- Brahmagupta–Fibonacci identity
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 closed under multiplication. The identity is a special case ("n" = 2) ofLagrange's identity .Specifically:

:$egin\{align\}left(a^2\; +\; b^2\; ight)left(c^2\; +\; d^2\; ight)\; \{\}=\; left(ac-bd\; ight)^2\; +\; left(ad+bc\; ight)^2\; qquadqquad(1)\; \backslash \; \{\}\; =\; left(ac+bd\; ight)^2\; +\; left(ad-bc\; ight)^2.qquadqquad(2)end\{align\}$

For example,

:$(1^2\; +\; 4^2)(2^2\; +\; 7^2)\; =\; 30^2\; +\; 1^2\; =\; 26^2\; +\; 15^2.,$

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 .In the

integer case this identity finds applications innumber theory for example when used in conjunction with one of 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 was discovered by

Brahmagupta (598–668), an Indian mathematician and astronomer. His "Brahmasphutasiddhanta " was translated fromSanskrit into Arabic byMohammad al-Fazari , which was subsequently translated intoLatin in 1126. [*George G. Joseph (2000). "The Crest of the Peacock", p. 306.*] The identity later appeared inPrinceton University Press . ISBN 0691006598.Fibonacci 's "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 toquaternions . There is a similar eight-square identity derived from the Cayley numbers which has connections toBott 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 ofcomplex numbers namely that::$|\; a+bi\; |\; |\; c+di\; |\; =\; |\; (a+bi)(c+di)\; |\; ,$

since

:$|\; a+bi\; |\; |\; c+di\; |\; =\; |\; (ac-bd)+i(ad+bc)\; |,,$

by squaring both sides

:$|\; a+bi\; |^2\; |\; c+di\; |^2\; =\; |\; (ac-bd)+i(ad+bc)\; |^2,,$

and by the definition of absolute value,

:$(a^2+b^2)(c^2+d^2)=\; (ac-bd)^2+(ad+bc)^2.\; ,$

**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 norm in the field**Q**("i") is "multiplicative". That is, we have: $N(a+bi)\; =\; a^2\; +\; b^2\; ext\{\; and\; \}N(c+di)\; =\; c^2\; +\; d^2,\; ,$

and also

: $N((a+bi)(c+di))\; =\; N((ac-bd)+i(ad+bc))\; =\; (ac-bd)^2\; +\; (ad+bc)^2.\; ,$

Therefore the identity is saying that

: $N((a+bi)(c+di))\; =\; N(a+bi)\; cdot\; N(c+di).\; ,$

**ee also***

Brahmagupta matrix

*Indian mathematics

*List of Indian mathematicians **References****External links*** [

*http://planetmath.org/encyclopedia/BrahmaguptasIdentity.html Brahmagupta's identity at*]PlanetMath

* [*http://www.pballew.net/fiboiden.html Brahmagupta-Fibonacci identity*]

* [*http://mathworld.wolfram.com/BrahmaguptaIdentity.html Brahmagupta Identity*] onMathWorld

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