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
- 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(ii)}$
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation