Absolute Value Calculator

What absolute value means

The absolute value of a number is its distance from $0$ on the number line — a measurement that is always zero or positive, no matter which direction the number sits in. Written with bars around the number, $|x|$, absolute value strips away the sign and keeps only the size:

$|7| = 7 \qquad |-7| = 7$

Both $7$ and $-7$ are exactly $7$ units away from $0$, just on opposite sides, so both have the same absolute value.

|x|: distance from zero

For a single number $x$, $|x|$ answers "how far is $x$ from $0$?":

• If $x$ is positive or $0$, it's already its own distance from $0$: $|x| = x$.
• If $x$ is negative, its distance from $0$ is the same size with the sign dropped: $|x| = -x$ (which is positive, since $x$ itself is negative).

|a − b|: distance between two numbers

Wrapping a subtraction in absolute value bars gives the distance between two numbers, regardless of which one is larger:

$|a - b| = \text{distance between } a \text{ and } b$

Plain subtraction $a - b$ can come out negative if $b$ is bigger than $a$, but the bars erase that sign, so the order never changes the answer:

$|3 - 10| = |-7| = 7 \qquad |10 - 3| = |7| = 7$

Both compute the same distance — $7$ units — between $3$ and $10$ on the number line. This calculator's "distance between two numbers" mode uses exactly this |a − b| form.

Absolute value is always ≥ 0

Because absolute value measures distance, it can never be negative. The smallest possible absolute value is $0$, which happens only when the input itself is $0$ — every other number, positive or negative, has a strictly positive distance from $0$. This calculator's answer is always zero or positive; if you ever see a negative result for an absolute value, something has gone wrong upstream.

Worked examples

Example 1 — single negative number: $|-7| = 7$, since $-7$ is $7$ units from $0$.

Example 2 — exact decimal: $|3.5| = 3.5$; a positive number is already its own absolute value.

Example 3 — distance between two numbers: $|3 - 10| = 7$ and $|10 - 3| = 7$ — swapping the order gives the same distance.

Example 4 — zero: $|0| = 0$, the only number that is its own (and only) distance from itself.

Where absolute value shows up

Absolute value appears throughout math and science: measuring error or deviation (how far a measurement is from a target, regardless of direction), computing distance on a number line or between coordinates, and defining functions piecewise around a sign change. It's closely related to the Square Root calculator — for any real number $x$, $|x| = \sqrt{x^2}$, since squaring erases the sign and the (principal) square root always returns a non-negative value.