Absolute Value Function is Completely Multiplicative/Proof 1
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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
Let $x = 0$ or $y = 0$.
Then:
| \(\ds x y\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \implies \ \ \) | \(\ds \left\vert{x y}\right\vert\) | \(=\) | \(\ds 0\) |
and either $\left\vert{x}\right\vert = 0$ or $\left\vert{y}\right\vert = 0$ and so:
| \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left\vert{x y}\right\vert\) | from $(1)$ above |
$\Box$
Let $x > 0$ and $y > 0$.
Then:
| \(\ds \left\vert{x}\right\vert\) | \(=\) | \(\ds x\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds \left\vert{y}\right\vert\) | \(=\) | \(\ds y\) | |||||||||||
| \(\ds \implies \ \ \) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) | \(=\) | \(\ds x y\) |
and:
| \(\ds x y\) | \(>\) | \(\ds 0\) | ||||||||||||
| \(\ds \left\vert{x y}\right\vert\) | \(=\) | \(\ds x y\) | Definition of Absolute Value | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) |
$\Box$
Let $x < 0$ and $y < 0$.
Then:
| \(\ds \left\vert{x}\right\vert\) | \(=\) | \(\ds -x\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds \left\vert{y}\right\vert\) | \(=\) | \(\ds -y\) | |||||||||||
| \(\ds \implies \ \ \) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) | \(=\) | \(\ds \left({-x}\right) \left({-y}\right)\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds x y\) |
and:
| \(\ds x y\) | \(>\) | \(\ds 0\) | ||||||||||||
| \(\ds \left\vert{x y}\right\vert\) | \(=\) | \(\ds x y\) | Definition of Absolute Value | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) |
$\Box$
Let $x < 0$ and $y > 0$.
Then:
| \(\ds \left\vert{x}\right\vert\) | \(=\) | \(\ds - x\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds \left\vert{y}\right\vert\) | \(=\) | \(\ds y\) | |||||||||||
| \(\ds \implies \ \ \) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) | \(=\) | \(\ds - x y\) |
and:
| \(\ds x y\) | \(<\) | \(\ds 0\) | ||||||||||||
| \(\ds \left\vert{x y}\right\vert\) | \(=\) | \(\ds - x y\) | Definition of Absolute Value | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left\vert{x}\right\vert \, \left\vert{y}\right\vert\) |
$\Box$
The same argument, mutatis mutandis, covers the case where $x > 0$ and $y < 0$.
$\blacksquare$