How the differences of the two squares a2-b2(a b)(a-b) relates to the product of a complex number and it's conjugate?

Answer 1

(a+b)(a-b) = a2 - b2

if we replace b by ib in this equation, then we get:

(a+ib)(a-ib) = a2 - (ib)2 = a2 + b2

if z=a+ib then its conjuguate is a-ib and their product is a2 + b2

Answer 2

In real number, when multiplying the difference of two squares, the middle term disappears:(a + b)*(a - b) = a^2 - ab + ab + b^2 = a^2 - b^2

Similarly, with the complex number and its conjugate, the imaginary term cancels out.

(a + ib)*(a - ib) = a^2 - iab + iab + (ib)^2 = a^2 - i^2b^2 = a^2 + b^2.