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} $\square$