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Absolute Value Calculator

Calculate the absolute value of any number. Find the distance from zero on the number line.

Single Number

Multiple Numbers

Absolute Value of -25

|-25| = 25

Distance from zero: 25 units

xOriginal Value
-25
|x|Absolute Value
25

Multiple Values Results

OriginalAbsolute Value
-55
1010
-1515
2020
-33
Sum53
Average10.60

Understanding Absolute Value

Definition

The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value.

|x| = x if x >= 0 |x| = -x if x < 0

Examples

  • |5| = 5
  • |-5| = 5
  • |0| = 0
  • |-100| = 100

What is Absolute Value?

The absolute value of a number is its distance from zero on the number line, regardless of direction. Absolute value is always non-negative - it measures magnitude without considering sign.

Definition:

  • |a| = a if a ≥ 0 (positive numbers stay the same)
  • |a| = -a if a < 0 (negative numbers become positive)
  • Think of it as "removing the negative sign" if present

Simple Examples:

Number Absolute Value Explanation
|7| 7 Already positive
|-7| 7 Distance from 0 is 7
|0| 0 Zero distance from 0
|-3.5| 3.5 Works for decimals
|-2/3| 2/3 Works for fractions

Properties of Absolute Value

Absolute value follows important mathematical rules:

Absolute Value Properties

Non-negativity: |a| ≥ 0 (always) Identity: |a| = 0 if and only if a = 0 Symmetry: |-a| = |a| Multiplication: |a × b| = |a| × |b| Division: |a / b| = |a| / |b| (b ≠ 0) Power: |aⁿ| = |a|ⁿ Square Root: |a| = √(a²) Triangle Inequality: |a + b| ≤ |a| + |b| Reverse Triangle: ||a| - |b|| ≤ |a - b|

Where:

  • |a|= Absolute value of a
  • a, b= Any real numbers

Solving Absolute Value Equations

To solve equations with absolute value, consider both positive and negative cases:

Basic Form: |x| = a

  • If a > 0: x = a or x = -a (two solutions)
  • If a = 0: x = 0 (one solution)
  • If a < 0: No solution (absolute value can't be negative)

General Form: |expression| = a

  1. Set expression = a, solve for x
  2. Set expression = -a, solve for x
  3. Check both solutions in original equation

Example: |x - 3| = 5

Case Equation Solution
Positive case x - 3 = 5 x = 8
Negative case x - 3 = -5 x = -2

Both x = 8 and x = -2 are valid solutions.

Absolute Value Inequalities

Absolute value inequalities require understanding of intervals:

Absolute Value Inequalities

Less than (|x| < a, where a > 0): -a < x < a Solution: Open interval between -a and a Greater than (|x| > a, where a > 0): x < -a OR x > a Solution: Two rays going outward Less than or equal (|x| ≤ a): -a ≤ x ≤ a Solution: Closed interval [-a, a] Greater than or equal (|x| ≥ a): x ≤ -a OR x ≥ a Solution: Two closed rays

Where:

  • a= Positive boundary value
  • x= Variable to solve for

How to Use This Calculator

Our absolute value calculator handles numbers, expressions, and equations:

  1. Enter Value: Input a number, expression, or equation
  2. Click Calculate: Get instant results
  3. View Results:
    • Absolute value of numbers
    • Solutions to equations
    • Interval notation for inequalities
    • Step-by-step solutions

Input Types:

  • Simple values: |-7|, |3.14|, |-2/3|
  • Equations: |x - 3| = 5, |2x + 1| = 7
  • Inequalities: |x| < 4, |x - 2| ≥ 3
  • Expressions: |a - b| (distance between a and b)

Features:

  • Solves both equations and inequalities
  • Shows interval notation and number line
  • Handles complex expressions
  • Verifies solutions

Real-World Applications

Absolute value appears in many practical contexts:

Distance and Difference:

  • |a - b| gives the distance between any two numbers a and b
  • Temperature change: |today - yesterday| = change in degrees
  • Stock market: |open - close| = daily price movement

Error and Tolerance:

  • Manufacturing tolerance: |actual - target| ≤ tolerance
  • Scientific measurement error
  • Quality control limits

Statistics:

  • Mean Absolute Deviation (MAD)
  • Mean Absolute Error (MAE)
  • Absolute residuals in regression

Programming:

  • Calculating differences regardless of order
  • Magnitude of numbers
  • Distance calculations in games/graphics
  • abs() function in most languages

