Absolute Value Calculator
Enter any number x to get |x|, with step-by-step working and a number line.
Number line: x and |x|
● x ● |x| ▮ distance from 0
Piecewise definition
The absolute value is defined in two pieces:
- If x ≥ 0, then |x| = x (the number stays as it is).
- If x < 0, then |x| = -x (negate it to make it positive).
Intuition: |x| is the distance from x to 0 on the number line, and distances are never negative.
Worked examples
| Input | Branch used | Working | Result |
|---|---|---|---|
| -7 | x < 0 | |x| = -x = -(-7) | 7 |
| 4.25 | x ≥ 0 | |x| = x | 4.25 |
| -0.5 | x < 0 | |x| = -x = -(-0.5) | 0.5 |
| 0 | x ≥ 0 | |x| = x | 0 |
| -12.8 | x < 0 | |x| = -x = -(-12.8) | 12.8 |
Properties of absolute value
| Property | Example (verified) |
|---|---|
| Non-negativity: |x| ≥ 0 for every x | |-12.8| = 12.8 ≥ 0 |
| Symmetry: |-x| = |x| | |-4.25| = |4.25| = 4.25 |
| Multiplicative: |a × b| = |a| × |b| | |-3 × 4| = |-12| = 12 = 3 × 4 |
| Triangle inequality: |a + b| is at most |a| + |b| | |-5 + 2| = |-3| = 3, and 3 ≤ 7 = 5 + 2 |
| Square root: √(x²) = |x| | √((-7)²) = √49 = 7 |
Frequently asked questions
What is the absolute value of a number?
The absolute value of a number is its distance from 0 on the number line, so it is never negative. Both -7 and 7 sit 7 units from 0, so |-7| = 7 and |7| = 7.
How do you calculate absolute value?
Use the piecewise definition: if x is zero or positive, |x| = x; if x is negative, |x| = -x. So |4.25| = 4.25, while |-12.8| = -(-12.8) = 12.8.
What is the absolute value of -7?
7. Because -7 is negative, apply |x| = -x: -(-7) = 7. On the number line, -7 is 7 units to the left of 0.
What is the absolute value of 0?
0. Zero is neither positive nor negative, and it is 0 units from itself, so |0| = 0.
Can an absolute value be negative?
No. By definition |x| is never below 0 for any real x. When x is negative, the minus sign in |x| = -x flips it positive, for example |-4.25| = 4.25.
What is the difference between absolute value and absolute difference?
Absolute value takes one number: |x| is its distance from 0, for example |-7| = 7. Absolute difference takes two numbers: |a - b| is the distance between them. This calculator handles single-number absolute value only.
What are the main properties of absolute value?
Four matter most: |x| is never negative; |-x| = |x|; |a x b| = |a| x |b|, for example |-3 x 4| = |-12| = 12 = 3 x 4; and the triangle inequality, where |a + b| is less than or equal to |a| + |b|, for example |-5 + 2| = 3 which is less than 7 = |-5| + |2|. Also, the square root of x squared equals |x|.
What is the absolute value of -0.5?
0.5. Since -0.5 is negative, negate it: -(-0.5) = 0.5.
Free tool by NRTechStudio