Calculate the absolute value of any number. Find the distance from zero, solve absolute value equations, and understand |x| notation. Free calculator.
The Absolute Value Calculator computes |x| — the distance of a number from zero on the number line. The absolute value is always non-negative: |5| = 5 and |−5| = 5.
Absolute value is a fundamental concept in mathematics with applications in distance calculations, error measurement, signal processing, and optimization. It removes the sign of a number, returning its magnitude.
This tool computes the absolute value, shows properties (|x| = |−x|, |x × y| = |x| × |y|), and helps solve absolute value equations and inequalities by breaking them into cases.
Quantifying this parameter enables meaningful comparison across time periods and projects, revealing trends that inform better decisions about personal productivity and resource management. This structured approach transforms vague productivity goals into measurable targets, making it easier to track improvement and stay motivated toward meaningful professional achievements.
Quantifying this parameter enables meaningful comparison across time periods and projects, revealing trends that inform better decisions about personal productivity and resource management.
While |x| is simple for a single number, this tool helps verify absolute value properties, compare magnitudes, and understand how absolute value equations work. Precise quantification supports meaningful goal-setting and accountability, ensuring that improvement efforts are focused on areas with the greatest potential impact on output. Data-driven tracking enables proactive schedule management, helping professionals protect focused work time and reduce the cognitive overhead of constant task-switching throughout the day.
|x| = x if x ≥ 0 |x| = −x if x < 0 Equivalently: |x| = √(x²)
Result: 42
|−42| = 42. The absolute value removes the negative sign, giving the distance from zero.
The distance between two points a and b on a number line is |a − b|. In two dimensions, distance uses the Pythagorean theorem, which extends absolute value to vectors.
Solving |2x − 3| = 7 requires two cases: 2x − 3 = 7 (x = 5) or 2x − 3 = −7 (x = −2). This case-splitting technique is essential in algebra.
Minimizing Σ|xᵢ − c| finds the median, while minimizing Σ(xᵢ − c)² finds the mean. Absolute-value optimization (L1 norm) produces sparser, more robust solutions.
Absolute value |x| is the distance of x from zero on the number line. It is always non-negative. |7| = 7 and |−7| = 7.
|a + b| ≤ |a| + |b|. The "shortcut" (going directly) is never longer than the "two-step route." This is a fundamental inequality in mathematics.
If a > 0, then x = a or x = −a. If a = 0, then x = 0. If a < 0, there is no solution since absolute value cannot be negative.
For a complex number a + bi, |a + bi| = √(a² + b²). This extends the concept of distance to the complex plane.
Mean absolute deviation (MAD) uses |xᵢ − mean| to measure spread. It is more robust to outliers than standard deviation.
For real numbers, they are the same. "Modulus" is sometimes used for complex numbers or in modular arithmetic, where it has a different meaning.