Complex Modulus equals Zero iff Zero

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Theorem

Let $z = a + i b$ be a complex number.

Let $\cmod z$ be the modulus of $z$.


Then:

$\cmod z = 0 \iff z = 0$


Proof

Necessary Condition

\(\ds z\) \(=\) \(\ds 0\)
\(\ds \) \(=\) \(\ds 0 + 0 i\)
\(\ds \leadsto \ \ \) \(\ds \cmod z\) \(=\) \(\ds \sqrt {0^2 + 0^2}\) Definition of Complex Modulus
\(\ds \) \(=\) \(\ds 0\)

$\Box$


Sufficient Condition

\(\ds \cmod z\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \cmod {a + b i}\) \(=\) \(\ds 0\) Definition of $z$
\(\ds \leadsto \ \ \) \(\ds \sqrt {a^2 + b^2}\) \(=\) \(\ds 0\) Definition of Complex Modulus
\(\ds \leadsto \ \ \) \(\ds a^2 + b^2\) \(=\) \(\ds 0\) squaring both sides
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds 0\) Square of Real Number is Non-Negative
\(\ds b\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds z\) \(=\) \(\ds 0\) Definition of $z$

$\blacksquare$


Sources