Basic Question on the real and imaginary part of a complex number I was introduced to the use of Re and Im to denote the real and imaginary parts of complex numbers today, but I'm still feeling unclear on how exactly they work.
I think understand them for a basic example: $$Re(a+ib)=a$$
But we were given this equality, which I'm unsure about:
$$Re\left(\frac{1}{z}\right)=Re\left(\frac{z}{{\lvert z\rvert}^2}\right)$$
If someone could help me out, maybe with a simple proof of this, it would be helpful!

$$Re(\frac{1}{z})=Re(\frac{1}{z}\times\frac{\bar{z}}{\bar{z}})=Re(\frac{\bar{z}}{|z|^2})=\frac{1}{|z|^2}Re(\bar{z})=\frac{1}{|z|^2}Re(z)=Re(\frac{z}{|z|^2})$$
