# Product of Imaginary Unit with Itself

## Theorem

Let $\tuple {0, 1}$ denote the imaginary unit.

Then:

$\tuple {0, 1} \times \tuple {0, 1} = \tuple {-1, 0}$

where $\times$ denotes complex multiplication.

## Proof

 $\ds \tuple {0, 1} \times \tuple {0, 1}$ $=$ $\ds \tuple {0 \times 0 - 1 \times 1, 0 \times 1 + 0 \times 1}$ Definition of Complex Multiplication $\ds$ $=$ $\ds \tuple {-1, 0}$

$\blacksquare$