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In [[algebra]], '''Brahmagupta's identity''' says that the product of two numbers of the form <math>a^2+nb^2</math> is itself a number of that form. In other words, the set of such numbers is [[closure (mathematics)|closed]] under multiplication. Specifically:

:<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 & & & (1) \\
                                               & {}= \left(ac+nbd\right)^2 + n\left(ad-bc\right)^2, & & & (2)
\end{align}</math>

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]].

==History==
The identity is a generalization of the so-called [[Brahmagupta–Fibonacci identity|Fibonacci identity]] (where ''n''=1) which is actually found in [[Diophantus]]' ''[[Arithmetica]]'' (III, 19).
That identity was rediscovered by [[Brahmagupta]] (598&ndash;668), an [[Indian mathematicians|Indian mathematician]] and [[Indian astronomy|astronomer]], who generalized it 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.

== Application to Pell's equation ==
In its original context, Brahmagupta applied his discovery to the solution of what was later called [[Pell's equation]], namely ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''Ny''<sup>2</sup>&nbsp;=&nbsp;1. Using the identity in the 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]]
* [[Brahmagupta–Fibonacci identity]]
* [[Indian mathematics]]
* [[List of Indian mathematicians]]

==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]

[[Category:Algebra]]
[[Category:Elementary algebra]]
[[Category:Mathematical identities]]
[[Category:Brahmagupta]]