harmonic waves that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
, therefore, let $W_1$ and $W_2$ be be expressed as:
where:
and $\epsilon$ is the phase of $\phi_1$ $\phi_2$.
Then:
Hence the amplitude $a$ of the slower component is seen to be:
:$A_b = \dfrac {\size {\omega_1 - \omega_2} } 2$
By definition of angular frequency:
Hence the result.
== Sources ==
* 
* 
Category:Beats
Ampère's Force Law

== Theorem ==
Let $s_1$ and $s_2$ be wires in a vacuum carrying steady currents $I_1$ and $I_2$.
Then the force between $s_1$ and $s_2$ is given by:
:$\ds \mathbf F \propto I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} } {\size {\mathbf r_1 - \mathbf r_2}^3} }$
where:
:$\d \mathbf l_1$ and $\d \mathbf l_2$ are infinitesimal vectors associated with $s_1$ and $s_2$ respectively
:$\mathbf r_1$ and $\mathbf r_2$ are the position vectors pointing from $\d \mathbf l_2$ towards $\d \mathbf l_1$.
=== SI Units ===
=== Unrationalized CGS Units ===
== Proof ==
== Also see ==
* Ampère-Maxwell Law‎
== Sources ==
* : Appendix $\text A$: Units
Category:Ampère's Force Law
Category:Electric Current
Category:Electromagnetism
Ampère's Force Law/SI Units

== Theorem ==
In SI units, Ampère's Force Law is expressed as:
:$\ds \mathbf F = \dfrac {\mu_0} {4 \pi} I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} } {\size {\mathbf r_1 - \mathbf r_2}^3} }$
where:
:$\mathbf F$ is expressed in newtons
:$\mathbf r_1 - \mathbf r_2$ is expressed in metres
:$I_1$ and $I_2$ are expressed in amperes
:$\mu_0$ denotes the vacuum permeability whose value is given by:
::$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6}$ henries per metre.
== Sources ==
* : Appendix $\text A$: Units
Category:Ampère's Force Law
Ampère's Law with Maxwell's Addition

== Physical Law ==
:$\nabla \times \mathbf B = \mu_0 \paren {\mathbf J + \varepsilon_0 \dfrac {\partial \mathbf E} {\partial t} }$
where:
:$\nabla \times \mathbf B$ denotes the curl of the magnetic flux density $\mathbf B$
:$\mathbf J$ denotes the electric current
:$\mu_0$ denotes the vacuum permeability
:$\varepsilon_0$ denotes the vacuum permittivity
:$\dfrac {\partial \mathbf E} {\partial t}$ denotes the partial derivative of the electric field strength $\mathbf E$ time.
== Also presented as ==
== Sources ==
* : Introduction: Electromagnetic Theory
Category:Ampère's Law with Maxwell's Addition
Category:Maxwell's Equations