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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 \(|z_1-z_2|\) 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:

1. $|z|=0$ if and only if $z=0$.
2. $|z_1-z_2|=|z_2-z_1|$ for all $z_1,z_2\in\mathbb C$ (symmetry)
3. $|z_1+z_2|\le |z_1| + |z_2|$ for all $z_1,z_2\in\mathbb C$ (triangle inequality)

Notes

• This is a generalization of a corresponding proposition for real numbers.
• Because $(\mathbb C,|~|)$ is a metric space and $(\mathbb C, + ,\cdot)$ is a field, we also say, that the field of complex numbers is a valuated field.

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

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983