Complex plane

Summary

The complex plane is the plane $\mathbb{R}^2$ in which the point $(a, b)$ represents the complex number $a + b i$. The horizontal axis is the real axis and the vertical axis is the imaginary axis. In this picture, addition of complex numbers is addition of vectors. The modulus $|z|$ is the distance from $z$ to $0$, and the argument of $z$ is the angle from the positive real axis to $z$.

Remarks

Multiplication by a complex number $w \neq 0$ scales the plane by $|w|$ and turns it by the argument of $w$. The complex conjugate is the reflection in the real axis.

Explanations

L1 human

Draw the real numbers on a horizontal line, and draw the multiples of $i$ on a vertical line through $0$. Then each complex number is a point of the plane: $3 + 2 i$ is three steps to the right and two steps up.