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

\(\ds \)

\(=\)

\(\ds \tuple {-1, 0}\)

$\blacksquare$

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: The abbreviated notation