Use Negatives & Opposites

If you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers.

Negative numbers are numbers less than 0. The negative numbers are to the left of zero on the number line (see Figure 2.1.1).

 

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number and no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line:

• Going from left to right — the numbers increase in value.
• Going from right to left — the numbers decrease in value.
• See Figure 2.1.2

 

Now, we need to extend the number line (which only showed the whole numbers) to include negative numbers. The numbers marked by points in Figure 2.1.3 are called the integers. The integers are the numbers …−3, −2, −1, 0, 1,  2, 3….

 

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and −2 are the same distance from zero, they are called opposites. The opposite of 2 is −2, and the opposite of −2 is 2.

Figure 2.1.5 illustrates the definition. The opposite of 3 is −3.

 

 

The whole numbers and their opposites are called the integers. The integers are the numbers …−3, −2, −1, 0, 1, 2, 3….

Simplify Expressions With Absolute Value

We saw that numbers, such as 3 and −3, are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

• −5 is 5 units away from 0, so |−5| = 5
• 5 is 5 units away from 0, so |5| = 5

Figure 2.1.6 illustrates this idea. The integers 5 and are 5 units away from 0.

 

The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from 0 to 0 on the number line is zero units.

In the next example, we will order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, we first simplify inside the absolute value bars as much as possible, and then we take the absolute value of the resulting number.

In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. However, doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two colour counters to model the addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one colour (blue) represent positive. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero (see Figure 2.1.7).

 

We will use the counters to show how to add the four addition facts using the numbers (5, −5) and (3, −3).

[latex]\begin{array}{cccccccccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill & & & \hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \end{array}[/latex]

The first example adds 5 positives and 3 positives — both positives.

The second example adds 5 negatives and 3 negatives — both negatives.

In each case, we got 8 — either 8 positives or 8 negatives.

When the signs were the same, the counters were all the same colour, so we added them (see Figure 2.1.8).

 

So, what happens when the signs are different? Let’s add (−5 + 3) and (5 + (−3)).

When we use counters to model the addition of positive and negative integers, it is easy to see whether there are more positive or negative counters. So, we know whether the sum will be positive or negative (see Figure 2.1.9).

 

Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.

When you need to add numbers such as (37 + (−53)), you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because (53 – 37 = 16), there are 16 more red counters.

Therefore, the sum of 37 + (−53) is −16.

[latex]37 + (-53) = -16[/latex]

Let’s try another one. We’ll add −74 + (−27). Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is −101.

[latex](-74) + (-27) = -101[/latex]

Let’s look again at the results of adding the different combinations of (5, −5) and (3, −3).

Visualize the model as you simplify the expressions in the following examples.

The techniques used up to now extend to more complicated problems, like the ones we have seen before. Remember to follow the order of operations!

Subtract Integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers, and the red counters represent negative numbers.

Perhaps when you were younger, you read “5 − 3” as “5 take away 3.” When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers 5 and 3.

[latex]\begin{array}{cccccccccc}\hfill 5-3\hfill & & & \hfill -5-(-3)\hfill & & & \hfill -5-3\hfill & & & \hfill 5-(-3)\hfill \end{array}[/latex]

In the first example, we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one colour, and the “take away” model of subtraction was easy to apply (see Figure 2.1.12).

 

What happens when we have to subtract one positive and one negative number? We will need to use both white and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels — the value is the same, but it looks different.

To subtract −5 − 3, we restate it as −5 take away 3.

We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.

Remember, a neutral pair has a value of zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.

Now, the fourth case, 5 − (−3). We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So, we add neutral pairs until we have 3 negatives to take away.

Have you noticed that you can subtract signed numbers by adding the opposite? In Example H, −3 − 1 is the same as −3 + (−1) and 3 − (−1) is the same as 3 + 1. You will often see this idea, the subtraction property, written as follows:

Look at these two examples in Figure 2.1.23.

 

[latex]6-4\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}6+\left(-4\right)[/latex].

Of course, when you have a subtraction problem that has only positive numbers, like 6 − 4, you just do the subtraction. You already knew how to subtract 6 − 4 long ago. But knowing that 6 − 4 gives the same answer as 6 + (−4) helps when you are subtracting negative numbers. Make sure that you understand how 6 − 4 and 6 + (−4) give the same results!

Look at what happens when we subtract a negative in Figure 2.1.24.

 

[latex]8-\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}8+5[/latex]

Subtracting a negative number is like adding a positive!

You will often see this written as [latex]a-\left(-b\right)=a+b[/latex].

What happens when there are more than three integers? We just use the order of operations as usual.

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show the multiplication of integers. Let’s look at this concrete model to see what patterns we notice.

We will use the same examples that we used for addition and subtraction. In Figure 2.1.25, we will use the model to help us discover the pattern.

 

We remember that a ⋅ b means to add a, b times. Here, we are using the model just to help us discover the pattern.

The next two examples in Figure 2.1.26 are more interesting.

 

What does it mean to multiply 5 by −3? It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. However, there is nothing to take away, so we start by adding neutral pairs to the workspace. Then we take away 5 three times.

