# 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

 $\displaystyle z$ $=$ $\displaystyle 0$ $\displaystyle$ $=$ $\displaystyle 0 + 0 i$ $\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 $z$

$\blacksquare$