Absolute Value#
\( \begin{aligned} |x| = \begin{cases} \phantom{-}x \text{ if } x \ge 0 \\ -x \text{ if } x \lt 0 \\ \end {cases} \end {aligned} \)
The absolute value of a real number is that number’s distance from \(0\) along the real number line, and the absolute value of the difference of two real numbers (their absolute difference) is the distance between them.
\(|x| = \sqrt{x^2}\)
Nonnegativity
\(\forall x \in \mathbb{R} \quad |x| \ge 0\)
Positive Definiteness
\( \begin{aligned} \forall x \in \mathbb{R} \quad x \ne 0 & \iff |x| \ne 0 \\ x = 0 & \iff |x| = 0 \end {aligned} \)
Multiplicativity
\(|xy| = |x| |y|\)
Subadditivity (triangle inequality)
\(|x + y| \le |x| + |y|\)
\(|4 + (-1)| \le |4| + |(-1)|\)
Terms#
[ w ] Absolute Value