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 $x = x + 0i \in \R$. By the definition of the complex modulus,
 * $\left\vert x + 0i \right\vert = \sqrt{x^2 + 0^2} = \sqrt{x^2}$

Which is equal to the absolute value of $x$ by the definition of the absolute value.