Complex Number equals Conjugate iff Wholly Real
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Theorem
Let $z \in \C$ be a complex number.
Let $\overline z$ be the complex conjugate of $z$.
Then $z = \overline z$ if and only if $z$ is wholly real.
Proof
Let $z = x + i y$.
Then:
| \(\ds z\) | \(=\) | \(\ds \overline z\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + i y\) | \(=\) | \(\ds x - i y\) | Definition of Complex Conjugate | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds +y\) | \(=\) | \(\ds -y\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 0\) |
Hence by definition, $z$ is wholly real.
$\Box$
Now suppose $z$ is wholly real.
Then:
| \(\ds z\) | \(=\) | \(\ds x + 0 i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds x - 0 i\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \overline z\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory