Modulus Larger than Real Part

Theorem

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

Then the modulus of $z$ is larger than the real part $\map \Re z$ of $z$:

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

Proof

By the definition of a complex number, we have:

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

Then:

\(\ds \cmod z\)

\(=\)

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

Definition of Complex Modulus

\(\ds \)

\(\ge\)

\(\ds \sqrt {\paren {\map \Re z}^2 }\)

Square of Real Number is Non-Negative, as $\map \Im z$ is real

\(\ds \)

\(=\)

\(\ds \cmod {\map \Re z}\)

Square of Real Number is Non-Negative, as $\map \Re z$ is real

$\blacksquare$

Also see

• Modulus Larger than Imaginary Part

Sources

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