Absolute value - modulus — definition, number line interpretation

Overview

The absolute value (or modulus) of a number measures its distance from zero on the number line, ignoring direction. It turns every number into its non-negative magnitude.

Core concept

Why this definition?

• For $x \geq 0$: The number is already its own distance from zero → $|x| = x$.
• For $x < 0$: The number is negative, but distance must be positive → we negate it: $|x| = -x$ (which is positive because $x$ is negative).

Number line interpretation

The number line gives us a geometric view:

$\text{Distance from } a \text{ to } b = |b - a|$

Derivation from first principles:

1. What is distance? The gap between two points, always non-negative.
2. If $b > a$, the gap is $b - a$ (already positive).
3. If $b < a$, the gap is $a - b$ (we reverse the subtraction to keep it positive).
4. Absolute value unifies both cases: $|b - a|$ handles the sign automatically.

Why subtract? Because position on the line is given by coordinates; subtracting coordinates gives displacement, and absolute value gives magnitude.

Properties (derived from definition)

Property 1: Non-negativity

$|x| \geq 0 \text{ for all } x \in \mathbb{R}$

Why? By definition, we either keep a non-negative number or flip a negative one. Both produce non-negative results.

Property 2: $|x| = 0$ if and only if $x = 0$

Why? Zero is the only number at distance zero from itself.

Property 3: $|-x| = |x|$

Derivation:

• If $x \geq 0$, then $-x \leq 0$, so $|-x| = -(-x) = x = |x|$. ✓
• If $x < 0$, then $-x > 0$, so $|-x| = -x = |x|$. ✓

Geometric intuition: Reflecting a point across zero doesn't change its distance from zero.

Property 4: Triangle inequality

$|x + y| \leq |x| + |y|$

Why does this hold?
Think of walking: if you walk $x$ meters, then $y$ meters, the straight-line distance from start to end (that's $|x+y|$) can never exceed the total path length ($|x| + |y|$). Equality holds when you walk in a straight line (same direction); inequality when you backtrack.

Algebraic derivation (outline):

• Consider four cases based on signs of $x, y$.
• In each case, verify $|x+y| \leq |x| + |y|$ by expanding definitions.

Worked examples

Common mistakes

Memory aids

Active recall practice

#flashcards/maths

Feynman technique

Connections

• Number line and ordering — absolute value is rooted in the geometry of the number line
• Distance and metric spaces — $|x - y|$ defines the standard metric on $\mathbb{R}$
• Inequalities — absolute value inequalities ($|x| < a$, $|x - c| < r$) describe intervals
• Complex numbers — modulus — extends to $|z| = \sqrt{a^2 + b^2}$ for $z = a + bi$
• Piecewise functions — absolute value is the simplest piecewise-defined function
• Triangle inequality — fundamental to analysis, norms, and metric spaces
• Solving absolute value equations — technique of case splitting based on sign

Summary: Absolute value $|x|$ is distance from zero. It's always non-negative, symmetric ($|-x| = |x|$), and satisfies the triangle inequality. On the number line, $|b - a|$ gives the distance between $a$ and $b$. Equations involving absolute value split into cases based on sign. This concept underpins inequalities, metrics, and analysis.

Concept Map

Hinglish (regional understanding)