square of complex numbers

Question

I have this equation from here:

but it is not equal to:

$$(a + bi)^2 = a^2 + 2abi + (bi)^2.$$

could someone explain me what is the difference between this two calcultion?

Answer 1

$|a+bi|^2 = a^2+b^2$, as you say. And $(a + bi)^2 = a^2 + 2abi + (bi)^2$, as you say.

You would expect that $|a+bi|^2 = |(a+bi)^2|$, and that's what happens:

$(a+bi)^2 = (a^2-b^2) + (2ab)i$

So |$(a+bi)^2| = \sqrt{(a^2-b^2)^2 + (2ab)^2} = \sqrt{a^4+b^4 + 2a^2b^2} = \sqrt{(a^2+b^2)^2} = a^2 + b^2$