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!

Recall $|z|^2=z\overline{z}$ where $\overline{z}$ is the complex conjugate of $z$ (that is, if $z=a+ib$. then $\overline{z}=a-ib$).
Then use $$ \frac{1}{a+ib}=\frac{1}{a+ib}\frac{a-ib}{a-ib}=\frac{a-ib}{a^2+b^2}=\frac{a}{a^2+b^2}-i\frac{b}{a^2+b^2}.$$
