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

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

$\Box$


Sufficient Condition

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

$\blacksquare$


Sources