Absolute Value Calculator Online
Calculate absolute value |x|, 1D distance |A - B|, complex modulus |a + bi|, and solve |ax + b| = c.
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Definition: The absolute value of a real number x is its non-negative distance from zero: |x| = x if x ≥ 0, and |x| = -x if x < 0.
Since -42 < 0, |-42| = -(-42) = 42
Distance from origin 0 on the number line: 42 units
Quick Answer & Understanding Your Result
The Absolute Value of a real number represents its non-negative distance from zero on the number line: |x| = x if x ≥ 0, and |x| = -x if x < 0. It strips directional sign, ensuring all outputs satisfy |x| ≥ 0.
The 4 Core Absolute Value Applications
- Single Absolute Value (|x|): The magnitude of a number without regard to its sign (e.g., |-42| = 42).
- Distance Between Points (|A - B|): The geometric length between two points on a 1D real number line.
- Complex Modulus (|a + bi|): The Euclidean magnitude in the complex plane: |z| = √(a² + b²).
- Absolute Value Equations (|ax + b| = c): Yields two symmetric linear branches when c > 0.
Fundamental Algebraic Properties
- Non-negativity: |x| ≥ 0 for all real x, and |x| = 0 if and only if x = 0.
- Multiplicative Identity: |a × b| = |a| × |b|.
- Triangle Inequality: |a + b| ≤ |a| + |b|.
- Preservation of Squares: |x|² = x² = (-x)².
Enter your numbers and select an operation mode above to compute absolute values and solve equations with full steps.
Formula & Calculation Logic
Mathematical FormulaThe absolute value measures magnitude regardless of sign. In complex numbers it equals the Euclidean distance from the origin.
Step-by-Step Guide
Select Operation Mode
Choose Single value |x|, Distance |A - B|, Complex modulus, or Solve equation.
Enter Parameters
Input numeric parameters (number x, point B, or target c).
Execute Calculation
Click calculate to evaluate magnitude or split equation branches.
Review Solution
View solution values, geometric distance, and algebraic verification.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Absolute Value of Negative |-42| | |-42| = -(-42) = 42. | ||
| Distance Between -7 and 15 | |-7 - 15| = |-22| = 22 units. | ||
| Solve Equation |2x - 4| = 8 | Case 1: 2x - 4 = 8 => x = 6. Case 2: 2x - 4 = -8 => x = -2. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
Wolfram MathWorld - Absolute Value
NIST Digital Library of Mathematical Functions - Real & Complex Modulus
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: |x| = x (if x ≥ 0) or -x (if x < 0)
Last Reviewed: August 21, 2026
Primary Source: Wolfram MathWorld - Absolute Value
Reference Document: NIST Digital Library of Mathematical Functions - Real & Complex Modulus
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.