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.

$\blacksquare$


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