# Complex Modulus equals Zero iff Zero

## 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 $z$

$\blacksquare$