In summary:

[latex]\begin{array}{cccccccc}\hfill 5 \cdot 3& =\hfill & 15\hfill & & & \hfill -5\left(3\right)& =\hfill & -15\hfill \\ \hfill 5\left(-3\right)& =\hfill & -15\hfill & & & \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\hfill \end{array}[/latex]

Notice that for the multiplication of two signed numbers, when the:

• Signs are the same, the product is positive.
• Signs are different, the product is negative.

We will put this all together below.

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1?

Let’s multiply a positive number and then a negative number by −1 to see what we get.

[latex]\begin{array}{ccccccc}& & & \hfill -1\cdot 4\hfill & & & \hfill -1\left(-3\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill -4\hfill & & & \hfill 3\hfill \\ & & & \hfill -4\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}4.\hfill & & & \hfill 3\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-3.\hfill \end{array}[/latex]

Each time we multiply a number by −1, we get its opposite!

Divide Integers

What about division? Division is the inverse operation of multiplication. So, 15 ÷ 3 = 5 because 5 ⋅ 3 = 15. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15.

Look at some examples of multiplying integers to figure out the rules for dividing integers.

[latex]\begin{array}{cccccccccccccc}\hfill 5\cdot3& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div 3\hfill & =\hfill & 5\hfill & & & & & \hfill -5\left(3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div3\hfill & =\hfill & -5\hfill \\ \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div \left(-3\right)\hfill & =\hfill & -5\hfill & & & & & \hfill 5\left(-3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div \left(-3\right)\hfill & =\hfill & 5\hfill \end{array}[/latex]

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

• Signs are the same, the quotient is positive.
• Signs are different, the quotient is negative.

Remember that we can always check the answer of a division problem by multiplying.

Simplify Expressions With Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers — addition, subtraction, multiplication, and division. Remember to follow the order of operations.

The next example reminds us to simplify inside parentheses first.

Key Concepts

Addition of Positive & Negative Integers

[latex]\begin{array}{cccc}5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}-5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}8\hfill & & & \phantom{\rule{3.45em}{0ex}}-8\hfill \\ \text{both positive,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{both negative,}\hfill \\ \text{sum positive}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum negative}\hfill \\ \\ \\ -5+3\hfill & & & \phantom{\rule{2.5em}{0ex}}5+\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-2\hfill & & & \phantom{\rule{3.45em}{0ex}}2\hfill \\ \text{different signs,}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{different signs,}\hfill \\ \text{more negatives}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{more positives}\hfill \\ \text{sum negative}\hfill & & & \phantom{\rule{2.5em}{0ex}}\text{sum positive}\hfill \end{array}[/latex]

Property of Absolute Value

[latex]|n|\ge 0[/latex] for all numbers. Absolute values are always greater than or equal to zero!

Subtraction of Integers

[latex]\begin{array}{cccc}5-3\hfill & & & -5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}2\hfill & & & \phantom{\rule{0.95em}{0ex}}-2\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{2 positives}\hfill & & & \text{2 negatives}\hfill \\ \\ \\ -5-3\hfill & & & 5-\left(-3\right)\hfill \\ \phantom{\rule{0.95em}{0ex}}-8\hfill & & & \phantom{\rule{0.95em}{0ex}}8\hfill \\ 5\phantom{\rule{0.2em}{0ex}}\text{negatives, want to}\hfill & & & 5\phantom{\rule{0.2em}{0ex}}\text{positives, want to}\hfill \\ \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \text{subtract}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \text{need neutral pairs}\hfill & & & \text{need neutral pairs}\hfill \end{array}[/latex]

Subtraction Property

Subtracting a number is the same as adding its opposite.

Multiplication & Division of Two Signed Numbers

• Same signs — Product is positive
• Different signs — Product is negative

Glossary

• Absolute value — The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as |n|.
• Integers — The whole numbers and their opposites are called the integers: …−3, −2, −1, 0, 1, 2, 3….
• Opposite — The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: -a means the opposite of the number. The notation −a is read “the opposite of a.”

2.1: Practice Questions

1. Order each of the following pairs of numbers, using < or >.
   1. [latex]9\_\_\_4[/latex]
   2. [latex]-3_\_\_6[/latex]
   3. [latex]-8\_\_\_-2[/latex]
   4. [latex]1\_\_\_-10[/latex]

    

2. Simplify.
   1. [latex]\mid-32\mid[/latex]
   2. [latex]\mid0\mid[/latex]
   3. [latex]\mid16\mid[/latex]

    

3. Fill in <, >, or = for each of the following pairs of numbers.
   1. [latex]-6\_\_\_\mid-6\mid[/latex]
   2. [latex]-\mid-3\mid\_\_\_-3[/latex]

    

4. Simplify.
   1. [latex]-(-5)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-|-5|[/latex]
   2. [latex]8\mid-7|[/latex]
   3. [latex]\mid15-7\mid-\mid14-6\mid[/latex]
   4. [latex]18-\mid2(8-3)\mid[/latex]

    

5. Simplify each expression.
   1. [latex]-21+(-59)[/latex]
   2. [latex]48+(-16)[/latex]
   3. [latex]-14+(-12)+4[/latex]
   4. [latex]135+(-110)+83[/latex]
   5. [latex]19+2(-3+8)[/latex]

