Integers, Opposites, and Absolute Value on a Number Line | Grade 7 Math | LevelKite

When you worked with factors and multiples, every number was a whole number, such as $0$ , $1$ , $2$ or $3$ . Whole numbers cannot describe everything. A winter temperature can be $3$ degrees below zero, and a game score can be $5$ points below zero. To describe amounts on both sides of zero, you need more numbers.

The integers are the whole numbers together with $-1$ , $-2$ , $-3$ and so on. Listed in order, they are $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ , where the dots show that the list continues without end in both directions.

A positive integer is greater than $0$ . You usually write it with no sign, so $3$ means $+3$ . A negative integer is less than $0$ . You write it with a minus sign in front and read $-3$ as "negative three", so $3$ degrees below zero is $-3$ degrees. Zero is an integer that is neither positive nor negative.

A number line shows where each integer sits. In this lesson, every number line is a horizontal line with evenly spaced tick marks. The distance from one tick to the next is $1$ unit. One tick is labeled $0$ . The positive integers $1, 2, 3, \dots$ label the ticks to the right of $0$ , and the negative integers $-1, -2, -3, \dots$ label the ticks to the left of $0$ .

Look at the number line below. It has markers on the ticks labeled $-4$ and $4$ . Start at the tick labeled $0$ and count ticks to the right: $1, 2, 3, 4$ . The marker at $4$ is $4$ units from $0$ . Now start at $0$ again and count ticks to the left: $-1, -2, -3, -4$ . The marker at $-4$ is also $4$ units from $0$ .

The markers at

-4

and

4

are each

4

units from

0

, on different sides of

0

.

Two different numbers are opposites when they are the same distance from $0$ on the number line but on different sides of $0$ . On the number line above, $-4$ and $4$ are both $4$ units from $0$ , one to the left and one to the right. So the opposite of $4$ is $-4$ , and the opposite of $-4$ is $4$ .

Every integer has exactly one opposite. Zero is the special case: it is $0$ units from $0$ and on neither side, so $0$ is its own opposite.

You can read a minus sign in front of a number as "the opposite of". So $-4$ means the opposite of $4$ , and $-(-4)$ means the opposite of $-4$ , which is $4$ .

Worked example

Write an integer and find its opposite

One winter morning the temperature in a town is $3$ degrees Celsius below zero. Write this temperature as an integer. Then find the opposite of that integer and say what temperature it stands for.

Choose the sign. Below zero is the negative side of 0, so the integer is negative. The amount is 3 degrees, so the integer is 3 units from 0.
$-3$
Locate -3 by tick labels. On a number line, counting ticks to the left of 0 gives -1, -2, -3, so -3 is 3 units to the left of 0.
Find the opposite. It is the same distance from 0 on the other side. Counting ticks to the right of 0 gives 1, 2, 3, so the opposite of -3 is 3.
$-(-3) = 3$
Give the meaning. The integer 3 is positive, and the positive side stands for above zero, so 3 stands for 3 degrees Celsius above zero.

Answer

The temperature is $-3$ degrees Celsius. Its opposite is $3$ , which stands for $3$ degrees Celsius above zero.

To graph an integer means to mark its point on the number line. Two things tell you where the point goes:

The sign tells you the side of $0$ . Positive integers are to the right of $0$ , and negative integers are to the left of $0$ .
The number without its sign tells you how many units the point is from $0$ .

Zero is neither positive nor negative. You graph it on the tick labeled $0$ .

The number line also shows order. On these number lines, the tick labels increase from left to right, so each integer is less than every integer to its right. Order follows position on the line, not the size of the digits. For example, reading tick labels from left to right gives $-5, -4, -3, -2$ . So $-5$ is to the left of $-2$ , and $-5$ is less than $-2$ , even though $5$ is greater than $2$ .

Worked example

Graph three integers

Graph the integers $3$ , $-2$ and $-5$ on a number line.

Choose the ticks. The line must include every integer you graph, so use ticks 1 unit apart from -6 to 6, with 0 in the middle. The figure below this example shows this line.
Graph 3. It is positive, so its point is to the right of 0. Count ticks from 0: 1, 2, 3. Put a marker on the tick labeled 3.
Graph -2. It is negative, so its point is to the left of 0. Count ticks from 0: -1, -2. Put a marker on the tick labeled -2.
Graph -5. It is also negative, so count ticks to the left of 0: -1, -2, -3, -4, -5. Put a marker on the tick labeled -5.
Check by position. Reading the figure from left to right, the markers are on -5, -2 and 3. The marker at -5 is left of the marker at -2 because -5 is 5 units to the left of 0 and -2 is only 2 units to the left of 0.

