Modulus Larger than Real Part

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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:

\(\displaystyle \cmod z\) \(=\) \(\displaystyle \sqrt {\paren {\map \Re z}^2 + \paren {\map \Im z}^2}\) Definition of Complex Modulus
\(\displaystyle \) \(\ge\) \(\displaystyle \sqrt {\paren {\map \Re z}^2 }\) Square of Real Number is Non-Negative, as $\map \Im z$ is real
\(\displaystyle \) \(=\) \(\displaystyle \cmod {\map \Re z}\) Square of Real Number is Non-Negative, as $\map \Re z$ is real

$\blacksquare$


Also see


Sources