# Definition:Field of Real Numbers

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## 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 $1$.

## Sources

- 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $3$. FIELD - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts