Absolute Value of Product/Proof 2

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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

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 $y$.

By Complex Modulus of Product of Complex Numbers:

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

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.

$\blacksquare$