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Proposition: The distance of complex numbers makes complex numbers a metric space.
Let \(z, z_1,z_2\in \mathbb C\) be elements of the field of complex numbers. The absolute value of the difference \(z_1z_2\) is called the distance of the complex numbers \(z_1\) and \(z_2\). It defines a metric on \(\mathbb C\). In other words, \((\mathbb C,~)\) is a metric space and the distance \(~\) fulfills the following properties:
 $z=0$ if and only if $z=0$.
 $z_1z_2=z_2z_1$ for all $z_1,z_2\in\mathbb C$ (symmetry)
 $z_1+z_2\le z_1 + z_2$ for all $z_1,z_2\in\mathbb C$ (triangle inequality)
Notes
Table of Contents
Proofs: 1
Mentioned in:
Chapters: 1
Definitions: 2 3
Explanations: 4
Proofs: 5 6 7 8 9 10 11 12 13
Propositions: 14
Theorems: 15
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983