Zero Pairs and Integers on a Number Line | Grade 7 Math | LevelKite

Many quantities have a direction: up or down, gained or lost, earned or spent. You already write them with integers, the numbers $\dots, -2, -1, 0, 1, 2, \dots$ that have no fraction part.

Picture a number line: a line with numbers at evenly spaced tick marks, getting greater as you go right. Two numbers are opposites when they are the same distance from $0$ on different sides of it, like $5$ and $-5$ . (The opposite of $0$ is $0$ itself.)

In this lesson, you will see what happens when opposite quantities combine. Then you will add and subtract integers by moving along a number line.

Opposite quantities have the same size but opposite directions: up and down, gained and lost, positive charge and negative charge. Say you climb $5$ stairs and then go down $5$ stairs. The second move takes back exactly what the first one gave, so you are back where you started. With up as positive, the sum (the result of adding) is $5 + (-5) = 0$ .

A pair of opposites like this is called a zero pair. A number's opposite is also called its additive inverse, because adding it to the number gives $0$ .

Two quantities cancel like this only when they measure the same thing, their sizes match, and their directions are opposite. Climbing $5$ stairs and going down $3$ does not bring you back. The table shows four more zero pairs. Notice what the $0$ means in each one.

Opposite quantities that combine to make zero

Situation	The two quantities as integers	Sum	What the zero means
You earn 8 dollars, then spend 8 dollars.	$8$ (earned) and $-8$ (spent)	$8 + (-8) = 0$	You have the same amount of money as before.
An elevator goes down 3 floors, then up 3 floors.	$-3$ (down) and $3$ (up)	$-3 + 3 = 0$	The elevator is back on the floor where it started.
In a quiz game, you win 10 points, then lose 10 points.	$10$ (won) and $-10$ (lost)	$10 + (-10) = 0$	Your score is the same as before.
A hydrogen atom has one proton with charge $+1$ and one electron with charge $-1$ .	$1$ (proton) and $-1$ (electron)	$1 + (-1) = 0$	The atom has no overall charge.

Worked example

Opposite changes that make zero

The temperature inside a freezer is measured in degrees Celsius. While the door is open, the temperature rises $6$ degrees. After the door is shut, it falls $6$ degrees. Write each change as an integer, add the two changes, and explain what the sum means.

Choose what a positive number means before you write anything. Let a rise in temperature be positive, so a fall is negative. Then the rise of 6 degrees is positive 6 and the fall of 6 degrees is negative 6.
$\text{rise} = 6 \qquad \text{fall} = -6$
Check that the two changes are opposite quantities. Both are measured in degrees Celsius, both have the same size (6 degrees), and one goes up while the other goes down. So 6 and negative 6 are opposites.
Add the two changes. The fall takes back exactly what the rise added, and a number plus its opposite is always 0.
$6 + (-6) = 0$
Say what the 0 means. The sum is the total change, so the temperature has changed by 0 degrees in all. This does not say the freezer is at 0 degrees Celsius. It says the temperature is the same as it was before the door opened.

Answer

The changes are $6$ and $-6$ , and $6 + (-6) = 0$ . The total change is $0$ degrees, so the freezer ends at the temperature it started with.

A number line turns addition into movement. Let $p$ and $q$ stand for any two integers. To find $p + q$ , start at $p$ and let $q$ tell you how to move.

How far? Move $|q|$ units. $|q|$ is the absolute value of $q$ , its distance from $0$ . For example, $|5| = 5$ and $|-5| = 5$ .
Which way? Move right if $q$ is positive: adding a gain makes the total greater, and greater numbers are to the right. Move left if $q$ is negative: adding a loss makes the total less. If $q = 0$ , stay at $p$ .

Try $-2 + 5$ : the temperature is $-2$ degrees Celsius and then rises $5$ degrees. Start at $-2$ . Because $5$ is positive and $|5| = 5$ , move $5$ units right, counting $-1$ , $0$ , $1$ , $2$ , $3$ . You land on $3$ , so $-2 + 5 = 3$ .

On the number line below, Point A marks the start, $-2$ , and Point B marks the sum, $3$ . Each tick mark is $1$ unit from the next, and Point B is $5$ tick marks to the right of Point A.

Point A is the start,

-2

. Point B is the sum,

3

, which is

5

units to the right of Point A.

Worked example

Adding a negative integer on a number line

At noon the temperature is $4$ degrees Celsius. By evening it has changed by $-6$ degrees, which means it has fallen $6$ degrees. Use a number line to find $4 + (-6)$ , the evening temperature.

