ThmDex – An index of mathematical definitions, results, and conjectures.
Result R403 on D371: Imaginary number
Square of the imaginary number
Formulation 0
Let $i$ be the D371: Imaginary number.
Then \begin{equation} i^2 = (-1, 0) \end{equation}
Formulation 1
Let $i$ be the D371: Imaginary number.
Then \begin{equation} i^2 = - 1 \end{equation}
Proofs
Proof 0
Let $i$ be the D371: Imaginary number.
We have \begin{equation} \begin{split} i^2 = i i & = (0, 1) (0, 1) \\ & = (0 \cdot 0 - 1 \cdot 1, 0 \cdot 1 + 1 \cdot 0) \\ & = (-1, 0) \\ & = -1 \end{split} \end{equation} $\square$