Real Numbers form Valued Field

Theorem

The set of real numbers $\R$ forms a valued field under addition, multiplication and absolute value: $\struct {\R, +, \times, \size{\,\cdot\,}}$.

Proof

From Real Numbers form Field, we have that $\struct {\R, +, \times}$ forms a field.

From Absolute Value is Norm, we have that $\size{\size{\,\cdot\,}}$ is a norm on $\struct {\R, +, \times}$.

Hence $\struct {\R, +, \times, \size{\,\cdot\,}}$ is a valued field by definition.

$\blacksquare$

Also see

• Definition:Field of Real Numbers
• Real Numbers form Ordered Field