Complex Modulus of Real Number equals Absolute Value

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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

$\blacksquare$


Sources