Definition:Real Number/Axiomatic Definition

Definition
Let $\left({R, +, \cdot, \le}\right)$ be a Dedekind complete totally ordered field.

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

Also defined as
Some sources additionally specify that $\left({R, \le}\right)$ be densely ordered.

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