The absolute value of a number is how far it’s from zero, ignoring direction. It’s written with two vertical bars, so |-7.5| = 7.5.
The same idea gives you the distance between any two numbers, and that’s why it keeps turning up in geometry, statistics and code.
How to use the Absolute Value Calculator#
- Choose a mode. Pick a single absolute value or a distance.
- Enter your number. Negative numbers are fine.
- Read the result. The absolute value is always zero or positive.
The definition#
If the number is already positive or zero, it stays the same. If it’s negative, you drop the minus sign. So |12| = 12 and |-12| = 12.
Distance between two numbers#
The distance between -4 and 3 on the number line is |-4 − 3| = 7. Order doesn’t matter, because |3 − (-4)| is also 7.
Where it’s used#
Absolute values appear in error measures, like the absolute error in the percent error calculator. They’re used in solving equations like |x − 2| = 5, which has two answers, 7 and -3, and in programming, where an abs function is standard.
Absolute value as distance#
The absolute value of a number is its distance from zero on the number line, and distance is never negative. That is why |−7| and |7| are both 7. The same idea gives the distance between two numbers: |x − y| is how far apart they are, whichever one is larger. The distance from −3 to 5 is |−3 − 5| = |−8| = 8.
Zero has an absolute value of zero, and it is the only number for which that is true. Every other number has a positive absolute value.
Equations and inequalities#
Absolute values appear in equations like |x − 3| = 5. This means x is 5 units away from 3, so x = 8 or x = −2. For an inequality such as |x| < 4, x has to be less than 4 units from zero, so −4 < x < 4. For |x| > 4, x is more than 4 units away, so x < −4 or x > 4. Solving these means splitting the problem into two cases, one for a positive inside and one for a negative inside.
| Expression | Value | Why |
|---|---|---|
| |−12| | 12 | 12 units from zero |
| |3 − 10| | 7 | |−7| = 7 |
| −|5| | −5 | The minus sign is outside the bars |
| |−2 × 6| | 12 | Work out the inside first |
| |0| | 0 | Zero is its own opposite |
In spreadsheets and code#
The function exists nearly everywhere: ABS() in Excel and Google Sheets, abs() in Python, Math.abs() in JavaScript and fabs() or abs() in C. A piecewise definition says the same thing as the formula: |x| equals x when x is zero or positive, and −x when x is negative. The graph of y = |x| is a V with its point at the origin. Programmers use it to compare numbers with a tolerance, for example to check that two decimals are close enough by testing whether |a − b| is under 0.0001.
Two rules, and absolute value inequalities#
The absolute value of a product is the product of the absolute values: |a × b| = |a| × |b|. So |−3 × 4| = 12 and |−3| × |4| = 12. But the absolute value of a sum is not the sum of the absolute values. The triangle inequality says |a + b| ≤ |a| + |b|, with equality only when a and b have the same sign. With a = −3 and b = 4, |a + b| = 1, while |a| + |b| = 7. Many algebra mistakes come from treating the bars as brackets that simply drop minus signs.
| a | b | |a + b| | |a| + |b| |
|---|---|---|---|
| 3 | 4 | 7 | 7 |
| −3 | 4 | 1 | 7 |
| −3 | −4 | 7 | 7 |
| 5 | −5 | 0 | 10 |
Inequalities work too. To solve |x − 3| < 2, ask which numbers are less than 2 away from 3. The answer is every x between 1 and 5. In real life, absolute values show up whenever direction does not matter, such as the size of an error, a temperature change or the distance between two towns on a road.
Frequently asked questions#
The same number without the minus sign.
Zero.
No. It’s always zero or positive.
x can be 5 or -5.
Subtract one from the other and take the absolute value. The distance between 4 and −6 is |4 − (−6)| = |10| = 10. It does not matter which number you subtract from the other, because |x − y| and |y − x| are equal. Use the second mode of the calculator and enter both numbers.