Understanding basic complex number rules

Question

Many definitions for complex number say

1. $Re(z) = \frac{1}{2}(z + \bar{z})$
2. $Im(z) = \frac{1}{2i}(z - \bar{z})$
3. $|z| = \sqrt{z\cdot\bar{z}}$

I do understand 1. as I can visualize it (the addition will eliminate the value of the y-axis which is just the real part of $z$), but why do the other two apply? Do you know any proof of them?

Answer 1

$$z=z_R+iz_I,\quad \bar{z}=z_R-iz_I$$

from which follows

$$z+\bar{z}=2z_R\quad\textrm{and}\quad z-\bar{z}=2iz_I$$

Furthermore you have

$$z\cdot\bar{z}=(z_R+iz_I)(z_R-iz_I)=z_R^2+z_I^2=|z|^2$$