Product of Wholly Real Numbers is Wholly Real

Theorem

Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers.

Then $x y$ is also wholly real.

Proof

We have:

\(\ds x y\)

\(=\)

\(\ds \tuple {a, 0} \tuple {b, 0}\)

\(\ds \)

\(=\)

\(\ds \tuple {a \times b - 0 \times 0, a \times 0 + 0 \times b}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \tuple {a \times b, 0}\)

$\blacksquare$

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