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:
| \(\displaystyle x y\) | \(=\) | \(\displaystyle 0\) | |||||||||||
| \(\text {(1)}: \quad\) | \(\displaystyle \implies \ \ \) | \(\displaystyle \left\vert{x y}\right\vert\) | \(=\) | \(\displaystyle 0\) |
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 $y < 0$.
$\blacksquare$
