Absolute Value Calculator
Calculate the absolute value of any real number, with a step-by-step explanation of the result.
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How to use this calculator
The absolute value of a number is its distance from zero on the number line, always non-negative. Positive numbers and zero are unchanged; negative numbers become positive.
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Enter any real number — positive, negative, or zero.
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The absolute value is displayed instantly along with a step-by-step explanation.
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The result is always greater than or equal to zero.
Frequently asked questions
What does the “Step” line show?
It writes out which branch of the definition applied. For −7 it reads “−7 < 0, so |−7| = −(−7) = 7”; for a value of zero or above it simply states that the number equals its own absolute value. Zero counts as non-negative, so |0| = 0.
Can the result ever be negative?
No. The absolute value is zero for 0 and positive for every other real number. If you see a negative result somewhere, a sign was lost earlier, for example by treating −|x| as |−x|. The expression −|7| equals −7, while |−7| equals 7.
Why is “Distance from zero” the same as the answer?
Because absolute value is defined as distance from 0 on the number line, ignoring direction. −12.5 and 12.5 are both 12.5 units from zero. To measure the distance between two numbers, subtract them and enter the difference: 3 − (−7) is 10, so entering −10 returns 10.
Does it solve equations like |x − 3| = 5?
No, it evaluates a single number. For |x − 3| = 5, set x − 3 equal to 5 and to −5, which gives x = 8 and x = −2. For |x − 3| < 5 the solutions lie between −2 and 8. You can evaluate the candidates here by entering them as numbers.
How are very small and very large numbers displayed?
Numbers with magnitude below 0.000001 appear in exponent form, so −0.000000001 returns 1e-9 instead of a misleading 0. Ordinary decimals show up to eight places. Extremely large values such as 1e22 are shown as 1e+22.
What happens if I enter something that is not a number?
A non-numeric entry returns “Enter a valid number”. Type fractions as decimals, for example 0.75 instead of 3/4. Only real numbers are supported.
Does it work for complex numbers or vectors?
No. For a complex number a + bi the analogous quantity, the modulus, is the square root of a² + b², and for a vector it is the length. Neither is computed here. This calculator is for real numbers only.
Absolute value of a real number
The definition the calculator follows
The absolute value of x equals x when x is zero or positive and equals −x when x is negative. Equivalently, |x| is the square root of x², and it measures how far x sits from zero on the number line without regard to direction. The calculator applies that definition, shows the branch it used, and returns the same value again as the distance from zero.
| Input | |x| | Branch used |
|---|---|---|
| −7 | 7 | negative, so −x |
| −12.5 | 12.5 | negative, so −x |
| 3.25 | 3.25 | non-negative, unchanged |
| 0 | 0 | non-negative, unchanged |
| −0.000123 | 0.000123 | negative, so −x |
| −1e-9 | 1e-9 | negative, shown in exponent form |
Worked example: a temperature gap
If a reading is 3 degrees and the target is −7 degrees, the gap is 3 − (−7) = 10, or −10 if you subtract the other way. Entering −10 returns 10 either way, and that is the size of the gap regardless of which value you treat as first. This is the practical reason absolute value exists: errors, tolerances and distances care about size, not direction.
Worked example: very small values
Entering −0.000123 returns 0.000123 with the step shown in full, so the working stays visible. Entering −1e-9 returns 1e-9 in exponent form; without that format, rounding to a fixed number of decimals would turn such a tiny value into 0 and the step line would claim 0 < 0. This matters for measurement error work, where differences of that scale are common.
Rules that carry over to algebra
Three properties are used constantly. The absolute value of a product equals the product of the absolute values, |a × b| = |a| × |b|. The triangle inequality says |a + b| ≤ |a| + |b|. And |x − a| < r is the same as a − r < x < a + r, which describes all points within r of a. For |x| = k with k positive there are two solutions, x = k and x = −k.
What it does not do
Only one real number is processed at a time. There is no equation or inequality solving, no complex modulus and no vector length. Fractions must be typed as decimals, and the output is rounded to eight places for ordinary magnitudes, so a repeating decimal is displayed as an approximation.
Learn more from an authoritative source:
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