Absolute Value Calculator

About the Absolute Value 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 calculator 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.

When This Page Helps

While |x| is simple for a single number, this calculator helps verify absolute value properties, compare magnitudes, and understand how absolute value equations work.

How to Use the Inputs

1. Enter any number (positive, negative, or zero).
2. View |x| (the absolute value).
3. See both the original value and its absolute value.
4. Check properties like |x| = |−x|.
5. Use for distance calculations or error measurement.

Example Calculation

Tips & Best Practices

• |x| is always ≥ 0.
• |x| = 0 only when x = 0.
• |a − b| gives the distance between a and b on the number line.
• |x × y| = |x| × |y| (multiplicative property).
• The triangle inequality: |a + b| ≤ |a| + |b|.
• In programming, use Math.abs(x) for absolute value.

Absolute Value in Distance

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.

Absolute Value Equations

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.

Absolute Value in Optimization

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.

Frequently Asked Questions

• What is absolute value?

  Absolute value |x| is the distance of x from zero on the number line. It is always non-negative. |7| = 7 and |−7| = 7.

• What is the triangle inequality?

  |a + b| ≤ |a| + |b|. The "shortcut" (going directly) is never longer than the "two-step route." This is a fundamental inequality in mathematics.

• How do you solve |x| = a?

  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.

• What is the absolute value of a complex number?

  For a complex number a + bi, |a + bi| = √(a² + b²). This extends the concept of distance to the complex plane.

• How is absolute value used in statistics?

  Mean absolute deviation (MAD) uses |xᵢ − mean| to measure spread. It is more robust to outliers than standard deviation.

• What is the difference between absolute value and modulus?

  For real numbers, they are the same. "Modulus" is sometimes used for complex numbers or in modular arithmetic, where it has a different meaning.