$${z = a + bi}$$
$${w = c + di}$$
$${|z| = \sqrt{a^2 + b^2} }$$
Complex conjugate:
$${\bar z = a - bi}$$
$${Re(z) = a}$$
$${Im(z) = b}$$
$${Re(z) = \frac{1}{2} (z + \bar z) }$$
$${Im(z) = \frac{1}{2i} (z - \bar z) }$$
$${|z|^2 = a^2 + b^2 = z \bar z }$$
$${|z| = \sqrt{z \bar z} }$$
$${\overline{z + w} = \bar z + \bar w }$$
$${\overline{z w} = \bar z \bar w }$$
$${z \bar w = \overline{\bar z w} }$$
$${\bar z w = \overline{z \bar w} }$$
$${|z \bar w| = |\bar z w| = |z| |w| }$$
$${Re(z \bar w) \leq |Re(z \bar w)| \leq |z \bar w| = |z| |w| }$$
Triangle Inequality:
$${|z + w| \leq |z| + |w| }$$