4.01 Add and subtract integers

We know that integers can be either positive, negative, or 0 (which is neither positive or negative). This is indicated by the sign on an integer. Integers with + (or no sign at all) are positive. Integers with - are negative.

The sign of an integer gives it a direction. We can imagine that for every integer on the number line there is an arrow going from 0 to that integer.

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 the sum. This is because, adding two (or more) integers always results in another integer.

The image 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?

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 shows that 4 + \left(-9\right) = -5, which is the same result that we get from \left(-9\right) + 4.

To think of this process more simply, we can plot the first integer on the number line and move the direction and number of spaces indicated by the second integer.

When we add a positive number we move to the right.

-1+4=3

When we add a negative number we move to the left.

4+\left(-3\right)=1

Examples

Example 1

Find the value of -7 + 13.

Worked Solution

Create a strategy

Draw a model with arrows using a number line.

Apply the idea

We start by drawing an arrow for -7 in the number line.

Adding a positive integer means we move to the right. So, to draw the arrow that represents adding 13, we need to count 13 units to the right starting at -7.

Finally, we draw a third arrow which starts at 0 and ends at the tip of the second arrow. This third arrow represents the sum of -7 and 13 which is 6.

\displaystyle -7 + 13

\displaystyle =

\displaystyle 6

Evaluate

Reflect and check

What if we start by drawing first the arrow that represents 13 then the arrow that represents adding -7?

Notice that when we draw the third arrow that represents the sum, we get the same answer.

This shows that -7 + 13 is the same as 13 + \left( -7 \right). The order of drawing the arrows does not affect the sum.

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Example 2

Find the value of 11 + \left(-6\right).

Worked Solution

Create a strategy

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

Apply the idea

Plot 11 on the number line:

From 11 we want to move 6 units to the left.

\displaystyle 11 + \left(-6\right)

\displaystyle =

\displaystyle 5

Evaluate

Reflect and check

We can combine the adjacent signs by writing:

\displaystyle 11 + \left(-6\right)	\displaystyle =	\displaystyle 11 - 6	Adding a negative 6 is the same as subtracting 6
	\displaystyle =	\displaystyle 5	Subtract

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Example 3

Find the value of- 12 + \left(-8\right).

Worked Solution

Create a strategy

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

Apply the idea

Plot -12 on a number line:

From -12, move 8 units to the left.

\displaystyle - 12 + \left(-8\right)

\displaystyle =

\displaystyle -20

Reflect and check

\displaystyle - 12 + \left(-8\right)

\displaystyle =

\displaystyle - 12 - 8

Adding a negative is the same as subtracting a positive

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We have looked at how to model addition with number lines. Now we will explore integer chips. Integer chips are another way to model integers and use the fact that the sum of two opposite integers is 0.

Now that we've explored subtraction with integer chips, let's explore subtraction on a number line. When we added integers on a number line we went in the direction indicated by the sign of the number. But the subtraction operation tells us to reverse the direction of the integer that follows.

When we subtract a positive number we move to the left, because we're reversing the direction indicated by positive 4.

-2-4=-6

When we subtract a negative number we move to the right, because we're reversing the direction indicated by -3.

4-\left(-3\right)=7

Let's look at 3 - 5. We can see from the number line that this is actually the same as 3 + \left(-5\right). In other words, subtracting 5 is the same as adding the opposite of 5.

Let's look at 7-\left(-2\right). Starting at 7, we move to the right 2 because we're reversing the direction of -2. From the number line we can see this gives us the same result as adding 7+2.

Examples

Example 4

Find the value of 8 - 7.

Worked Solution

Create a strategy

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

Apply the idea

Plot 8 on a number line:

From 8, move 7 units to the left.

\displaystyle 8 -7

\displaystyle =

\displaystyle 1

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Example 5

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 - \left(-9\right)

\displaystyle =

\displaystyle 3 + 9

Add its opposite

We start by drawing an arrow to represent 3 and another arrow to representing adding 9 in the number line. Then, we draw a third arrow that represents the sum which is 12.

\displaystyle 3 - \left(-9\right)

\displaystyle =

\displaystyle 12

Evaluate

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Example 6

The image shows how the location of a miner traveling up and down a mine shaft relates to an integer on the number line:

a

If Nadia is initially 2\text{ m} above the surface, and descends 6\text{ m} in the elevator, what integer represents her end point?

Worked Solution

Create a strategy

The height of 2\text{ m} above the surface is represented by the integer 2 on the number line. Descending 6\text{ m} means we move in the negative direction on the number line by 6 units.

Apply the idea

\displaystyle \text{Location}	\displaystyle =	\displaystyle 2 - 6	Set up the equation
\displaystyle \text{}	\displaystyle =	\displaystyle -4	Evaluate

Nadia ends up below the surface, so the integer representing the end point is -4.

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b

If Nadia is at a location represented by the integer -4, and ascends 3\text{ m}, which option describes her new location?

A

7\text{ m} below the surface

B

1\text{ m} below the surface

C

3\text{ m} above the surface

D

7\text{ m} above the surface

Worked Solution

Create a strategy

Ascending 3\text{ m} in the elevator corresponds to moving in the positive direction on the number line by 3 units.

Apply the idea

\displaystyle \text{Location}	\displaystyle =	\displaystyle -4 + 3	Set up the equation
\displaystyle \text{ }	\displaystyle =	\displaystyle -1	Evaluate

Nadia's new location is 1\text{ m} below the surface. So, the correct answer is B.

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