The **Complex product identity** expresses the product of any two numbers, each expressed as the sum of two squares, as the sum of two squares.

- (a
^{2}+b^{2})(A^{2}+B^{2}) = (aA+bB)^{2}+(aB−bA)^{2}

## Derivation from complex numbersEdit

If u=a−bi and U=A+Bi, then from

- |u||U|=|uU|,

we get

- (a
^{2}+b^{2})(A^{2}+B^{2}) = (aA+bB)^{2}+(aB−bA)^{2}

In defining *u*, the sign of *b* is arbitrary; We could have just as easily defined u=a+bi (or interchanged the meanings of *A* and *B*), so the identity can also be stated,

- (a
^{2}+b^{2})(A^{2}+B^{2}) = (aA−bB)^{2}+(aB+bA)^{2}

## GeneralizationEdit

By defining $ u=a-\sqrt{n}bi\, $ and $ U=A+\sqrt{n}Bi\, $, then from

- |u||U|=|uU|,

we get

- (a
^{2}+nb^{2})(A^{2}+nB^{2}) = (aA+nbB)^{2}+n(aB−bA)^{2}

## ApplicationEdit

A simple application of this identity is

- 2(A^2+B^2) = (A+B)^2+(A-B)^2

## See alsoEdit

- Quaternion identity, that the product of two numbers expressed as the sum of four squares can be expressed as the sum of four squares.