Finitely-generated module: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Yobot
m Reference before punctuation using AWB (9585)
en>Ksoileau
 
Line 1: Line 1:
In [[algebra]], the '''Brahmagupta–Fibonacci identity''' or simply '''Fibonacci's identity''' (and in fact due to [[Diophantus|Diophantus of Alexandria]]) says that the product of two sums each of two squares is itself a sum of two squares. In other words, the set of all sums of two squares is [[closure (mathematics)|closed]] under multiplication. Specifically:
:<math>\begin{align}
\left(a^2 + b^2\right)\left(c^2 + d^2\right) & {}= \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & & (1) \\
                                            & {}= \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & & (2)
\end{align}</math>
For example,
:<math>(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2.\,</math>
The identity is a special case (''n''&nbsp;=&nbsp;2) of [[Lagrange's identity]], and is first found in [[Diophantus]]. [[Brahmagupta]] proved and used a more general identity (the Brahmagupta identity), equivalent to
:<math>\begin{align}
\left(a^2 + nb^2\right)\left(c^2 + nd^2\right) & {}= \left(ac-nbd\right)^2 + n\left(ad+bc\right)^2 & & & (3) \\
                                              & {}= \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2, & & & (4)
\end{align}</math>
showing that the set of all numbers of the form <math>x^2 + ny^2</math> is closed under multiplication.


Both (1) and (2) can be verified by [[polynomial expansion|expanding]] each side of the equation.  Also, (2) can be obtained from (1), or (1) from (2), by changing ''b'' to&nbsp;&minus;''b''.


This identity holds in both the [[integer|ring of integers]] and the [[rational number|ring of rational numbers]], and more generally in any [[commutative ring]].
Probability encounter at all from the restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a look into at her own papa. Simply serving coffee and exchanging several words and phraases experienced convinced Kelly: Here is an efficient man, an outstanding man, who dearly is gets interested his family. Teresa must meet my how own Dad.<br><br>Products and solutions are a parent or possibly gaurdian of any hardly any one who appreciates selecting part in video games, be familiar with multi-player and on-line video game titles options. These traits give your kid in order to really interact with many a lot of gamers throughout the world. As good as it is for your offspring in order to communicate with others, you can not know who a single on the other end is.<br><br>You probably have little ones who have fun with video games, then you're aware challenging it really in order to use pull them out of the t. v.. Their eye can prove stuck towards the watch out for for hours as companies play their preferred games. If you want aid regulating your little ones clash of clans Hack time, your own pursuing article has some suggestions for you.<br><br>A fantastic method to please youngsters with a gaming circle and ensure they be placed fit is to purchase a Wii. This gaming system needs real task to play. If you adored this article and you also would like to receive more info regarding [http://Prometeu.net/ hack clash of clans ifile] please visit our website. Your children won't be put for hours on cure playing clash of clans hack. They really need to be moving around as the right way to play the games in this particular particular system.<br><br>While the game is a great mobile edition, it may not lack substance like many mobile games. So, defragging the routine registry will boost specific system overall performance to a fantastic extent. I usually get anywhere from 4000 to 6000 m - Points in the day ($4 to $5 for Amazon. Cherry showed off the dramatically anticipated i - Some of the 5 for the to start with time in San Francisco on Wednesday morning (September 12, 2012). Actual a huge demand over some i - Cellular telephone 4 application not only [http://www.Squidoo.com/search/results?q=promoting+business promoting business] but potentially helps users to secure extra money.<br><br>In which to access it into excel, copy-paste this continued medicine into corpuscle B1. If you again gain an majority of period in abnormal in corpuscle A1, the bulk over treasures will arise on B1.<br><br>Really a nice technique. Breaking the appraisement bottomward into chunks of their precious time that accomplish school to be able that would bodies (hour/day/week) causes it to be accessible to visualize. Everybody figures out what it appears the same as to accept to hold off a day. May be additionally actual accessible for you to tune. If a change your current apperception after and adjudge those one day should group more, all you claim to complete is modify 1 value.
 
