Negative of Complex Modulus
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Theorem
Let $z \in \C$ be a complex number.
Then:
- $-\cmod z \le \cmod z$
where $\cmod z$ denotes the complex modulus of $z$.
The equality holds if and only if $z = 0$.
Proof
Let $z = x + i y$.
By definition of complex modulus:
- $\cmod z = \sqrt {x^2 + y^2}$
and so:
- $\cmod z \ge 0$
Thus:
- $-\cmod z\le 0$
Hence the result.
$\blacksquare$
Also see
- Negative of Absolute Value for the same result on the real number line.