Absolute Value Measures Distance from Zero | Grade 6 Math | LevelKite

Think about two numbers: $-4$ and $4$ . They are not the same number. One is negative and sits to the left of $0$ on a number line. The other is positive and sits to the right of $0$ . The number line below shows both of them.

Still, they have something in common. Each one is the same number of steps away from $0$ . That distance has a name, and it is the big idea of this lesson.

You already work with rational numbers: numbers that can be written as a fraction of two integers, such as $-4$ , $\frac{3}{4}$ and $-1.75$ . Here you will learn what the distance from $0$ means, how to find it for any rational number, and how to compare two of these distances.

Count equal steps from

0

:

4

steps to the left reach

-4

, and

4

steps to the right reach

4

.

Look at the number line. The small marks on it are called ticks. Here every tick has a printed number, and the printed numbers go up by $1$ . So one step from a tick to the next tick is $1$ unit, which is the distance from $0$ to $1$ .

The two markers show $-4$ and $4$ . Each marker sits exactly on a tick, so you can count whole steps from $0$ .

Start at $0$ and count steps to the marker labelled $4$ : one, two, three, four. That is $4$ steps to the right.
Go back to $0$ and count steps to the marker labelled $-4$ : one, two, three, four. That is $4$ steps to the left.

Each step is $1$ unit, so each number is $4 \times 1 = 4$ units from $0$ . The directions are different. The distance is the same.

Key concept

Absolute value is distance from zero

The absolute value of a number is its distance from $0$ on the number line.

When the number sits exactly on a tick of a number line with equal steps, you can find that distance by counting. Count the steps from $0$ to the number. Then multiply the count by the size of one step.

Steps to the left count the same as steps to the right, because a distance has no direction. A count of steps is $0$ or more, so an absolute value is never negative.

To write an absolute value, put a vertical bar on each side of the number. You read $\lvert -4 \rvert$ as the absolute value of negative four.

You counted $4$ steps of $1$ unit from $0$ to $-4$ , and $4$ steps of $1$ unit from $0$ to $4$ . In symbols:

\lvert -4 \rvert = 4 \quad \text{and} \quad \lvert 4 \rvert = 4

Steps are not always whole units. On the next number line, only some ticks have printed numbers. That is still enough to find the scale.

Pick two printed numbers that are next to each other. Count the equal steps between them. Then divide the difference between the two numbers by that count. The result is the size of one step, and you can count from $0$ just as before.

A number is printed at every fourth tick. The marker at

-1.75

sits on the seventh tick to the left of

0

.

Worked example

Count equal steps on a number line

The number line above has a marker at $-1.75$ . Count equal steps from $0$ to find $\lvert -1.75 \rvert$ .

Find the size of one step. Not every tick has a printed number, so use two printed numbers that are next to each other: 0 and 1. They are 1 unit apart, and there are 4 equal steps between them. So one step is 1 divided by 4.
$1 \div 4 = 0.25$
Count the steps from 0 to the marker labelled -1.75. The marker sits exactly on a tick. Counting left from 0, the ticks are -0.25, -0.5, -0.75, -1, -1.25, -1.5 and -1.75. The marker is on the 7th tick, so it is 7 steps from 0.
Multiply the number of steps by the size of one step, because the distance is 7 steps of 0.25 each.
$7 \times 0.25 = 1.75$
Write the distance as an absolute value. The steps went to the left, but a distance only counts how many steps there are, so the result is positive.
$\lvert -1.75 \rvert = 1.75$

Answer

The absolute value of $-1.75$ is $1.75$ , because $-1.75$ is $7$ steps of $0.25$ from $0$ .

You will not always have a number line to count on. That is fine. The way a number is written already tells you how far it is from $0$ .

A positive number such as $9$ is $9$ steps of $1$ unit to the right of $0$ . Its distance from $0$ is $9$ , so $\lvert 9 \rvert = 9$ . The absolute value of a positive number is the number itself.
A negative number such as $-9$ is $9$ steps of $1$ unit to the left of $0$ . Its distance from $0$ is also $9$ , so $\lvert -9 \rvert = 9$ . When a negative number is written with a negative sign in front, its absolute value is that number without the sign.
Zero is $0$ steps from $0$ , so $\lvert 0 \rvert = 0$ .

The steps do not have to be whole units. The table below uses the same reasoning with decimal steps and fraction steps.

