---
tag: 0FGENPMPJX
kind: statement
role: theorem
name: The reals form a field
msc: [12D99]
uses: [08J2M1JYAG]
sources:
  - url: https://en.wikipedia.org/wiki/Real_number
    locator: Arithmetic
---

## Statement

The [[real numbers|08J2M1JYAG]] $\mathbb{R}$, with their addition and multiplication, form a field.
That is, $\mathbb{R}$ is a commutative ring with $0 \neq 1$, and each real number $x \neq 0$ has a multiplicative inverse $x^{-1}$ with $x \cdot x^{-1} = 1$.

## Remarks

The field $\mathbb{R}$ contains the field $\mathbb{Q}$ of rational numbers as a subfield.
In the axiomatic approach, this is part of the axioms. In a construction of $\mathbb{R}$, for example with [[Cauchy sequences|0D8HE681SC]], it is a theorem.

## Explanations

### L1 human

The usual rules of arithmetic hold for the real numbers.
You can add, subtract and multiply real numbers, and you can divide by each real number that is not zero.