Product of Absolute Values of Integers

Theorem

Let $a, b \in \Z$ be integers.

Let $\size a$ denote the absolute value of $a$:

$\size a = \begin {cases} a & : a \ge 0 \\ -a & : a < 0 \end {cases}$


Then:

$\size a \times \size b = \size {a \times b}$


Proof

From Integers form Ordered Integral Domain, $\Z$ is an ordered integral domain.

The result follows from Product of Absolute Values on Ordered Integral Domain.

$\blacksquare$


Sources

  • 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers: Exercise $2$
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