# 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

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.2$. The Algebraic Theory