Physics:

  • Magnitude of vectors
  • Speed (absolute value of velocity)
  • Energy calculations

Absolute Value of Complex Numbers

For complex numbers z = a + bi, absolute value (also called modulus) is defined differently:

Complex Modulus

For z = a + bi: |z| = √(a² + b²) This is the distance from the origin in the complex plane. Properties: |z₁ × z₂| = |z₁| × |z₂| |z₁ / z₂| = |z₁| / |z₂| |z̄| = |z| (conjugate has same modulus) z × z̄ = |z|² Example: |3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5

Where:

  • a= Real part of complex number
  • b= Imaginary part of complex number
  • z̄= Complex conjugate (a - bi)

Worked Examples

Solve Absolute Value Equation

Problem:

Solve |2x - 4| = 10

Solution Steps:

  1. 1Split into two cases:
  2. 2Case 1: 2x - 4 = 10
  3. 32x = 14, so x = 7
  4. 4Case 2: 2x - 4 = -10
  5. 52x = -6, so x = -3
  6. 6Verify: |2(7) - 4| = |10| = 10 ✓
  7. 7Verify: |2(-3) - 4| = |-10| = 10 ✓

Result:

x = 7 or x = -3

Solve Absolute Value Inequality

Problem:

Solve |x - 5| < 3

Solution Steps:

  1. 1Form: |expression| < a means -a < expression < a
  2. 2-3 < x - 5 < 3
  3. 3Add 5 to all parts:
  4. 4-3 + 5 < x < 3 + 5
  5. 52 < x < 8
  6. 6Interval notation: (2, 8)
  7. 7All x values between 2 and 8 (exclusive)

Result:

2 < x < 8 or (2, 8)

Distance Between Numbers

Problem:

Find the distance between -7 and 4 on the number line

Solution Steps:

  1. 1Distance = |a - b| (absolute difference)
  2. 2Distance = |-7 - 4|
  3. 3= |-11|
  4. 4= 11
  5. 5Or: |4 - (-7)| = |11| = 11 (same result)
  6. 6The order doesn't matter due to absolute value

Result:

Distance = 11 units

Tips & Best Practices

  • ✓Think of absolute value as 'distance from zero' on the number line
  • ✓|a - b| gives the distance between any two numbers a and b
  • ✓For |x| = a: split into x = a OR x = -a (when a > 0)
  • ✓For |x| < a: solution is -a < x < a (values 'between')
  • ✓For |x| > a: solution is x < -a OR x > a (values 'outside')
  • ✓Always check: if |expression| = negative, there's no solution
  • ✓The absolute value of a product equals the product of absolute values: |ab| = |a||b|

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Frequently Asked Questions

No, never. By definition, absolute value represents distance from zero, and distance is always non-negative. |a| ≥ 0 for all real numbers a. If you get a negative result when calculating absolute value, check your work - you've made an error somewhere.
They're mathematically equivalent for real numbers: |x| = √(x²). Both always give a non-negative result. For example, |(-3)| = √((-3)²) = √9 = 3. The √(x²) form is useful in calculus and proofs, while |x| is cleaner for everyday use.
Because both a positive number and its negative are the same distance from zero. If |x| = 5, then x could be 5 (distance 5 to the right of 0) or -5 (distance 5 to the left of 0). However, equations like |x| = 0 have one solution (x = 0), and |x| = -3 has no solution (impossible for absolute value to be negative).
The triangle inequality |a + b| ≤ |a| + |b| says that the direct distance is never longer than going 'around.' It's called the triangle inequality because in a triangle, any side is shorter than the sum of the other two sides. For example: |3 + (-5)| = |-2| = 2, while |3| + |-5| = 8. Indeed, 2 ≤ 8.
The basic graph of y = |x| is a V-shape with vertex at (0,0). For y = |x - h| + k: the vertex moves to (h, k). For y = a|x|: |a| > 1 makes it steeper, |a| < 1 makes it wider, negative a flips it upside down. The graph always has a sharp corner at the vertex where the inside of the absolute value equals zero.
Most programming languages have an abs() function: abs(-5) returns 5. In Python: abs(x). In JavaScript: Math.abs(x). In C/C++: abs(x) for integers, fabs(x) for floats. You can also compute it as: x if x >= 0 else -x, or using sqrt(x*x), or using (x ^ (x >> 31)) - (x >> 31) for bit manipulation on integers.

Sources & References

Last updated: 2026-01-22

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.

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