# Rational Numbers form Subfield of Real Numbers

## Theorem

The (ordered) field $\struct {\Q, +, \times, \le}$ of rational numbers forms a subfield of the field of real numbers $\struct {\R, +, \times, \le}$.

That is, the field of real numbers $\struct {\R, +, \times, \le}$ is an extension of the rational numbers $\struct {\Q, +, \times, \le}$.

## Proof

Recall that Rational Numbers form Ordered Field.

$\Q \subseteq \R$

Hence the result by definition of subfield.

$\blacksquare$