Why is |z|² equal to z times its complex conjugate?

[jaejoon89]

Why is |z|^2 = z*z?

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

So it must have something to do with the absolute value, but I don't understand what or why.

 

[praharmitra]

|z| does not define the "Absolute Value" of a complex number. The notation |z| refers to the modulus of z, which is by definition

|z| = sqrt(a^2 + b^2)

Geometrically it gives the distance of the complex number from the origin on the Argand Plane.

And quite obviously |z|^2 is NOT EQUAL TO z^2

 

[Dick]

|z|^2 is conjugate(z)*z=a^2+b^2. It's not equal to z^2.

 

[jaejoon89]

although the modulus of z when z is of the form a + 0i (i.e. it is only in the reals), then wouldn't that be essentially like an absolute value?

 

[Dick]

Sure. Modulus of z is |z| is sqrt(a^2+b^2). It's still not the same as z^2. What's the question again?