Absolute Value
ArithmeticThe absolute value of a number is its distance from zero on the number line, always a non-negative result.
Formula
|x| = x \text{ if } x \ge 0; \ -x \text{ if } x < 0
Definition
The absolute value of a number is how far it is from zero, regardless of direction. It is always positive or zero. We write it with vertical bars: $|5| = 5$ and $|-5| = 5$.
Example
$|-8| = 8$ ($8$ steps from zero). $|3| = 3$ ($3$ steps from zero). $|0| = 0$ (zero is zero steps from itself).
Key Insight
Absolute value is like asking "how far?" not "which direction?" A temperature of $-10$ degrees and $+10$ degrees are both $10$ degrees away from zero.
Definition
For a real number $x$, $|x| = x$ if $x \ge 0$ and $|x| = -x$ if $x < 0$. Key properties: $|x| \ge 0$, $|xy| = |x||y|$, $|x + y| \le |x| + |y|$ (triangle inequality), $|x - y| =$ distance between $x$ and $y$ on the number line.
Example
$|5 - 12| = |-7| = 7$ (distance from $5$ to $12$). $|(-3)(4)| = |-12| = 12 = |-3| \times |4| = 3 \times 4$.
Key Insight
The triangle inequality $|x + y| \le |x| + |y|$ is one of the most important inequalities in mathematics. It generalizes to vectors, functions, and metric spaces, and is the foundation of all distance-based analysis.
Definition
On $\mathbb{R}$, $|x|$ defines a norm: $|x| \ge 0$, $|x| = 0$ iff $x = 0$, $|cx| = |c||x|$, $|x+y| \le |x|+|y|$. This norm generates a metric $d(x,y) = |x-y|$, making $\mathbb{R}$ a complete metric space. On $\mathbb{C}$, $|z| = \sqrt{a^2 + b^2}$ for $z = a+bi$ is the Euclidean norm. On $\mathbb{Q}$, the $p$-adic absolute value $|x|_p = p^{-v_p(x)}$ gives a completely different metric with counterintuitive properties.
Example
In the $3$-adic metric, the sequence $1, 3, 9, 27, 81, \ldots$ converges to $0$ because $|3^n|_3 = 3^{-n} \to 0$. The $p$-adic numbers $\mathbb{Q}_p$ are the completion of $\mathbb{Q}$ under the $p$-adic absolute value, used in modern number theory.
Key Insight
The choice of absolute value (real, complex, or $p$-adic) determines the geometry of a number system. Ostrowski's theorem classifies all absolute values on $\mathbb{Q}$: they are exactly the real absolute value, the $p$-adic absolute values, and the trivial one.