---
tag: 0CTE7A31XQ
kind: concept
name: Complex plane
aliases: [argand-diagram, argand-plane, gauss-plane]
msc: [30A99]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_plane
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Complex_plane
    locator: Argand diagram
same_as:
  wikidata: Q328998
  wikipedia: Complex plane
---

## Summary

The complex plane is the plane $\mathbb{R}^2$ in which the point $(a, b)$ represents the [[complex number|09FTAET5Z4]] $a + b i$.
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|0KW2X0TFBX]] $|z|$ is the distance from $z$ to $0$, and the [[argument|0ZZRSX91QR]] 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|0BJV24C4SY]] 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.