Real and Imaginary Parts in Terms of z and Its Conjugate

Remember that if $z=a+bi$, the conjugate of $z$, denoted as $\overline{z}$ here, is $a-bi$.

If we want the real part of $z$ in terms of $z, \overline{z}$ we find that $\text{Re}[z]=\frac{z+\overline{z}}{2}$. This isn't hard to prove, we just plug stuff in.

If we want the imaginary part, it is simply $\text{Im}[z]=\frac{z-\overline{z}}{2i}$.

Finally, the magnitude of $z$ is $\sqrt{z \cdot \overline{z}}$. (The magnitude, or distance from a point to the origin, is $\sqrt{a^2+b^2}$ ).