Each absolute value comes from counting equal steps from zero

Number	Side of zero	Equal steps from zero	Absolute value
$-12$	left	$12$ steps of $1$	$\lvert -12 \rvert = 12$
$2.5$	right	$5$ steps of $0.5$	$\lvert 2.5 \rvert = 2.5$
$-0.6$	left	$6$ steps of $0.1$	$\lvert -0.6 \rvert = 0.6$
$\frac{7}{3}$	right	$7$ steps of $\frac{1}{3}$	$\left\lvert \frac{7}{3} \right\rvert = \frac{7}{3}$
$-\frac{2}{5}$	left	$2$ steps of $\frac{1}{5}$	$\left\lvert -\frac{2}{5} \right\rvert = \frac{2}{5}$

Now you can compare absolute values. Comparing two absolute values means comparing two distances from $0$ . You are asking which number is farther from $0$ , or whether both are the same distance away.

Work in two parts. First, find each absolute value. Second, compare those two distances the way you already compare rational numbers. Decide whether the first is less than, greater than or equal to the second.

A number line helps you see the comparison. On one number line with equal steps, the marker that is more steps from $0$ has the greater absolute value. Two different numbers can also tie. You saw that $-4$ and $4$ are both $4$ steps of $1$ unit from $0$ , so $\lvert -4 \rvert = \lvert 4 \rvert$ .

A number is printed at every second tick, so each step is

0.5

. The marker at

-3.5

is

7

steps from

0

, and the marker at

2

is

4

steps from

0

.

Worked example

Compare absolute values on a number line

The number line above has markers at $-3.5$ and $2$ . Compare $\lvert -3.5 \rvert$ and $\lvert 2 \rvert$ .

Find the size of one step. The printed numbers 0 and 1 are next to each other and 1 unit apart, with 2 equal steps between them. So one step is 1 divided by 2.
$1 \div 2 = 0.5$
Count the steps from 0 to the marker labelled -3.5. It sits on the 7th tick to the left of 0. Seven steps of 0.5 give its distance from 0.
$\lvert -3.5 \rvert = 7 \times 0.5 = 3.5$
Count the steps from 0 to the marker labelled 2. It sits on the 4th tick to the right of 0. Four steps of 0.5 give its distance from 0.
$\lvert 2 \rvert = 4 \times 0.5 = 2$
Compare the two distances. Both counts use the same step size, so 7 steps is farther than 4 steps. The values agree: 3.5 is greater than 2.
$3.5 > 2$
Replace each distance with the absolute value it came from, because 3.5 is the absolute value of -3.5 and 2 is the absolute value of 2.
$\lvert -3.5 \rvert > \lvert 2 \rvert$

Answer

The absolute value of $-3.5$ is greater than the absolute value of $2$ , because $-3.5$ is $3.5$ units from $0$ and $2$ is only $2$ units from $0$ .

Worked example

Compare absolute values without a number line

Compare $\left\lvert -\frac{3}{4} \right\rvert$ and $\lvert -0.8 \rvert$ .

Find the first absolute value. The number -3/4 is 3 steps of one fourth to the left of 0, so its distance from 0 is three fourths.
$\left\lvert -\frac{3}{4} \right\rvert = \frac{3}{4}$
Find the second absolute value. The number -0.8 is 8 steps of one tenth to the left of 0, so its distance from 0 is 0.8.
$\lvert -0.8 \rvert = 0.8$
Write three fourths as a decimal so both distances have the same form. Multiply the top and bottom by 25. That is the same as multiplying by 1, so the value does not change.
$\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$
Compare the decimals by place value. Both have 0 ones. In the tenths place, 7 tenths is less than 8 tenths.
$0.75 < 0.8$
Replace each distance with the absolute value it came from, because 0.75 is the absolute value of -3/4 and 0.8 is the absolute value of -0.8.
$\left\lvert -\frac{3}{4} \right\rvert < \lvert -0.8 \rvert$

Answer

The absolute value of $-\frac{3}{4}$ is less than the absolute value of $-0.8$ , so $-0.8$ is farther from $0$ than $-\frac{3}{4}$ is.

Keep this idea: absolute value is distance from $0$ . To find it, count equal steps from $0$ to the number and leave the direction out. That is why $\lvert -4 \rvert$ and $\lvert 4 \rvert$ are both $4$ , and why an absolute value is never negative. To compare absolute values, find each distance first and then compare the distances.

Next, you will use absolute value to describe the size of amounts in real-world situations. You will also see how comparing absolute values is different from putting the numbers themselves in order.