Integers

A negative numbers is a number less than 0. The negative numbers are to the left of zero on the number line. See [link].

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

are the same distance from zero, each one is called the opposite of the other. The opposite of

2

are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

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.

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

Our work with opposites gives us a way to define the integers. The whole numbers and their opposites are called the integers. The integers are the numbers

\dots −3, −2, −1, 0, 1, 2, 3 …

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

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

We let one color (blue) represent positive. The other color (red) will represent the negatives.

Figure show two circles labeled positive blue and negative red.

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.

When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.

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

We will continue to use counters to model the subtraction. 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!

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

Figure on the left is labeled 5 minus 3 equals 2. There are 5 blue circles. Three of these are encircled and an arrow indicates that they are taken away. The figure on the right is labeled minus 5 minus open parentheses minus 3 close parentheses equals minus 2. There are 5 red circles. Three of these are encircled and an arrow indicates that they are taken away.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add 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.

| |

| | Model the first number. |

| {: valign=”top”}| We now add the needed neutral pairs. |

| {: valign=”top”}| We remove the number of counters modeled by the second number. |

| {: valign=”top”}| Count what is left. |

| {: valign=”top”}| |

| {: valign=”top”}{: summary=”Calculations of two expressions are shown. The expression on the left is minus 5 minus 3. The expression on the right is 5 minus open parentheses minus 3 close parentheses. We first model the first number. So, we have 5 red counters on the left and 5 white counters on the right. We now add three neutral pairs for both expressions. So we have 8 red and 3 white counters on the left and 8 white and 3 red counters on the right. We now remove the number of counters modeled by the second number. Three white counters are removed from the left and three red counters are removed from the right. We now have eight red counters on the left and 8 white counters on the right. Hence, minus 5 minus 3, on the left, is minus 8, whereas 5 minus minus 3, on the right, is 8.” .unnumbered .unstyled .can-break data-label=””}

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example,

−3 - 1

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

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show 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. Here, we are using the model just to help us discover the pattern.

The figure on the left is labeled 5 dot 3. Here, we need to add 5, 3 times. Three rows of five blue counters each are shown. This makes 15 positives. Hence, 5 times 3 is 15. The figure on the right is labeled minus 5 open parentheses 3 close parentheses. Here we need to add minus 5, 3 times. Three rows of five red counters each are shown. This makes 15 negatives. Hence, minus 5 times 3 is minus 15.

The next two examples are more interesting. What does it mean to multiply 5 by

−3 ?

times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

What about division? Division is the inverse operation of multiplication. So,

15 \div 3 = 5

In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

When we multiply a number by 1, the result is the same number. Each time we multiply a number by

−1,

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 Please Excuse 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.

This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Key Concepts

| Different signs | Result | {: valign=”top”}|———- | • Positive and negative | Negative | {: valign=”top”}| • Negative and positive | Negative | {: valign=”top”}{: summary=”.” .unnumbered data-label=””}

Practice Makes Perfect

Simplify Expressions with Absolute Value

In the following exercises, fill in $<, >,$

or $=$

for each of the following pairs of numbers.

ⓐ $\| −7 \| \_\_\_- \| −7 \|$

ⓑ $6 \_\_\_- \| −6 \|$

ⓓ $− (−13) \_\_\_- \| −13 \|$

ⓐ $>$

ⓑ $>$

ⓓ $>$

ⓐ $− \| −9 \| \_\_\_\| −9 \|$

ⓑ $−8 \_\_\_\| −8 \|$

ⓓ $− (−14) \_\_\_- \| −14 \|$

ⓐ $− \| 2 \| \_\_\_- \| −2 \|$

ⓑ $−12 \_\_\_- \| −12 \|$

ⓓ $\| −19 \| \_\_\_- (−19)$

ⓐ $=$

ⓑ $=$

ⓓ $=$

ⓐ $− \| −4 \| \_\_\_- \| 4 \|$

ⓑ $5 \_\_\_- \| −5 \|$

ⓓ $− \| −0 \| \_\_\_- (−0)$

In the following exercises, simplify.

