Product of Wholly Real Numbers is Wholly Real
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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