    

6. Simplify.
   1. [latex]8-2[/latex]
   2. [latex]-5-4[/latex]
   3. [latex]8-(-4)[/latex]
   4. [latex]44-28[/latex]
   5. [latex]44+(-28)[/latex]
   6. [latex]27-(-18)[/latex]
   7. [latex]27+18[/latex]

    

7. Simplify.
   1. [latex]15-(-12)[/latex]
   2. [latex]48-87[/latex]
   3. [latex]-17-42[/latex]
   4. [latex]-103-(-52)[/latex]
   5. [latex]-45-(54)[/latex]
   6. [latex]8-3-7[/latex]
   7. [latex]-5-4+7[/latex]
   8. [latex]-14-(-27)+9[/latex]
   9. [latex](2-7)-(3-8)(2)[/latex]
   10. [latex]-(6-8)-(2-4)[/latex]
   11. [latex]25-[10-(3-12)][/latex]
   12. [latex]6.3-4.3-7.2[/latex]
   13. [latex]{5}^{2}-{6}^{2}[/latex]

    

8. Multiply.
   1. [latex]-4\cdot8[/latex]
   2. [latex]-1\cdot6[/latex]
   3. [latex]9(-7)[/latex]
   4. [latex]-1(-14)[/latex]

    

9. Divide.
   1. [latex]-24\div6[/latex]
   2. [latex]-180\div15[/latex]
   3. [latex]-52\div(-4)[/latex]

    

10. Simplify each expression.
    1. [latex]5(-6)+7(-2)-3[/latex]
    2. [latex]{(-2)}^{6}[/latex]
    3. [latex]-{4}^{2}[/latex]
    4. [latex]-3(-5)(6)[/latex]
    5. [latex](8-11)(9-12)[/latex]
    6. [latex]26-3(2-7)[/latex]
    7. [latex]65\div(-5)+(-28)\div(-7)[/latex]
    8. [latex]9-2[3-8(-2)][/latex]
    9. [latex]{(-3)}^{2}-24\div(8-2)[/latex]

     

11. Solve the following exercises.
    1. Temperature: On January 15, the high temperature in Lytton, British Columbia, was 84° . That same day, the high temperature in Fort Nelson, British Columbia was -12°. What was the difference between the temperature in Lytton and the temperature in Embarrass?
    2. Checking Account: Ester has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?
    3. Checking Account: Kevin has a balance of -$38 in his checking account. He deposits $225 to the account. What is the new balance?
    4. Provincial Budgets: For 2019 the province of Quebec estimated it would have a budget surplus of $5.6 million. That same year, Alberta estimated it would have a budget deficit of $7.5 million. Use integers to write the budge of:
       1. Quebec
       2. Alberta

2.1 Practice Answers

1. Order each of the following pairs of numbers, using < or >.
   1. >
   2. <
   3. <
   4. >

    

2. Simplify.
   1. 32
   2. 0
   3. 16

    

3. Fill in <, >, or = for each of the following pairs of numbers.
   1. <
   2. =

    

4. Simplify.
   1. 5, -5
   2. 56
   3. 0
   4. 8

    

5. Simplify each expression.
   1. -80
   2. 32
   3. -22
   4. 108
   5. 29

    

6. Simplify.
   1. 6
   2. -9
   3. 12
   4. 16
   5. 16
   6. 45
   7. 45

    

7. Simplify.
   1. 27
   2. -39
   3. -59
   4. -51
   5. -99
   6. -2
   7. -2
   8. 22
   9. -15
   10. 4
   11. 6
   12. -5.2
   13. -11

    

8. Multiply.
   1. -32
   2. -6
   3. -63
   4. 14

    

9. Divide.
   1. -4
   2. -12
   3. 13

    

10. Simplify each expression.
    1. -47
    2. 64
    3. -16
    4. 90
    5. 9
    6. 41
    7. -9
    8. -29
    9. 5

     

11. Solve the following exercises.
    1. 96°
    2. -$28
    3. $187
       1. $5.6 million
       2. -$7.5 million

References

Utah State University. (n.d.a). Color chips – addition [Interactive activity]. National Library of Virtual Manipulatives. http://nlvm.usu.edu/en/nav/frames_asid_161_g_1_t_1.html?from=topic_t_1.html.

Utah State University. (n.d.b). Color chips – subtraction [Interactive activity]. National Library of Virtual Manipulatives. http://nlvm.usu.edu/en/nav/frames_asid_162_g_2_t_1.html?from=topic_t_1.html.

Attributions

All figures are from 1.2 Integers in Business/Technical Mathematics (BCcampus) by Izabela Mazur and Kim Moshenko (2021).

This chapter has been adapted from 1.2 Integers in Business/Technical Mathematics (BCcampus) by Izabela Mazur and Kim Moshenko (2021), which is under a CC BY 4.0 license.

The original chapter was adapted from 1.3 Add and Subtract Integers in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith (2020), which is under a CC BY 4.0 license. Adapted by Izabela Mazur.