Product of Absolute Values

Theorem

Let $\left({D, +, \times}\right)$ be an ordered integral domain.

For all $a \in D$, let $\left \vert{a}\right \vert$ denote the absolute value of $a$.

Then:

$\left \vert{a}\right \vert \times \left \vert{b}\right \vert = \left \vert{a \times b}\right \vert$

Corollary

For the number systems $\Z, \Q, \R$:

$\left \vert{a}\right \vert \cdot \left \vert{b}\right \vert = \left \vert{a b}\right \vert$

Proof

Let $P$ be the positivity property on $D$, let $<$ be the ordering induced by it, and let $N$ be the negativity property on $D$.

We consider all possibilities in turn.

$(1): \quad a = 0$ or $b = 0$

In this case, both the LHS $\left \vert{a}\right \vert \times \left \vert{b}\right \vert$ and the RHS are equal to zero.

So:

$\left \vert{a}\right \vert \times \left \vert{b}\right \vert = \left \vert{a \times b}\right \vert$

$(2): \quad P \left({a}\right), P \left({b}\right)$

First:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({a}\right), P \left({b}\right)\)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert = a, \left \vert{b}\right \vert = b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert \times \left \vert{b}\right \vert = a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \quad P \left({a}\right), P \left({b}\right)\)	\(\implies\)	\(\displaystyle P \left({a \times b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Positivity property: $(2)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a \times b}\right \vert = a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value

So:

$\left \vert{a}\right \vert \times \left \vert{b}\right \vert = \left \vert{a \times b}\right \vert$

$(3): \quad P \left({a}\right), N \left({b}\right)$

First:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle P \left({a}\right), N \left({b}\right)\)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert = a, \left \vert{b}\right \vert = -b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert \times \left \vert{b}\right \vert = - a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Negative Product

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \quad P \left({a}\right), N \left({b}\right)\)	\(\implies\)	\(\displaystyle N \left({a \times b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Properties of Negativity: $(5)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle P \left({- a \times b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of negativity property
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a \times b}\right \vert = -a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value

So:

$\left \vert{a}\right \vert \times \left \vert{b}\right \vert = \left \vert{a \times b}\right \vert$

Similarly $N \left({a}\right), P \left({b}\right)$.

$(4): \quad N \left({a}\right), N \left({b}\right)$

First:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle N \left({a}\right), N \left({b}\right)\)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert = -a, \left \vert{b}\right \vert = -b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a}\right \vert \times \left \vert{b}\right \vert = a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Negative Product

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \quad N \left({a}\right), N \left({b}\right)\)	\(\implies\)	\(\displaystyle P \left({a \times b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Properties of Negativity: $(4)$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle P \left({a \times b}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of negativity property
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left \vert{a \times b}\right \vert = a \times b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of absolute value

So:

$\left \vert{a}\right \vert \times \left \vert{b}\right \vert = \left \vert{a \times b}\right \vert$

In all cases the result holds.

$\blacksquare$

Proof of Corollary

Follows directly from:

• Integers form Ordered Integral Domain

• Rational Numbers form Ordered Integral Domain

• Real Numbers form Ordered Integral Domain

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 2.7$: Theorem $11 \ \text{(i)}$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.16$