Absolute value

Absolute value refers to the non-negative value of a real number, regardless of its sign. It is denoted by two vertical lines surrounding the number or expression. For example, the absolute value of -3 and 3 are both 3, represented as |-3| = 3 and |3| = 3.

Definition

The absolute value of a real number a is defined as:

• If a is greater than or equal to zero, then the absolute value of a is a.
• If a is less than zero, then the absolute value of a is -a.

In mathematical notation, this is expressed as |a| = a if a ≥ 0, and |a| = -a if a < 0.

Properties

The absolute value function has several key properties, including:

• Non-negativity: For any real number a, |a| ≥ 0.
• Identity of Positives: For any positive real number a, |a| = a.
• Symmetry: For any real number a, |-a| = |a|.
• Triangle Inequality: For any real numbers a and b, |a + b| ≤ |a| + |b|.
• Multiplicative: For any real numbers a and b, |a × b| = |a| × |b|.

Applications

The concept of absolute value is used in a wide range of mathematical and scientific fields, including algebra, calculus, complex analysis, geometry, and physics. It is particularly useful in situations where only the magnitude of a quantity matters, and not its direction.

See also

• Real number
• Mathematical notation
• Algebra
• Calculus
• Complex analysis
• Geometry
• Physics