Definition:Real Number/Axiomatic Definition

Definition
Let $\struct {R, +, \times, \le}$ be a Dedekind complete ordered field.

Then $R$ is called the (field of) real numbers.

Also defined as
Some sources additionally specify that $\struct {R, \le}$ be densely ordered.

This condition, while conceptually important, is superfluous, by Totally Ordered Field is Densely Ordered.