Absolute Value Function is Completely Multiplicative/Proof 1

From ProofWiki
Jump to navigation Jump to search

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

Let $x = 0$ or $y = 0$.

Then:

\(\displaystyle x y\) \(=\) \(\displaystyle 0\)
\((1):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x y}\right\vert\) \(=\) \(\displaystyle 0\)

and either $\left\vert{x}\right\vert = 0$ or $\left\vert{y}\right\vert = 0$ and so:

\(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) \(=\) \(\displaystyle 0\)
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{x y}\right\vert\) from $(1)$ above

$\Box$


Let $x > 0$ and $y > 0$.

Then:

\(\displaystyle \left\vert{x}\right\vert\) \(=\) \(\displaystyle x\)
\(\, \displaystyle \land \, \) \(\displaystyle \left\vert{y}\right\vert\) \(=\) \(\displaystyle y\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) \(=\) \(\displaystyle x y\)


and:

\(\displaystyle x y\) \(>\) \(\displaystyle 0\)
\(\displaystyle \left\vert{x y}\right\vert\) \(=\) \(\displaystyle x y\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\)

$\Box$


Let $x < 0$ and $y < 0$.

Then:

\(\displaystyle \left\vert{x}\right\vert\) \(=\) \(\displaystyle -x\)
\(\, \displaystyle \land \, \) \(\displaystyle \left\vert{y}\right\vert\) \(=\) \(\displaystyle -y\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) \(=\) \(\displaystyle \left({-x}\right) \left({-y}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle x y\)


and:

\(\displaystyle x y\) \(>\) \(\displaystyle 0\)
\(\displaystyle \left\vert{x y}\right\vert\) \(=\) \(\displaystyle x y\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\)

$\Box$


Let $x < 0$ and $y > 0$.

Then:

\(\displaystyle \left\vert{x}\right\vert\) \(=\) \(\displaystyle - x\)
\(\, \displaystyle \land \, \) \(\displaystyle \left\vert{y}\right\vert\) \(=\) \(\displaystyle y\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) \(=\) \(\displaystyle - x y\)


and:

\(\displaystyle x y\) \(<\) \(\displaystyle 0\)
\(\displaystyle \left\vert{x y}\right\vert\) \(=\) \(\displaystyle - x y\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{x}\right\vert \, \left\vert{y}\right\vert\)

$\Box$


The same argument, mutatis mutandis, covers the case where $x > 0$ and $y < 0$.

$\blacksquare$