# Definition:Field of Real Numbers

## Definition

The field of real numbers $\struct {\R, + \times, \le}$ is the set of real numbers under the two operations of addition and multiplication, totally ordered by the ordering $\le$.

When the totally ordering $\le$ is subordinate or irrelevant in the context in which it is used, $\struct {\R, + \times}$ is usually seen.

## Also see

Thus:

$\struct {\R, +}$ is the additive group of real numbers
$\struct {\R_{\ne 0}, \times}$ is the multiplicative group of real numbers
The zero of $\struct {\R, + \times}$ is $0$
The unity of $\struct {\R, + \times}$ is $1$.