Negative of Complex Modulus

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

Then:
 * $-\left\vert{z}\right\vert \le \left\vert{z}\right\vert$

where $\left\vert{z}\right\vert$ denotes the complex modulus of $z$.

The equality holds iff $z = 0$.

Proof
Let $z = x + i y$.

By definition of complex modulus:
 * $\left\vert{z}\right\vert = \sqrt{x^2 + y^2}$

and so:
 * $\left\vert{z}\right\vert \ge 0$

Thus:
 * $-\left\vert{z}\right\vert \le 0$

Hence the result.

Also see

 * Negative of Absolute Value for the same result on the real number line.