Why does the imaginary part disappear when you take the magnitude square of a complex number?

Question

I am looking at the following example problem dealing with complex numbers:

$$\int_{-\infty }^{\infty }\left | e^{-4t +j\frac{\pi }{4}} \right |^{2}dt$$

In this example when you take the magnitude square of a complex the integral becomes:

$$\int_{-\infty }^{\infty } e^{-8t} dt$$

I am not understanding how the imaginary part disappears when you take the magnitude square of a complex number. I understand that $j^2$ = -1 but it doesn't make sense how the imaginary part goes away. Any assistance in understanding this step is greatly appreciated.

Answer 1

For real $a,b$ we have

$$|e^{a+jb}|=|e^ae^{jb}|=e^a|e^{jb}|=e^a,$$

since $|e^{jb}|=1.$

Answer 2

By definition, if $z$ is a complex number then $|z|$ is its distance in the complex plane from $0$. This distance is a non-negative real number since it is just the length of some line segment. Its square is also a non-negative real number.