# Absolute Value of Product

## Theorem

Let $x, y \in \R$ be real numbers.

Then:

$\size {x y} = \size x \size y$

where $\size x$ denotes the absolute value of $x$.

## Proof 1

Let either $x = 0$ or $y = 0$, or both.

We have that $\size 0 = 0$ by definition of absolute value.

Hence:

$\size x \size y = 0 = x y = \size {x y}$

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

Then:

 $\displaystyle x y$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {x y}$ $=$ $\displaystyle x y$ Definition of Absolute Value

and:

 $\displaystyle x$ $=$ $\displaystyle \size x$ Definition of Absolute Value $\displaystyle y$ $=$ $\displaystyle \size y$ $\displaystyle \leadsto \ \$ $\displaystyle \size x \size y$ $=$ $\displaystyle x y$ $\displaystyle$ $=$ $\displaystyle \size {x y}$

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

Then:

 $\displaystyle x y$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {x y}$ $=$ $\displaystyle x y$ Definition of Absolute Value

and:

 $\displaystyle -x$ $=$ $\displaystyle \size x$ Definition of Absolute Value $\displaystyle -y$ $=$ $\displaystyle \size y$ $\displaystyle \leadsto \ \$ $\displaystyle \size x \size y$ $=$ $\displaystyle \paren {-x} \paren {-y}$ $\displaystyle$ $=$ $\displaystyle xy$ $\displaystyle$ $=$ $\displaystyle \size {x y}$

The final case is where one of $x$ and $y$ is positive, and one is negative.

Without loss of generality, let $x < 0$ and $y > 0$.

Then:

 $\displaystyle x y$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle \size {x y}$ $=$ $\displaystyle -\paren {x y}$ Definition of Absolute Value

and:

 $\displaystyle -x$ $=$ $\displaystyle \size x$ Definition of Absolute Value $\displaystyle y$ $=$ $\displaystyle \size y$ $\displaystyle \leadsto \ \$ $\displaystyle \size x \size y$ $=$ $\displaystyle \paren {-x} y$ $\displaystyle$ $=$ $\displaystyle -\paren {x y}$ $\displaystyle$ $=$ $\displaystyle \size {x y}$

The case where $x > 0$ and $y < 0$ is the same.

$\blacksquare$

## Proof 2

Let $x$ and $y$ be considered as complex numbers which are wholly real.

That is:

$x = x + 0 i, y = y + 0 i$

From Complex Modulus of Real Number equals Absolute Value, the absolute value of $x$ and $y$ equal the complex moduli of $x + 0 i$ and $y + 0 i$.

Thus $\cmod x \cmod y$ can be interpreted as the complex modulus of $x$ multiplied by the complex modulus of $y$.

$\cmod x \cmod y = \cmod {x y}$

As $x$ and $y$ are both real, so is $x y$.

Thus by Complex Modulus of Real Number equals Absolute Value, $\cmod {x y}$ can be interpreted as the absolute value of $x y$ as well as its complex modulus.

$\blacksquare$