➖ Absolute Value Calculator: Find |x| Instantly

What is the Absolute Value Calculator?

Absolute value answers a simple question: how far is this number from zero, ignoring which side it sits on? On a number line, -7 and 7 are both seven units away from the origin, so both have an absolute value of 7. Formally, |x| equals x when x is zero or positive, and equals -x (the negation, which makes it positive) when x is negative. The notation uses two vertical bars around the value, and you will also hear it called the modulus of a number. This calculator handles one value at a time or a long list, and it accepts integers, decimals, fractions written as division, and expressions to be evaluated inside the bars.

The key property is that the result is never negative. |0| is 0, and for every other number the output is strictly positive. This is why absolute value shows up wherever only the size of a quantity matters and the direction does not: the gap between two temperatures, the magnitude of an error, the speed of an object regardless of which way it travels. It turns a signed quantity into a plain non-negative size, which is exactly what you want when you compare distances or measure how far off an estimate was.

A point that trips people up is the difference between |-x| and -|x|. The first takes the absolute value of -x, which is positive; the second takes the absolute value of x and then negates it, which is negative (or zero). The bars act on whatever sits inside them first, so |3 - 8| = |-5| = 5, not -5. When an expression appears inside the bars, evaluate the inside completely, then take the distance from zero of that result. The same rule explains why absolute value does not split across addition or subtraction: |a + b| is generally not |a| + |b|.

Absolute value obeys a small set of algebraic rules that are worth memorising. It distributes over multiplication and division, so |a times b| = |a| times |b| and |a / b| = |a| / |b| when b is not zero. It does not distribute over addition or subtraction; instead the triangle inequality holds, |a + b| is less than or equal to |a| + |b|. Two more identities come up constantly: |x| = the square root of (x squared), and |x| = |-x|. These let you rewrite messy expressions and check answers quickly.

The idea extends beyond plain numbers. In algebra, an absolute value equation such as |X| = k splits into two cases, X = k and X = -k, because two different points on the number line can be the same distance from zero. Absolute value inequalities split too: |X| less than k means -k is less than X is less than k (a band around zero), while |X| greater than k means X greater than k or X less than -k (everything outside the band). Graphed, the function y = |x| forms a V shape with its vertex at the origin, since the output rises in both directions away from zero.

For complex numbers the same word means the modulus, the distance of the point a + bi from the origin in the plane, computed as the square root of (a squared plus b squared). That is why engineers, statisticians and physicists all lean on absolute value: it is the cleanest way to express size, magnitude or error without worrying about sign or direction. This tool focuses on real numbers and lists of them, which covers the vast majority of homework, spreadsheet and everyday needs.

When to use it

• Finding how far a number is from zero, regardless of its sign, for a homework or algebra problem.
• Measuring the size of a difference or error, such as |measured - expected|, where only the magnitude matters.
• Converting a column of signed values (temperature swings, profit and loss, sensor readings) into their non-negative sizes.
• Setting up an absolute value equation or inequality and checking the two cases it splits into.
• Computing distances between two points on a number line as |a - b| for geometry or statistics work.
• Checking your own working when simplifying an expression that contains absolute value bars.

How to use the Absolute Value Calculator

1. Type a single number into the box, for example -7, to see its absolute value.
2. Or paste a list of numbers separated by commas, spaces or new lines to convert them all at once.
3. You can also enter an expression such as 3 - 8 and the tool evaluates inside the bars first.
4. Read the absolute value (the distance from zero) from the result panel.
5. For a list, check the per-value table that shows each input next to its absolute value.

Formula & method

Worked examples

Absolute value of common inputs

Properties and rules of absolute value

Solving absolute value equations and inequalities

Common mistakes to avoid

• Thinking absolute value can be negative. Absolute value is a distance, so it is never negative. If you ever get a negative answer, you have likely written -|x| (negate after) instead of |x|. The result of the bars alone is always 0 or positive.
• Splitting the bars across an operation. In |a - b| you must evaluate a - b first and then take the absolute value. |3 - 8| is |-5| = 5, not |3| - |8| = -5. The bars treat everything inside as one quantity, and absolute value does not distribute over addition or subtraction.
• Confusing |-x| with -|x|. |-x| is the absolute value of -x, which is positive. -|x| is the absolute value of x with a minus sign placed in front, which is negative or zero. They are not the same thing.
• Forgetting the second case in an equation. When you solve |X| = k for k greater than 0, there are two answers, X = k and X = -k. Many learners report only the positive case and miss the negative one. Always set up both branches.
• Solving an equation that has no solution. If you isolate the absolute value and it equals a negative number, such as |x| = -3, there is no solution, because a distance can never be negative. Do not force an answer.
• Mixing up the bar notation with brackets. The two vertical bars |x| are not parentheses or rounding. They specifically mean distance from zero, so |2.7| stays 2.7, it does not round to 3.

Glossary

Absolute value

The distance of a number from zero on the number line, written |x|. It is always zero or positive.

Modulus

Another name for absolute value: the modulus of a number is its size with the sign removed. For complex numbers it is the distance from the origin in the plane.

Number line

A straight line where every real number has a position; absolute value measures how far a number sits from the zero point.

Magnitude

The size of a quantity ignoring its direction or sign, which is exactly what absolute value reports.

Negation

Multiplying a number by -1. For a negative number, negation produces its positive absolute value.

Triangle inequality

The rule that |a + b| is less than or equal to |a| + |b|, which limits how large the absolute value of a sum can be.

Absolute value equation

An equation containing absolute value bars, such as |X| = k, which usually splits into two cases, X = k and X = -k.

Argument

The expression sitting inside the absolute value bars, which you evaluate or solve before applying the distance from zero.

Frequently asked questions