Complex Number equals Negative of Conjugate iff Wholly Imaginary

Theorem

Let $z \in \C$ be a complex number.

Let $\overline z$ be the complex conjugate of $z$.

Then $\overline z = -z$ if and only if $z$ is wholly imaginary.

Proof

Let $z = x + i y$.

Then:

\(\ds \overline z\)

\(=\)

\(\ds -z\)

\(\ds \leadsto \ \ \)

\(\ds x - i y\)

\(=\)

\(\ds -\left({x + i y}\right)\)

Definition of Complex Conjugate

\(\ds \leadsto \ \ \)

\(\ds +x\)

\(=\)

\(\ds -x\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds 0\)

Hence by definition, $z$ is wholly imaginary.

$\Box$

Now suppose $z$ is wholly imaginary.

Then:

\(\ds \overline z\)

\(=\)

\(\ds 0 - i y\)

\(\ds \)

\(=\)

\(\ds -i y\)

\(\ds \)

\(=\)

\(\ds -\left({0 + i y}\right)\)

\(\ds \)

\(=\)

\(\ds -z\)

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory