Complex Number equals Conjugate iff Wholly Real

Theorem
Let $$z \in \mathbb{C}$$ be a complex number.

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

Then $$z = \overline z$$ iff $$z$$ is wholly real.

Proof
Let $$z = x + \imath y$$.


 * Suppose $$z = \overline z$$.

Then by definition of complex conjugate, $$x + \imath y = x - \imath y$$.

Thus $$+y = -y$$ and so $$y = 0$$.

Hence by definition, $$z$$ is wholly real.


 * Now suppose $$z$$ is wholly real.

Then $$z = x + 0 \imath = x = x - 0 \imath = \overline z$$.