Absolute Value

Distance, without the direction

Every integer’s position on the number line carries two pieces of information. One is how far the number is from zero. The other is which side of zero it is on. The sign is what tells you that: a positive number sits to the right, a negative number to the left.

Often a problem cares only about the first piece. How far did the temperature move, up or down? To answer that, we need a way to keep the distance and let the side go.

The absolute value of a number is its distance from zero on the number line.

Distance is a count of equal steps, and you cannot take a negative number of steps along a line. So an absolute value is never negative.

The notation

To write “the distance of a number from zero,” we wrap the number in a pair of vertical bars. The absolute value of $-5$ is written $\lvert -5 \rvert$, and we read it aloud as “the absolute value of negative five.” Since $-5$ is five units from zero,

The bars are a command: measure how far what is inside sits from zero, and report that distance. A few quick readings, straight off the line:

The number $7$ is seven units from zero, and so is $-7$, so both have absolute value $7$. Zero is no distance from itself, so $\lvert 0 \rvert = 0$. Notice the output of the bars is just a number: once you have measured the distance, the bars have done their job and disappear.

Why the rule splits into two cases

If absolute value simply throws away a minus sign, why bother with a careful definition? Because “throw away the minus sign” describes what you see on the page. It is not a reason, and it stops working the moment a letter stands where the number was.

Read the two sides of zero off the line instead, one at a time.

A number that is already positive, or zero, sits at its own distance with nothing to undo. The point $7$ is seven units to the right of zero, so $\lvert 7 \rvert = 7$, the number itself.

A negative number is the interesting case. The point $-5$ lies five units to the left of zero, so its distance is $5$. That $5$ is the opposite of $-5$, and taking the opposite of a negative number gives a positive result: $-(-5) = 5$. So $\lvert -5 \rvert = 5$, the opposite of the number.

Neither reading used the size of the number. What mattered was only which side of zero it sat on, so the same two readings work for any integer $x$. Writing them down needs one new symbol. The sign $\ge$ is read “is greater than or equal to”, so $x \ge 0$ says that $x$ is zero or above. It is the familiar $>$ with the equal case allowed in as well.

The second case looks like it should be negative, and it is not. Here $-x$ means the opposite of $x$, and $x$ is negative in that case, so its opposite is positive.

A number and its opposite share one absolute value

Opposites, as the first lesson of this chapter showed, are the same distance from zero on opposite sides. Absolute value reports exactly that distance, so it cannot tell $8$ apart from $-8$: both give $8$. That holds for every number and its opposite:

A small but useful consequence is that the only number with an absolute value of $0$ is $0$ itself. Any other number sits some positive distance from zero, so its absolute value is positive.

The bars are grouping symbols

The absolute-value bars are grouping symbols. Like parentheses, they tell you what to finish first. They also do one job parentheses do not: once the inside is a single number, the bars measure how far that number is from zero. So a pair of bars is two instructions in one, and the grouping happens first.

Compare two expressions that look almost identical:

In the first, the bars group the sum, so add inside first and then take the absolute value:

In the second, each number has its own bars, so take each absolute value first and then add:

The two are not equal. In the first expression the bars enclose $(-3) + 5$ and reduce it to the single number $2$. The second never forms that sum at all, so it measures $-3$ and $5$ separately and adds the two distances. So finish the arithmetic inside a pair of bars first. That leaves one number, and its distance from zero is the answer.

A minus sign written outside the bars

The bars measure only what stands between them. A minus sign written in front of a pair of bars is outside them, so it takes no part in the measuring. It waits until the measuring is done, and then takes the opposite of the result:

The second answer is negative, and nothing has gone wrong, because it is not an absolute value. It is the opposite of one. “An absolute value is never negative” is a promise about what comes out from between the bars, and this minus sign was never between them.

Worked examples

Distance between two numbers

Absolute value also answers a question you might not expect: how far apart are two numbers on the line?

Walk from $3$ to $8$ on the line and it takes five one-unit steps. The two subtractions give opposite results, $-5$ and $5$. Taking the absolute value turns both of them into that same five:

The subtraction $3 - 8$ measures the signed gap, and its sign only records which number was taken from which. The bars throw that sign away and keep the size of the gap, which is what “how far apart” should mean. Since $3 - 8$ and $8 - 3$ are opposites, and opposites share one absolute value, the order of subtraction cannot change the answer.

So the gap between any two integers $a$ and $b$ is the absolute value of their difference,

This works across zero too. The distance between $-2$ and $4$ is

and the walk splits neatly at zero: two steps from $-2$ up to zero, then four more out to $4$.

Common mistakes