Sum 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:

\(\ds x + y\) \(=\) \(\ds \tuple {a, 0} + \tuple {b, 0}\)
\(\ds \) \(=\) \(\ds \tuple {a + b, 0 + 0}\) Definition of Complex Addition
\(\ds \) \(=\) \(\ds \tuple {a + b, 0}\)

$\blacksquare$


Sources