Modulus Larger than Imaginary Part

Theorem

Let $z \in \C$ be a complex number.

Then the modulus of $z$ is larger than the imaginary part $\map \Im z$ of $z$:

$\cmod z \ge \size {\map \Im z}$

By the definition of a complex number, we have:

$z = \map \Re z + i \map \Im z$

$\ds \cmod z$

$=$

$\ds \sqrt {\paren {\map \Re z}^2 + \paren {\map \Im z}^2}$

Definition of Complex Modulus

$\ds $

$\ge$

$\ds \sqrt {\paren {\map \Im z}^2 }$

$\ds $

$=$

$\ds \size {\map \Im z}$

$\blacksquare$

1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.4$ Inequalities: $(1)$