Definition:Field of Real Numbers

Definition
The field of real numbers $\left({\R, + \times, \le}\right)$ 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, $\left({\R, + \times}\right)$ is usually seen.

Also see

 * Real Numbers form Totally Ordered Field

Thus:
 * $\left({\R, +}\right)$ is the additive group of real numbers
 * $\left({\R_{\ne 0}, \times}\right)$ is the multiplicative group of real numbers
 * The zero of $\left({\R, + \times}\right)$ is $0$
 * The unity of $\left({\R, + \times}\right)$ is $1$.