Complex Numbers and the Argand Plane

A complex number is $z = a + bi$ where $a, b \in \mathbb{R}$ and $i$ is defined by $i^2 = -1$. We call $a = \operatorname{Re}(z)$ the real part and $b = \operatorname{Im}(z)$ the imaginary part. Complex numbers add componentwise and multiply by expanding and using $i^2 = -1$.

Every $z$ is a point $(a, b)$ in the Argand plane. Its modulus (distance from the origin) is

and its conjugate is $\bar z = a - bi$ (a reflection across the real axis).

Worked example. With $z = 3 + 4i$: $|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.

Multiplication: $(1 + i)(1 - i) = 1 - i + i - i^2 = 1 - (-1) = 2$. And powers of $i$ cycle with period $4$: $i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1$, then it repeats.