Absolute Value of a Number

What the Absolute Value Calculator Does

This Absolute Value Calculator returns the absolute value of any number you enter, written as |x|. The absolute value is the distance of a number from zero on the number line, so the result is always zero or positive. Enter a value like -7 and it returns 7; enter 4.2 and it returns 4.2.

It is useful for students checking algebra and pre-calculus homework, engineers and finance users measuring how far a reading deviates from a target, and anyone who needs a quick magnitude without worrying about direction or sign.

How Absolute Value Works (The Formula)

Absolute value strips the sign from a number while keeping its size. The standard piecewise definition is:

|x| = x when x is greater than or equal to 0, and |x| = -x when x is less than 0.

The second case can look confusing: when x is negative, -x is actually positive. For example, if x = -5, then -x = -(-5) = 5. An equivalent formula used in programming and spreadsheets is |x| = sqrt(x^2), since squaring removes the sign and the square root returns the non-negative root.

Worked Example

Suppose you want |-12.5|. Because -12.5 is less than 0, you apply the second rule: |x| = -x = -(-12.5) = 12.5. The number sits 12.5 units to the left of zero, so its distance from zero is 12.5.

Check it with the square-root method: sqrt((-12.5)^2) = sqrt(156.25) = 12.5. Both approaches agree, which is exactly what the calculator returns.

Working With Expressions Inside the Bars

If the input is an expression, evaluate everything inside the bars first, then take the absolute value. Treat the bars like parentheses for order of operations.

For instance, |3 - 9| = |-6| = 6, not 3 - 9 = -6. Likewise, a coefficient outside the bars multiplies after you take the absolute value: 2 x |-4| = 2 x 4 = 8.

Common Mistakes and Tips

A few errors come up repeatedly. Keep these points in mind when using absolute value:

• Absolute value is never negative. |x| = 0 only when x = 0; otherwise the result is positive.
• It does not simply delete the minus sign in every step. |-x| is not always x unless you know x is non-negative; |-(−5)| = 5, but the rule depends on the actual value.
• Distribute carefully: |a + b| is generally not |a| + |b|. For example, |3 + (-7)| = 4, but |3| + |-7| = 10.
• For an absolute value equation like |x| = 6, remember there are two solutions, x = 6 and x = -6.
• When measuring error or deviation, absolute value gives the magnitude of the gap regardless of whether the reading was above or below target.

Where Absolute Value Is Used

Beyond homework, absolute value appears anywhere direction does not matter but size does. In statistics it underlies the mean absolute deviation; in quality control and instrumentation it measures how far a measurement strays from a setpoint; in finance it can express the size of a gain or loss without its sign.

On a number line you can read it directly: the absolute value of any point is simply how many units it sits from zero, in either direction.