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, with an ordering $\le$ compatible with the ring structure of $\R$..

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

Also see

 * Real Numbers form Ordered Field

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 $1$.