Complex Modulus of Real Number equals Absolute Value

Theorem

Let $x \in \R$ be a real number.

Then the complex modulus of $x$ equals the absolute value of $x$.

Proof

Let $x = x + 0 i \in \R$.

Then:

\(\ds \cmod {x + 0 i}\)

\(=\)

\(\ds \sqrt {x^2 + 0^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {x^2}\)

\(\ds \)

\(=\)

\(\ds \size x\)

Definition 2 of Absolute Value

$\blacksquare$

Sources

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