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 $1$.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $2$
• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields
• 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