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 numbers[]
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
Generalization[]
By defining and , then from
- |u||U|=|uU|,
we get
- (a2+nb2)(A2+nB2) = (aA+nbB)2+n(aB−bA)2
Application[]
A simple application of this identity is
- 2(A^2+B^2) = (A+B)^2+(A-B)^2
See also[]
- Quaternion identity, that the product of two numbers expressed as the sum of four squares can be expressed as the sum of four squares.