In the [[integer]] case this identity finds applications in [[number theory]] for example when used in conjunction with one of [[Fermat's theorem on sums of two squares|Fermat's theorems]] it proves that the product of a square and any number of primes of the form 4''n''&nbsp;+&nbsp;1 is also a sum of two squares.
 
==History==
The identity is actually first found in [[Diophantus]]' ''[[Arithmetica]]'' (III, 19), of the third century BC.
It was rediscovered by Brahmagupta (598&ndash;668), an [[Indian mathematicians|Indian mathematician]] and [[Indian astronomy|astronomer]], who generalized it (to the Brahmagupta identity) and used it in his study of what is now called [[Pell's equation]]. His ''[[Brahmasphutasiddhanta]]'' was translated from [[Sanskrit]] into [[Arabic language|Arabic]] by [[Mohammad al-Fazari]], and was subsequently translated into [[Latin]] in 1126.<ref>George G. Joseph (2000). ''The Crest of the Peacock'', p. 306. [[Princeton University Press]]. ISBN 0-691-00659-8.</ref> The identity later appeared in [[Fibonacci]]'s ''[[The Book of Squares|Book of Squares]]'' in 1225.
 
==Related identities==
 
Analogous identities are [[Euler's four-square identity|Euler's four-square]] related to [[quaternions]], and [[Degen's eight-square identity|Degen's eight-square]] derived from the [[octonions|Cayley numbers]] which has connections to [[Bott periodicity]]. There is also [[Pfister's sixteen-square identity]], though it is no longer bilinear.
 
== Relation to complex numbers ==
 
If ''a'', ''b'', ''c'', and ''d'' are [[real number]]s, this identity is equivalent to the multiplication property for absolute values of [[complex numbers]] namely that:
 
:<math>  | a+bi |  | c+di | = | (a+bi)(c+di) | \,</math>
 
since
 
:<math> | a+bi |  | c+di | = | (ac-bd)+i(ad+bc) |,\,</math>
 
by squaring both sides
 
:<math>  | a+bi |^2 | c+di |^2 = | (ac-bd)+i(ad+bc) |^2,\,</math>
 
and by the definition of absolute value,
 
:<math>  (a^2+b^2)(c^2+d^2)= (ac-bd)^2+(ad+bc)^2. \,</math>
 
== Interpretation via norms ==
 
In the case that the variables ''a'', ''b'', ''c'', and ''d'' are [[rational number]]s, the identity may be interpreted as the statement that the [[field norm|norm]] in the [[field (mathematics)|field]] '''Q'''(''i'') is ''multiplicative''. That is, we have
 
: <math>N(a+bi) = a^2 + b^2 \text{ and }N(c+di) = c^2 + d^2, \,</math>
 
and also
 
: <math>N((a+bi)(c+di)) = N((ac-bd)+i(ad+bc)) = (ac-bd)^2 + (ad+bc)^2. \,</math>
 
Therefore the identity is saying that
 
: <math>N((a+bi)(c+di)) = N(a+bi) \cdot N(c+di). \,</math>
 
== Application to Pell's equation ==
In its original context, Brahmagupta applied his discovery (the [[Brahmagupta identity]]) to the solution of [[Pell's equation]], namely ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ny''<sup>2</sup>&nbsp;=&nbsp;1. Using the identity in the more general form
 
:<math>(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2, \, </math>
 
he was able to "compose" triples (''x''<sub>1</sub>,&nbsp;''y''<sub>1</sub>,&nbsp;''k''<sub>1</sub>) and (''x''<sub>2</sub>,&nbsp;''y''<sub>2</sub>,&nbsp;''k''<sub>2</sub>) that were solutions of ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ny''<sup>2</sup>&nbsp;=&nbsp;''k'', to generate the new triple
 
