The complex numbers form a field

Statement

The complex numbers $\mathbb{C}$, 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 $a + b i \neq 0$ 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 $i \cdot (-i) = -i^2 = 1$.