Corollary: The absolute value makes the set of rational numbers a metric space.

(related to Definition: Absolute Value of Rational Numbers)

Let \(x=\frac ab,y=\frac cd\in \mathbb Q\). The distance \(|\frac ab-\frac cd|\) of two rational numbers \(\frac ab\) and \(\frac cd\) defines a metric on \(\mathbb Q\). In other words, \((\mathbb Q,|~|)\) is a metric space and the distance \(|~|\) fulfills the following properties:

\((i)\) \(|\frac ab|=0\) if and only if \(a=0\) for all \(\frac ab\in\mathbb Q\).

\((ii)\) \(|\frac ab-\frac cd|=|\frac cd-\frac ab|\) for all \(\frac ab,\frac cd\in\mathbb Q\) (symmetry)

\((iii)\) \(|\frac ab+\frac cd|\le |\frac ab| + |\frac cd|\) for all \(\frac ab,\frac cd\in\mathbb Q\) (triangle inequality)

Proofs: 1

Definitions: 1 2
Explanations: 3
Proofs: 4 5 6 7 8
Propositions: 9

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  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984