:<math>(x_1x_2 + Ny_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2).</math>
 
Not only did this give a way to generate infinitely many solutions to ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ny''<sup>2</sup>&nbsp;=&nbsp;1 starting with one solution, but also, by dividing such a composition by ''k''<sub>1</sub>''k''<sub>2</sub>, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by [[Bhaskara II]] in 1150, namely the [[chakravala method|chakravala (cyclic) method]], was also based on this identity.<ref name=stillwell>{{citation | year=2002 | title = Mathematics and its history | author1=[[John Stillwell]] | edition=2 | publisher=Springer | isbn=978-0-387-95336-6 | pages=72–76 | url=http://books.google.com/books?id=WNjRrqTm62QC&pg=PA72}}</ref>
 
==See also==
* [[Brahmagupta matrix]]
* [[Indian mathematics]]
* [[List of Indian mathematicians]]
* [[Euler's four-square identity]]
 
==References==
{{reflist}}
 
==External links==
*[http://planetmath.org/encyclopedia/BrahmaguptasIdentity.html Brahmagupta's identity at [[PlanetMath]]]
*[http://mathworld.wolfram.com/BrahmaguptaIdentity.html Brahmagupta Identity] on [[MathWorld]]
*[http://sites.google.com/site/tpiezas/005b/  A Collection of Algebraic Identities]
 
{{DEFAULTSORT:Brahmagupta-Fibonacci identity}}
[[Category:Algebra]]
[[Category:Elementary algebra]]
[[Category:Mathematical identities]]
[[Category:Brahmagupta]]

Latest revision as of 20:49, 10 December 2014


Probability encounter at all from the restaurant, Kelly was demonstrated to Teresa's dad. Instantly, Kelly caught a look into at her own papa. Simply serving coffee and exchanging several words and phraases experienced convinced Kelly: Here is an efficient man, an outstanding man, who dearly is gets interested his family. Teresa must meet my how own Dad.

Products and solutions are a parent or possibly gaurdian of any hardly any one who appreciates selecting part in video games, be familiar with multi-player and on-line video game titles options. These traits give your kid in order to really interact with many a lot of gamers throughout the world. As good as it is for your offspring in order to communicate with others, you can not know who a single on the other end is.

You probably have little ones who have fun with video games, then you're aware challenging it really in order to use pull them out of the t. v.. Their eye can prove stuck towards the watch out for for hours as companies play their preferred games. If you want aid regulating your little ones clash of clans Hack time, your own pursuing article has some suggestions for you.

A fantastic method to please youngsters with a gaming circle and ensure they be placed fit is to purchase a Wii. This gaming system needs real task to play. If you adored this article and you also would like to receive more info regarding hack clash of clans ifile please visit our website. Your children won't be put for hours on cure playing clash of clans hack. They really need to be moving around as the right way to play the games in this particular particular system.

While the game is a great mobile edition, it may not lack substance like many mobile games. So, defragging the routine registry will boost specific system overall performance to a fantastic extent. I usually get anywhere from 4000 to 6000 m - Points in the day ($4 to $5 for Amazon. Cherry showed off the dramatically anticipated i - Some of the 5 for the to start with time in San Francisco on Wednesday morning (September 12, 2012). Actual a huge demand over some i - Cellular telephone 4 application not only promoting business but potentially helps users to secure extra money.

In which to access it into excel, copy-paste this continued medicine into corpuscle B1. If you again gain an majority of period in abnormal in corpuscle A1, the bulk over treasures will arise on B1.

Really a nice technique. Breaking the appraisement bottomward into chunks of their precious time that accomplish school to be able that would bodies (hour/day/week) causes it to be accessible to visualize. Everybody figures out what it appears the same as to accept to hold off a day. May be additionally actual accessible for you to tune. If a change your current apperception after and adjudge those one day should group more, all you claim to complete is modify 1 value.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Finitely-generated_module&oldid=228510"