complex plane

The complex plane is a flat map on which every complex number gets its own address. Real numbers live on a one-dimensional number line, but complex numbers need a second dimension because they carry two pieces of information — a real part and an imaginary part. So instead of a line, we use a plane.

The horizontal axis is the real axis and records the real part; the vertical axis is the imaginary axis and records the imaginary part. The number a + bi is plotted at the point with coordinates (a, b), exactly like an ordered pair in the ordinary coordinate plane. The origin is 0, the point one step right is 1, and the point one step up is i.

This picture turns algebra into geometry. Adding complex numbers becomes adding arrows tip-to-tail, the conjugate becomes a reflection across the real axis, and multiplying becomes a combination of stretching and rotating. Seeing complex numbers as points on this plane is the single most powerful idea for building intuition about them.