Rational Numbers form Subfield of Real Numbers

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Theorem

The 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 Totally Ordered Field.

Then from Rational Numbers form Subset of Real Numbers:

$\Q \subseteq \R$

Hence the result by definition of subfield.

$\blacksquare$


Sources