Proposition: Imaginary Unit

The complex multiplication of the complex number \((0,1)\) is \((-1,0)\), i.e.

\[(0,1)\cdot(0,1)=(-1,0).\] We set \[i:=(0,1)\in\mathbb C\] and call the complex number \(i\) the imaginary unit. Because the complex number \((-1,0)\in\mathbb C\) can be identified with the real number \(-1\in\mathbb R\), we have \[i^2=-1.\]

Proofs: 1

Algorithms: 1
Definitions: 2
Explanations: 3
Lemmas: 4 5 6
Proofs: 7 8 9 10

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013