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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.

(a2+b2)(A2+B2) = (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

(a2+b2)(A2+B2) = (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,

(a2+b2)(A2+B2) = (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

(a2+nb2)(A2+nB2) = (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.