Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication

For each \(x\in\mathbb C\), \(x\neq 0\), there exists a number \(x^{-1}\in\mathbb C\) with \(x\cdot x^{-1}=1\).

In particular, the inverse of $x\neq 0$ can be calculated using the formula $$\frac 1x=\frac{\Re(x)-\Im(x)}{|x|^2}$$

In the following interactive figure, you can experiment with the value of \(x\) (i.e. its position in the complex plane) and see, how it influences the value of \(x^{-1}\), the inverse complex number with respect to multiplication:

Proofs: 1

Proofs: 1

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