Proof

(related to Proposition: Imaginary Unit)

We first verify that the complex multiplication of the complex number \((0,1)\) indeed equals \((-1,0)\):

\[(0,1)\cdot(0,1)=(0\cdot 0-1\cdot 1,0\cdot 1+1\cdot 0)=(-1,0).\] Because the field of real numbers is embedded in the field of complex numbers, we can identify the complex number \((-1,0)\) with the real number \(-1\). By setting \[i:=(0,1)\in\mathbb C,\] we get \[i^2=-1.\]


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References

Bibliography

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