Complex Modulus is Non-Negative

Theorem

Let $z = a + i b \in \C$ be a complex number.

Let $\cmod z$ be the modulus of $z$.

Then:

$\cmod z \ge 0$

Proof

\(\ds \cmod z\)

\(=\)

\(\ds \cmod {a + b i}\)

Definition of $z$

\(\ds \)

\(=\)

\(\ds +\sqrt {a^2 + b^2}\)

Definition of Complex Modulus

\(\ds \)

\(\ge\)

\(\ds 0\)

Definition of Positive Square Root

$\blacksquare$

Sources

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