# Rational Numbers form Ordered Field

## Theorem

The set of rational numbers $\Q$ forms an ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$.

## Proof

Recall that by Integers form Ordered Integral Domain, $\struct {\Z, +, \times, \le}$ is an ordered integral domain

By Rational Numbers form Field, $\struct {\Q, +, \times}$ is a field.

In the formal definition of rational numbers, $\struct {\Q, +, \times}$ is the field of quotients of $\struct {\Z, +, \times, \le}$

By Total Ordering on Field of Quotients is Unique, it follows that $\struct {\Q, +, \times}$ has a unique total ordering on it that is compatible with its ring structure.

Thus $\struct {\Q, +, \times, \le}$ is an ordered field.

$\blacksquare$