Rational Numbers form Subfield of Real Numbers

From ProofWiki
Jump to navigation Jump to search

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.

Then from Rational Numbers form Subset of Real Numbers:

$\Q \subseteq \R$

Hence the result by definition of subfield.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Rational_Numbers_form_Subfield_of_Real_Numbers&oldid=520911"