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:

 $\displaystyle \cmod {x + 0 i}$ $=$ $\displaystyle \sqrt {x^2 + 0^2}$ Definition of Complex Modulus $\displaystyle$ $=$ $\displaystyle \sqrt {x^2}$ $\displaystyle$ $=$ $\displaystyle \size {x}$ Absolute Value Equals Square Root of Square

$\blacksquare$