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
| Original | Absolute Value |
|---|---|
| -5 | 5 |
| 10 | 10 |
| -15 | 15 |
| 20 | 20 |
| -3 | 3 |
| Sum | 53 |
| Average | 10.60 |
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
- Set expression = a, solve for x
- Set expression = -a, solve for x
- 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:
- Enter Value: Input a number, expression, or equation
- Click Calculate: Get instant results
- 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:
- 1Split into two cases:
- 2Case 1: 2x - 4 = 10
- 32x = 14, so x = 7
- 4Case 2: 2x - 4 = -10
- 52x = -6, so x = -3
- 6Verify: |2(7) - 4| = |10| = 10 ✓
- 7Verify: |2(-3) - 4| = |-10| = 10 ✓
Result:
x = 7 or x = -3
Solve Absolute Value Inequality
Problem:
Solve |x - 5| < 3
Solution Steps:
- 1Form: |expression| < a means -a < expression < a
- 2-3 < x - 5 < 3
- 3Add 5 to all parts:
- 4-3 + 5 < x < 3 + 5
- 52 < x < 8
- 6Interval notation: (2, 8)
- 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:
- 1Distance = |a - b| (absolute difference)
- 2Distance = |-7 - 4|
- 3= |-11|
- 4= 11
- 5Or: |4 - (-7)| = |11| = 11 (same result)
- 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|
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