Absolute Value Function is Completely Multiplicative/Proof 3

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Theorem

The absolute value function on the real numbers $\R$ is completely multiplicative:

$\forall x, y \in \R: \left\vert{x y}\right\vert = \left\vert{x}\right\vert \, \left\vert{y}\right\vert$

where $\left \vert{a}\right \vert$ denotes the absolute value of $a$.


Proof

\(\ds \size {x y}\) \(=\) \(\ds \sqrt {\paren {x y}^2}\) Definition 2 of Absolute Value
\(\ds \) \(=\) \(\ds \sqrt {x^2 y^2}\)
\(\ds \) \(=\) \(\ds \sqrt {x^2} \sqrt{y^2}\)
\(\ds \) \(=\) \(\ds \size x \cdot \size y\)

$\blacksquare$


Sources