Edit filter log

'In [[algebra]], '''Brahmagupta's identity''', also sometimes called '''Fibonacci's identity''', implies that the product of two sums 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. The identity is a special case (''n''&nbsp;=&nbsp;2) of [[Lagrange's identity]]. 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 \ \qquad\qquad(1) \\ & {} = \left(ac+bd\right)^2 + \left(ad-bc\right)^2.\qquad\qquad(2) \end{align}</math> For example, :<math>(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2.\,</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 &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]]. 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 was discovered by [[Brahmagupta]] (598&ndash;668), an [[Indian mathematicians|Indian mathematician]] and [[Indian astronomy|astronomer]]. 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 0691006598.</ref> The identity later appeared in [[Fibonacci]]'s ''[[The Book of Squares|Book of Squares]]'' in 1225. ==Related identities== [[Euler's four-square identity]] is an analogous identity involving four squares instead of two that is related to [[quaternions]]. There is a similar [[Degen's eight-square identity|eight-square identity]] derived from the [[octonions|Cayley numbers]] which has connections to [[Bott periodicity]]. == 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> ==See also== * [[Brahmagupta matrix]] * [[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://www.pballew.net/fiboiden.html Brahmagupta-Fibonacci identity] *[http://sites.google.com/site/tpiezas/005b/ A Collection of Algebraic Identities] [[Category:Algebra]] [[Category:Elementary algebra]] [[Category:Mathematical identities]] [[Category:Brahmagupta]] [[ar:مطابقة براهماغوبتا-فيبوناتشي]] [[de:Brahmagupta-Identität]] [[es:Identidad de Brahmagupta]] [[fr:Identité de Brahmagupta]] [[it:Identità di Brahmagupta]] [[ja:ブラーマグプタの二平方恒等式]] [[sl:Brahmaguptova enakost]] [[zh:婆罗摩笈多-斐波那契恒等式]]'