Absolute Value Calculator
Calculate the absolute value (magnitude) of any number. Find the distance from zero, work with expressions, or process multiple values at once.
|x| = x if x >= 0, |x| = -x if x < 0Absolute Value
|x|
5
Calculate Absolute Value
Can be positive, negative, or zero
Number Line Visualization
Distance from 0: 5 units
Step-by-Step Solution
Step 1: Identify the number
x = -5
Step 2: Determine if positive, negative, or zero
x = -5 < 0 (negative)
Step 3: Apply the absolute value rule
|x| = -x when x < 0, so |-5| = -(-5) = 5
|-5| = 5
Absolute Value Properties
| Property | Formula | Description |
|---|---|---|
| Non-negativity | |x| >= 0 | Always zero or positive |
| Definiteness | |x| = 0 iff x = 0 | Only zero has zero absolute value |
| Symmetry | |-x| = |x| | Sign does not affect result |
| Multiplicative | |xy| = |x||y| | Product of absolutes |
| Triangle Inequality | |x+y| <= |x|+|y| | Sum of sides >= third side |
| Reverse Triangle | ||x|-|y|| <= |x-y| | Difference of absolutes |
Common Absolute Values
| x | |x| | Explanation |
|---|---|---|
| -10 | 10 | 10 units left of zero |
| -5 | 5 | 5 units left of zero |
| -1 | 1 | 1 unit left of zero |
| 0 | 0 | At zero |
| 1 | 1 | 1 unit right of zero |
| 5 | 5 | 5 units right of zero |
| 10 | 10 | 10 units right of zero |
Absolute Value
|x|
5
?What is Absolute Value?
The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. Written as |x|, it removes the sign from a number. For example, |5| = 5 and |-5| = 5. For any number x: |x| = x if x >= 0, and |x| = -x if x < 0.
Understanding Absolute Value
Absolute value (denoted |x|) represents the magnitude of a real number without regard to its sign. Geometrically, it is the distance of a number from zero on the number line. The absolute value function converts negative numbers to positive while leaving positive numbers unchanged. It is fundamental in mathematics for measuring distance, error, and deviation.
Key Facts About Absolute Value
- Absolute value |x| = distance from zero
- Always non-negative (zero or positive)
- |x| = x when x >= 0
- |x| = -x when x < 0
- |a x b| = |a| x |b| (multiplicative property)
- |a + b| <= |a| + |b| (triangle inequality)
- |a - b| represents distance between a and b
- Common in error calculations, physics, and statistics
Quick Answer
The absolute value of a number is its distance from zero on the number line, always expressed as a non-negative value. Written as |x|, it removes the sign from a number. For example, |5| = 5 and |-5| = 5. For any number x: |x| = x if x >= 0, and |x| = -x if x < 0.
Frequently Asked Questions
Absolute value |x| is the distance of a number from zero on the number line. It is always non-negative (zero or positive). For example, |5| = 5 and |-5| = 5. Both 5 and -5 are the same distance (5 units) from zero.
To find the absolute value of a negative number, simply remove the negative sign. Mathematically, multiply the negative number by -1. For example, |-7| = -(-7) = 7.
Absolute value is always non-negative, meaning it is zero or positive. The only number whose absolute value is zero is zero itself. |0| = 0. All other numbers have a positive absolute value.
1) |x| >= 0 (non-negativity), 2) |x| = 0 only if x = 0, 3) |xy| = |x||y| (multiplicative), 4) |x/y| = |x|/|y|, 5) |x + y| <= |x| + |y| (triangle inequality), 6) ||x| - |y|| <= |x - y|.
Absolute value is used in many applications: measuring distance between points, calculating error and deviation in statistics, defining norms in linear algebra, solving inequalities, physics (magnitude of vectors), and financial calculations (showing loss/gain magnitude).
Last updated: 2025-01-15
Absolute Value
|x|
5