en>Ksoileau: /* Formal definition */

2014-12-10T19:49:35Z

Formal definition

en>Yobot: Reference before punctuation using AWB (9585)

2013-11-07T11:43:49Z

Reference before punctuation using AWB (9585)

← Older revision		Revision as of 13:43, 7 November 2013
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	~~Greetings~~. The ~~author~~'s ~~name~~ is ~~Lindsey Hoelscher~~ when is ~~not~~ his ~~birth name. Connecticut~~ is ~~where she~~'s ~~lived for various~~. ~~What his family~~ and ~~him love end up being~~ to ~~do ceramics~~ and ~~nevertheless never cease~~. He is ~~currently~~ a ~~production~~ and ~~planning officer. See what~~'s ~~new on his website here~~: ~~http~~://~~www~~.~~pinterest~~.~~com~~/~~seodress~~/~~sacramento~~-~~realtor~~-~~let~~-me-~~help~~-~~you~~-~~buy~~-or-~~sell~~-~~you/~~		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'' = 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 −''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]].

			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'' + 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 (598–668), 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> − ''Ny''<sup>2</sup> = 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>, ''y''<sub>1</sub>, ''k''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>, ''k''<sub>2</sub>) that were solutions of ''x''<sup>2</sup> − ''Ny''<sup>2</sup> = ''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> − ''Ny''<sup>2</sup> = 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]]

en>Quondum: /* lead */ Linking generating set

2012-09-02T10:06:08Z

lead: Linking generating set

New page

Greetings. The author's name is Lindsey Hoelscher when is not his birth name. Connecticut is where she's lived for various. What his family and him love end up being to do ceramics and nevertheless never cease. He is currently a production and planning officer. See what's new on his website here: http://www.pinterest.com/seodress/sacramento-realtor-let-me-help-you-buy-or-sell-you/

Finitely-generated module - Revision history

en>Ksoileau: /* Formal definition */

en>Yobot: Reference before punctuation using AWB (9585)

en>Quondum: /* lead */ Linking generating set