topic badge

7.02 Adding and subtracting integers

Lesson

Introduction

Now that we are familiar with locating integers on the number line, we can think about how to use the number line to add and subtract with integers.

When we were adding and subtracting whole numbers, the result was always another whole number. In particular, the result of a subtraction was always greater than or equal to zero. If we imagine the expression 5 - 3 = 2 to represent taking 3 apples from a pile of 5 apples, then it doesn’t make much sense to ask how many apples are left after performing the subtraction 3-5.

It turns out that apple arithmetic is fine for whole numbers, but it is not good enough if we want to make full use of the integers, which include positive and negative numbers. To understand the addition and subtraction of integers we can instead use the arithmetic of arrows on the number line or integer chips (yellow for positive and red for negative). This will give us a way to move back and forth along the number line, and to make sense of expressions like 3 - 5.

Arrows on the number line - addition

To begin, we can imagine that for every integer on the number line there is a corresponding arrow going from 0 to that integer.

Exploration

Move the point on the number line below.

Loading interactive...

For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

The addition of integers can be represented by adding their arrows on the number line. When we combine the lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to an integer.

The image below shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the order of addition does not affect the result?

A number line with a blue arrow from 0 to 6, a green arrow from 6 to 8, and a red arrow from 0 from 8.

What if we want to add a negative integer? We use the same approach, the only difference being that the arrows are pointing in different directions. The image below shows that 4 + (-9) = -5, which is the same result that we get from (-9) + 4.

A number line with a blue arrow from 0 to 4, a green arrow from 4 to negative 5, and a red arrow from 0 to negative 5.

The examples above show how we can combine positive and negative integers using addition to produce any other integer we like.

Examples

Example 1

Find the value of -7 + 13.

Worked Solution
Create a strategy

Plot the necessary points on the number line to solve the expression.

Apply the idea

We are starting at -7 and we want to add 13 to this.

-10-8-6-4-20246810

Adding a positive integer means we move to the right. So we need to move 13 units to the right so we get to 8:

-10-8-6-4-20246810
\displaystyle -7 + 13 \displaystyle =\displaystyle 6Evaluate
Idea summary

Adding a positive integer means we move to the right.

Arrows on the number line - subtraction

Let's go back to our example of 3 - 5, this is actually the same as 3 + (-5) = -2, as shown below. In other words, subtracting 5 is the same as adding the opposite of 5.

A number line with green arrows from 0 to 5,  0 to negative 5, 3 to -2, a blue arrow from 0 to 3 and a red arrow from 0 to -2

Finally, we can use the idea that subtracting a number is the same as adding its opposite to make sense of the expression 7 - (-2). Taking away -2 is the same as adding the opposite of -2, which we can write as 7 + (-(-2)). Now, this number (-(-2) is “the opposite of the opposite of 2”, which we know is just 2. So we have 7 - (-2) = 7 + 2, which gives 9 from our now familiar addition of arrows.

A number line with green, blue and red arrows. Ask your teacher for more information.

Exploration

Use the applet below to explore how to add and subtract integers on the number line. What kind of arrows add together to give 0? What is the result of adding 0 to any other integer?

Loading interactive...

Same with addition, for a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.

Examples

Example 2

Find the value of 3 - (-9).

Worked Solution
Create a strategy

Subtracting a negative integer is the same as adding its opposite.

Apply the idea

We are starting at 3 and we want to subtract -9 from this.

\displaystyle 3 - (-9)\displaystyle =\displaystyle 3 + 9Add its opposite

Plot 3 on the number line:

0123456789101112131415

From 3 we want to add 9, so we are going to move to the right.

0123456789101112131415
\displaystyle 3 - (-9)\displaystyle =\displaystyle 12Evaluate
Idea summary

Subtracting a negative integer means moving it to the right, and is the same as adding its opposite integer.

Adjacent signs

We have seen that any way we might want to move back or forth along the number line can be expressed as a sum of integers. Subtraction is just a particular kind of addition.

However, the concept of subtraction is still useful; in many cases it can make more sense to take away an integer rather than add its opposite. So it is common to see expressions involving the addition of a negative integer rewritten as the subtraction of a positive integer. This process is called combining adjacent signs.

For a number line with the positive direction to the right, we find the following results.

  • Adding a positive integer means we move to the right:3 + (+5) \quad = \quad 8 \quad = \quad 3 + 5
  • Adding a negative integer means we move to the left:3 + (-5) \quad = \quad -2 \quad = \quad 3 - 5
  • Subtracting a positive integer means we move to the left:3 - (+5) \quad = \quad -2 \quad = \quad 3 - 5
  • Subtracting a negative integer means we move to the right:3 - (-5) \quad = \quad 8 \quad = \quad 3 + 5

In an expression like 3 + (-5), the + and - are adjacent signs which are combined into one subtraction operation to give 3 - 5. This is a common way to simplify expressions. However, notice that these adjacent signs have different meanings. The first tells us the operation (either addition or subtraction), while the second tells us the sign of the integer (either positive or negative).

With this in mind, an expression like 3 - (-5) would best be read as "3 subtract negative 5", or "3 take away negative 5", or even "3 minus negative 5". We would like to avoid reading it as "3 minus minus 5", since "minus" is an operation, not the sign of the integer -5.

Examples

Example 3

Find the value of -7 + 12 + (-8).

Worked Solution
Create a strategy

Adding a negative integer means we move to the left on the number line.

Apply the idea
\displaystyle -7 + 12 + (-8)\displaystyle =\displaystyle -7 + 12 - 8Rewrite the expression

Plot -7 on a number line:

-10-8-6-4-20246810

From -7, move up 12 units to the right and we end up at 5.

-10-8-6-4-20246810
\displaystyle -7 + 12 + (-8)\displaystyle =\displaystyle 5 - 8Perform -7 + 12

Starting at 5, move down 8 units to the left.

-10-8-6-4-20246810
\displaystyle -12 + 16 + (-18)\displaystyle =\displaystyle -3Perform 5-8
Idea summary

For a number line with the positive direction to the right, we find the following results.

  • Adding a positive integer means we move to the right.
  • Adding a negative integer means we move to the left:.
  • Subtracting a positive integer means we move to the left.
  • Subtracting a negative integer means we move to the right.

Outcomes

VCMNA241

Compare, order, add and subtract integers.

What is Mathspace

About Mathspace