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algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, Brahmagupta's identity says that, for given n, the product of two numbers of the form a^2+nb^2 is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: :\begin \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 Both (1) and (2) can be verified by 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 ring of integers and the ring of rational numbers, and more generally in any commutative ring.


History

The identity is a generalization of the so-called Fibonacci identity (where ''n''=1) which is actually found in Diophantus' '' Arithmetica'' (III, 19). That identity was rediscovered by Brahmagupta (598–668), an
Indian mathematician Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...
and
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
, who generalized it and used it in his study of what is now called
Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
. His '' Brahmasphutasiddhanta'' was translated from
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
into
Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...
by Mohammad al-Fazari, and was subsequently translated into
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
in 1126.George G. Joseph (2000). ''The Crest of the Peacock'', p. 306.
Princeton University Press Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financia ...
. .
The identity later appeared in
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Wester ...
's '' 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 Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, ...
, namely ''x''2 − ''Ny''2 = 1. Using the identity in the form :(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, he was able to "compose" triples (''x''1, ''y''1, ''k''1) and (''x''2, ''y''2, ''k''2) that were solutions of ''x''2 − ''Ny''2 = ''k'', to generate the new triple :(x_1x_2 + Ny_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2). Not only did this give a way to generate infinitely many solutions to ''x''2 − ''Ny''2 = 1 starting with one solution, but also, by dividing such a composition by ''k''1''k''2, 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 (cyclic) method, was also based on this identity.


See also

*
Brahmagupta matrix In mathematics, the following matrix was given by Indian mathematician Brahmagupta: :B(x,y) = \begin x & y \\ \pm ty & \pm x \end. It satisfies :B(x_1,y_1) B(x_2,y_2) = B(x_1 x_2 \pm ty_1 y_2,x_1 y_2 \pm y_1 x_2).\, Powers of the matrix are def ...
* Brahmagupta polynomials *
Brahmagupta–Fibonacci identity In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity say ...
*
Brahmagupta's interpolation formula Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta (598–668 CE) in the early 7th century CE. The Sanskrit couplet describing the formula can be ...
* Gauss composition law * Indian mathematics *
List of Indian mathematicians chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and mathematicians ...


References


External links


Brahmagupta's identity
at
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Brahmagupta Identity
on
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

A Collection of Algebraic Identities
{{Webarchive, url=https://web.archive.org/web/20120306122543/http://sites.google.com/site/tpiezas/005b , date=2012-03-06 Algebraic identities Brahmagupta