(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 \[i^2=-1.\]
