Enter a number or algebraic expression (e.g., -7, x, x - 5, -3x + 2)
Provide a value to evaluate the expression
Numbers: -5, 3.14, 0
Variables: x, y
Linear expressions: x - 5, 2x + 3, -3x
Note: The | symbols around expressions are optional
The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is denoted by vertical bars around the number, such as |x|. Since distance is always non-negative, the absolute value of any real number is always zero or positive. For example, both 5 and -5 are 5 units away from zero, so |5| = |-5| = 5.
Absolute value is fundamental in mathematics because it allows us to measure magnitude without considering sign. This concept is essential in calculus for defining limits and continuity, in algebra for solving equations and inequalities, and in real-world applications like calculating distances, errors, and deviations.
Non-negativity
|a| ≥ 0 for all real numbers a
Identity of Indiscernibles
|a| = 0 if and only if a = 0
Symmetry
|−a| = |a| for all real numbers a
Multiplicativity
|ab| = |a| × |b| for all real numbers
Triangle Inequality
|a + b| ≤ |a| + |b| for all real numbers
Preservation of Division
|a/b| = |a|/|b| when b ≠ 0
This absolute value calculator applies standard mathematical definitions. Results depend on correct input formatting and assumptions. For complex expressions, verify the interpretation matches your intended input. This tool is for educational purposes and should be used alongside proper mathematical understanding.