Absolute Value

The absolute value of a real number is its numerical value without regard to its sign. In other words, absolute value converts negative numbers to positive numbers, while leaving positive numbers and zero unchanged.

Definition

The absolute value of a real number a is denoted by |a|. It is defined as follows:

|a| = a, if a ≥ 0
|a| = -a, if a < 0

Thus, for any real number a, the absolute value |a| is always non-negative.

Properties

Here are some important properties of absolute values:

  1. Non-negativity: For any real number a, |a| ≥ 0.
  2. Definite Positivity: For any non-zero real number a, |a| > 0.
  3. Symmetry: For any real number a, |-a| = |a|.
  4. Triangle Inequality: For any real numbers a and b, |a + b| ≤ |a| + |b|.
  5. Preservation of Zero: |0| = 0.

Applications

Absolute value is a fundamental concept in mathematics with wide-ranging applications. It is used in various fields of study including algebra, calculus, geometry, and trigonometry. In real-world terms, absolute value is often used when negative values of a variable are either not possible or should not count as less than zero, such as in calculations involving distances, lengths, or magnitudes.