Find the start, the distance and the direction. The sum starts at 4. The number being added is negative 6. Its absolute value is 6, so the distance is 6 units. It is negative, so the direction is left.
$|-6| = 6$
Move the first 4 units left. This lands on 0, because 4 and negative 4 are a zero pair: opposites add to 0.
$4 + (-4) = 0$
Move the units that are still to go. You have used 4 of the 6 units, so 2 are left. Moving 2 units left from 0 lands on negative 2. Moving 4 units left and then 2 units left is the same as moving 6 units left, so negative 2 is the sum.
$4 + (-6) = 4 + (-4) + (-2) = 0 + (-2) = -2$
Match your work to the number line below. Point A is the start, 4. Point B is 0, where you are after the first 4 units. Point C is the sum, negative 2. Point C is 6 tick marks to the left of Point A.

Answer

$4 + (-6) = -2$ , so the evening temperature is $-2$ degrees Celsius.

From Point A at

4

, it is

4

units left to Point B at

0

and

2

more units left to Point C at

-2

.

Subtraction asks a question. For integers $p$ and $q$ , $p - q$ is the number that gives back $p$ when you add $q$ to it. For example, $9 - 4 = 5$ because $5 + 4 = 9$ . So subtracting $q$ has to undo adding $q$ .

Adding $q$ moves you $|q|$ units, right or left depending on the sign of $q$ . To undo that move, go the same distance the opposite way. That is exactly what adding the opposite of $q$ does. So to subtract $q$ on a number line, start at $p$ and add the opposite of $q$ . It works because $q$ and its opposite make a zero pair: add $q$ back and you return to $p$ .

Try $3 - 5$ . The opposite of $5$ is $-5$ , so $3 - 5 = 3 + (-5)$ . Start at $3$ and move $5$ units left. The first $3$ units take you to $0$ , and the other $2$ take you to $-2$ . So $3 - 5 = -2$ . Check by adding back: $-2 + 5 = 3$ , the number you started with.

Key concept

Adding is moving, and opposites make zero

Let $p$ and $q$ be any integers.

Add: $p + q$ is the number $|q|$ units from $p$ : to the right when $q$ is positive, to the left when $q$ is negative. When $q = 0$ , the sum is $p$ itself.
Opposites: $p$ and $-p$ are the same distance from $0$ . Starting at $p$ and adding $-p$ moves you that distance back to $0$ , so the sum is $0$ .
Subtract: subtracting $q$ undoes adding $q$ , so it is the same as adding $-q$ , the opposite (additive inverse) of $q$ .

p + (-p) = 0 \qquad p - q = p + (-q)

Worked example

Subtracting a negative integer on a number line

Use a number line to find $2 - (-5)$ .

Say what the subtraction asks for. It asks for the number that gives 2 when you add negative 5 to it. So subtracting negative 5 has to undo adding negative 5.
$\text{answer} + (-5) = 2$
Turn the subtraction into an addition. Adding negative 5 moves 5 units left. To undo that, move 5 units right, which is what adding 5 does. The opposite of negative 5 is 5, so subtracting negative 5 is the same as adding 5.
$2 - (-5) = 2 + 5$
Add on the number line. Start at 2. The number being added, 5, is positive and its absolute value is 5, so move 5 units right. You land on 7.
$2 + 5 = 7$
Check by adding back what you subtracted. Start at 7 and add negative 5: move 5 units left. You land on 2, the number you began with, so 7 is correct.
$7 + (-5) = 2$
Match your work to the number line below. Point A is the start, 2. Point B is the result, 7, which is 5 tick marks to the right of Point A. Point C belongs to the note that follows the number line.

Answer

$2 - (-5) = 2 + 5 = 7$ . Subtracting $-5$ moves you $5$ units to the right, from $2$ to $7$ .

Point A is the start,

2

. Point B is the result,

7

. Point C, at

-3

, is the mistaken result in the note below. Points B and C are each

5

units from Point A.

Here is the idea to keep. Opposite quantities, like a rise of $6$ degrees and a fall of $6$ degrees, combine to make $0$ , because one takes back exactly what the other gave. On a number line, adding $q$ to $p$ means moving $|q|$ units from $p$ : right when $q$ is positive, left when $q$ is negative. Subtracting $q$ undoes that move, so you add the opposite of $q$ instead.

Next, you will build on this to add integers and to subtract integers by adding the additive inverse.