# 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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### References

#### Bibliography

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