---
tag: 0FRSR10DWF
kind: statement
role: theorem
name: The complex numbers form a field
msc: [12D99]
uses: [09FTAET5Z4, 0FGENPMPJX]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Complex conjugate, absolute value, argument and division
  - url: https://en.wikipedia.org/wiki/Complex_number
    locator: Abstract and algebraic definitions
---

## Statement

The [[complex numbers|09FTAET5Z4]] $\mathbb{C}$, with their addition and multiplication, form a field.
That is, $\mathbb{C}$ is a commutative ring with $0 \neq 1$, and each complex number $z \neq 0$ has a multiplicative inverse $z^{-1}$ with $z z^{-1} = 1$.

## Remarks

The [[inverse of a complex number|0FQRAV5SKH]] $a + b i \neq 0$ is $\dfrac{a - b i}{a^2 + b^2}$.
The field $\mathbb{C}$ contains $\mathbb{R}$ as a subfield. It has dimension $2$ as a vector space over $\mathbb{R}$, with basis $1, i$.
The field $\mathbb{C}$ is algebraically closed, but it is not an ordered field.

## Explanations

### L1 human

You can add, subtract, multiply and divide complex numbers, as with real numbers.
The only exception is division by $0$.
For example, $\dfrac{1}{i} = -i$, because $i \cdot (-i) = -i^2 = 1$.