Answer

The markers go on the ticks labeled $-5$ , $-2$ and $3$ . From left to right, the graphed integers are $-5$ , $-2$ and $3$ .

The graph of

-5

,

-2

and

3

: two markers are left of

0

and one marker is right of

0

.

When you graphed $-5$ on the number line above, you counted $5$ units from $0$ . That count has a name. The absolute value of a number is its distance from $0$ on the number line. You write it by putting the number between two bars, and you read $|-5|$ as "the absolute value of negative five".

A distance tells you how far, not which side, so it is never negative. The number $-5$ is $5$ units from $0$ , so $|-5| = 5$ . Counting ticks to the right of $0$ gives $1, 2, 3, 4, 5$ , so $5$ is also $5$ units from $0$ and $|5| = 5$ . Zero is $0$ units from itself, so $|0| = 0$ .

Notice the opposites $-5$ and $5$ . They are on different sides of $0$ , but they have the same absolute value.

Key concept

Sign gives the side, absolute value gives the distance

Every integer has a position on the number line, and that position has two parts. The sign tells you the side of $0$ : positive integers are to the right and negative integers are to the left. Zero is neither positive nor negative and sits between them. The absolute value tells you the distance from $0$ . For every integer $a$ , $|a|$ is the number of units between $a$ and $0$ , so $|a|$ is never negative. Opposites are the same distance from $0$ , so they have the same absolute value.

|a| \ge 0

Worked example

Read absolute values from a number line

The number line below this example has markers on the ticks labeled $-6$ , $0$ and $4$ . Use it to find $|-6|$ and $|4|$ .

State what the bars ask for. The absolute value of -6 is the distance from -6 to 0, so measure from the marker at -6 to the marker at 0.
Count units from -6 to 0 by tick labels: -5, -4, -3, -2, -1, 0. That is 6 ticks, and the ticks are 1 unit apart, so the distance is 6 units.
$|-6| = 6$
Count units from 4 to 0 the same way: 3, 2, 1, 0. That is 4 ticks, so the distance is 4 units.
$|4| = 4$
Check the results. Each one is a distance, so neither is negative. Also notice that -6 is farther from 0 than 4 is, yet -6 is less than 4, because -6 is to the left of 4. Absolute value gives distance from 0. Position gives order.

Answer

The absolute values are $|-6| = 6$ , because $-6$ is $6$ units from $0$ , and $|4| = 4$ , because $4$ is $4$ units from $0$ .

The marker at

-6

is

6

units from the marker at

0

, and the marker at

4

is

4

units from the marker at

0

.

Worked example

Work backward from an absolute value

Which integers have an absolute value of $7$ ? Is there an integer with an absolute value of $-7$ ?

Turn the first question into a distance. Let n stand for an integer you are looking for. An absolute value of 7 means n is 7 units from 0. A distance does not name a side, so check both sides of 0.
$|n| = 7$
Check the right side of 0. Counting ticks from 0 gives 1, 2, 3, 4, 5, 6, 7, so the integer 7 is 7 units from 0.
$|7| = 7$
Check the left side of 0. Counting ticks from 0 gives -1, -2, -3, -4, -5, -6, -7, so the integer -7 is 7 units from 0.
$|-7| = 7$
Make sure there are no others. Each side of 0 has only one tick that is exactly 7 units from 0, so 7 and -7 are the only answers. They are opposites.
Answer the second question. An absolute value is a distance from 0, and a distance is never negative, so no integer has an absolute value of -7.

Answer

The integers $7$ and $-7$ both have an absolute value of $7$ . No integer has an absolute value of $-7$ , because a distance from $0$ is never negative.

Takeaway. Every integer has its own tick on the number line. For any integer other than $0$ , the sign tells you the side of $0$ . The absolute value tells you the distance from $0$ , so it is never negative. Opposites, such as $-4$ and $4$ , are the same distance from $0$ on different sides. Order follows position on the line, not distance from $0$ .

Next, you will compare and order integers and practice finding the absolute value and the opposite of an integer.