Absolute Value

Definition

The absolute value of a real number $a$, denoted $|a|$, is its distance from $0$ on the real number line. Formally: $|a| = \begin{cases} a & \text{if } a \ge 0 \\ -a & \text{if } a < 0 \end{cases}$

• How to read: “The absolute value of a equals a if a is greater than or equal to zero, and negative a if a is less than zero.”
• Meaning: Strips the sign from $a$, leaving only its magnitude (distance from zero). Negative numbers become positive; zero and positive numbers stay unchanged.

Why It Matters

Magnitude is often more important than direction; whether a bridge is 10 meters “short” or 10 meters “long,” the deviation is what causes the failure. Absolute value provides the mathematical tool to quantify errors, tolerances, and gaps regardless of their orientation.

Core Concepts

• Magnitude without Direction: Absolute value strips away the sign, leaving only the “size” of the number.

• Distance between Points: The distance $d$ between two points $P(a)$ and $Q(b)$ on a number line is given by: $d(P, Q) = |b - a|$

  • How to read: “The distance between P and Q equals the absolute value of b minus a.”
  • Meaning: Distance is always non-negative. Order doesn’t matter: $|b-a| = |a-b|$. Use whenever you need the gap between two real-number positions.

• Properties:

  • $|a| \ge 0$
  • How to read: “The absolute value of a is greater than or equal to zero.”
  • Meaning: Distance cannot be negative; $|a| = 0$ only when $a = 0$.
  • $|-a| = |a|$
  • How to read: “The absolute value of negative a equals the absolute value of a.”
  • Meaning: Flipping sign does not change magnitude.
  • $|ab| = |a||b|$
  • How to read: “The absolute value of a times b equals the absolute value of a times the absolute value of b.”
  • Meaning: Multiplication commutes with taking absolute value—useful when simplifying products inside bars.
  • $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$ ($b \neq 0$)
  • How to read: “The absolute value of a over b equals the absolute value of a divided by the absolute value of b.”
  • Meaning: Division inside absolute value splits into separate magnitudes; requires $b \neq 0$.

Connected Concepts

• Absolute Value Equations
• Absolute Value Inequalities
• Distance Formula
• Midpoint Formula
• Vector Algebra
• Complex Numbers: Operations
• Norms of Functions
• Inner Products of Functions
• Precise Definition of a Limit
• Scale