Absolute Value Calculator: Find |x| Instantly

The absolute value of a number is its distance from zero on the number line, ignoring direction. The symbol is two vertical bars: |x|. So |7| is 7 and |-7| is also 7, because both 7 and -7 sit 7 units away from zero, just on opposite sides.

That is why absolute value is never negative. Distance cannot be less than zero, and the absolute value of zero itself is zero, since zero is no distance from zero. In symbols, |x| equals x when x is positive or zero, and equals -x when x is negative, which flips the sign to make it positive.

For roots and powers, a square root calculator handles the related idea that squaring removes sign, since the square root of x squared is |x|, not x.

The definition splits into two cases. If the number is zero or positive, the absolute value is the number itself: |5| = 5 and |0| = 0. If the number is negative, the absolute value is the number with its sign flipped: |-5| = -(-5) = 5.

The minus sign in -x does not mean the result is negative. It is the operation of negating x, and negating a negative number gives a positive number. That is the step beginners trip over most, and it is worth checking with a concrete pair: |3| = 3 and |-3| = -(-3) = 3.

Example 1: |-7| = 7, because -7 is 7 units from zero. Example 2: |42| = 42, because a positive number keeps its value. Example 3: |0| = 0. Example 4: |-3.25| = 3.25, because flipping the sign of -3.25 gives 3.25.

Example 5: |100 - 140| = |-40| = 40. This pattern appears whenever you measure a difference: the error between a measured value and a true value is reported as an absolute error, so a reading 40 units low and a reading 40 units high both have absolute error 40.

An equation like |x| = 5 has two solutions, x = 5 and x = -5, because both numbers are 5 units from zero. In general, |x| = a with a positive has the two solutions x = a and x = -a, while |x| = 0 has only x = 0, and |x| = -3 has no solution, since an absolute value can never be negative.

An inequality like |x| < 5 means x is within 5 units of zero, so -5 < x < 5. The reverse, |x| > 5, means x is more than 5 units from zero, so x < -5 or x > 5. Drawing the two cases on a number line turns an abstract inequality into a visible interval.

The most common mistake is writing |-5| = -5, keeping the negative sign. Absolute value strips the sign, it never preserves it, so |-5| = 5. A second mistake is treating -x as automatically negative: if x is -4, then -x is 4, not -4.

A third mistake is assuming |x| = x is always true. It holds only for x at or above zero. A fourth is forgetting the second solution of |x| = a: both a and -a satisfy it, and dropping one loses half the answer.

Absolute value measures errors, differences, and distances wherever direction does not matter. A temperature 3 degrees below forecast and 3 degrees above it are both 3 degrees off. A stock that falls $2 and one that rises $2 both moved $2 in magnitude. GPS and mapping use absolute differences in coordinates to find distances.

In programming, absolute value appears in sorting by closeness, in graphics for distances between points, and in statistics for mean absolute error, where overestimates and underestimates count equally. Whenever a problem asks how far apart two values are, absolute value is the tool.