Definition

1. Noun:
   • The magnitude of a real number without regard to its sign: The "absolute value" of a number is its distance from zero on the number line, always expressed as a non-negative value.

Usage

• The "absolute value" function is denoted by vertical bars, e.g., |x|.
• It is used to express distance, magnitude, or a value that must be non-negative in mathematical and real-world contexts.

Examples

• The absolute value of 5 is 5, written as |5| = 5.
• The absolute value of -7 is 7, written as |-7| = 7.
• In the equation |x| = 3, the solutions are x = 3 and x = -3.
• The distance between two points on a number line is the absolute value of their difference.

Advanced Usage

• Properties: Key properties include |a| ≥ 0, |a| = |-a|, and |a * b| = |a| * |b|.
• In Equations and Inequalities: Solving equations like |x - 2| = 5 involves considering both positive and negative cases for the expression inside the absolute value.
• Complex Numbers: While primarily for real numbers, the concept of "absolute value" (often called modulus) extends to complex numbers, representing their distance from the origin in the complex plane.

Variants and Related Words

• Modulus: A term often used synonymously with "absolute value," especially in the context of complex numbers.
• Magnitude: A more general term for size or length, which can be equivalent to absolute value for real numbers.

Synonyms

• Numerical value
• Modulus (in specific contexts)

Related Phrases

• Absolute value function: The function f(x) = |x|, which outputs the absolute value of its input.
• Absolute value inequality: An inequality involving an absolute value expression, such as |x| < 4.

Related Idioms

No common idioms are directly based on the term "absolute value." It is a precise mathematical term.

Noun

1. a real number regardless of its sign