What is absolute value
The absolute value of a number x, written |x|, is its distance from zero on the number line — always zero or positive. For example, |−7| = 7 and |7| = 7, because both are 7 units from zero.
Distance has no direction, so 7 and −7 sit the same number of steps from the origin and share one absolute value. Absolute value is also what makes distance work on a number line — the one-dimensional version of the distance formula.
How absolute value is defined
Formally, |x| = x when x is zero or positive, and |x| = −x when x is negative. Taking the negative of a negative number makes it positive, which is why |−7| = 7. It strips the sign while keeping the magnitude.
One common mistake is treating the bars as ordinary brackets: |−3 + 1| is |−2| = 2, not 3 + 1. Always finish the arithmetic inside the bars first, and remember the output is never below zero. Because |x| equals the positive square root of x squared, you can get the same answer by squaring the input and using the square root calculator.
- |x| = x for x greater than or equal to zero
- |x| = −x for x less than zero
- |x| = 0 only when x = 0
Where absolute value is used
Absolute value measures size regardless of direction — useful for distances, errors and deviations. It appears in the percent difference calculator, which uses |a − b| to compare two values.
In measurement work you often care only about how far off a reading is, not whether it was high or low, so error terms are wrapped in bars before being averaged. In geometry, the distance between two points on a line is |a − b|. A larger absolute value means a bigger gap; a value near zero means the two quantities are nearly identical.
Frequently asked questions
- What is the absolute value of −7?
- The absolute value of −7 is 7, since it is 7 units away from zero.
- Can absolute value be negative?
- No. Absolute value is always zero or positive because it represents a distance.
- What is the absolute value of zero?
- The absolute value of zero is zero.
- How is |x| written in math?
- Absolute value is written with vertical bars around the number, like |x| or |−7|.
- Does |a − b| give the same answer as |b − a|?
- Yes, both give the same result. The bars remove the sign, so the order of subtraction does not matter and |3 − 10| = |10 − 3| = 7. That is why absolute value is used for distances and gaps between two values.