\| 15 - 7 \| - \| 14 - 6 \|

\| 17 - 8 \| - \| 13 - 4 \|

18 - \| 2 (8 - 3) \|

15 - \| 3 (8 - 5) \|

18 - \| 12 - 4 (4 - 1) + 3 \|

27 - \| 19 + 4 (3 - 1) - 7 \|

10 - 3 \| 9 - 3 (3 - 1) \|

13 - 2 \| 11 - 2 (5 - 2) \|

Add and Subtract Integers

In the following exercises, simplify each expression.

ⓐ $−7 + (−4)$

ⓑ $−7 + 4$

ⓐ $−11$

ⓑ $−3$

ⓐ $−5 + (−9)$

ⓑ $−5 + 9$

48 + (−16)

34 + (−19)

−14 + (−12) + 4

−22

−17 + (−18) + 6

19 + 2 (−3 + 8)

29

24 + 3 (−5 + 9)

ⓐ $13 - 7$

ⓑ $−13 - (−7)$

ⓓ $13 - (−7)$

ⓐ 6 ⓑ $−6$

ⓓ $20$

ⓐ $15 - 8$

ⓑ $−15 - (−8)$

ⓓ $15 - (−8)$

−17 - 42

−59

−58 - (−67)

−14 - (−27) + 9

64 + (−17) - 9

ⓐ $44 - 28$

ⓑ $44 + (−28)$

ⓐ 16 ⓑ16

ⓐ $35 - 16$

ⓑ $35 + (−16)$

ⓐ $27 - (−18)$

ⓑ $27 + 18$

ⓐ 45 ⓑ45

ⓐ $46 - (−37)$

ⓑ $46 + 37$

(2 - 7) - (3 - 8)

(1 - 8) - (2 - 9)

− (6 - 8) - (2 - 4)

− (4 - 5) - (7 - 8)

25 - [10 - (3 - 12)]

32 - [5 - (15 - 20)]

Multiply and Divide Integers

In the following exercises, multiply or divide.

ⓐ $−4 \cdot 8$

ⓑ $13 (−5)$

ⓓ $−52 \div (−4)$

ⓐ $−32$

ⓑ $−65$

ⓓ $13$

ⓐ $−3 \cdot 9$

ⓑ $9 (−7)$

ⓓ $−84 \div (−6)$

ⓐ $−28 \div 7$

ⓑ $−180 \div 15$

ⓓ $−1 (−14)$

ⓐ $−4$

ⓑ $−12$

ⓓ $14$

ⓐ $−36 \div 4$

ⓑ $−192 \div 12$

ⓓ $−1 (−19)$

Simplify and Evaluate Expressions with Integers

In the following exercises, simplify each expression.

ⓐ ${(−2)}^{6}$

ⓑ $− 2^{6}$

ⓐ $64$

ⓑ $−64$

ⓐ ${(−3)}^{5}$

ⓑ $− 3^{5}$

5 (−6) + 7 (−2) - 3

−47

8 (−4) + 5 (−4) - 6

−3 (−5) (6)

90

−4 (−6) (3)

(8 - 11) (9 - 12)

9

(6 - 11) (8 - 13)

26 - 3 (2 - 7)

41

23 - 2 (4 - 6)

65 \div (−5) + (−28) \div (−7)

−9

52 \div (−4) + (−32) \div (−8)

9 - 2 [3 - 8 (−2)]

−29

11 - 3 [7 - 4 (−2)]

8 - \| 2 - 4 (4 - 1) + 3 \|

1

7 - \| 5 - 3 (4 - 1) - 6 \|

9 - 3 \| 2 (2 - 6) - (3 - 7) \|

−3

5 - 2 \| 2 (1 - 4) - (2 - 5) \|

{(−3)}^{2} - 24 \div (8 - 2)

5

{(−4)}^{2} - 32 \div (12 - 4)

In the following exercises, evaluate each expression.

y + (−14)

when* * *

ⓐ $y = −33$

ⓑ $y = 30$

ⓐ $−47$

ⓑ $16$

x + (−21)

when* * *

ⓐ $x = −27$

ⓑ $x = 44$

{(x + y)}^{2}

when* * *

x = −3, y = 14

121

{(y + z)}^{2}

when* * *

y = −3, z = 15

9 a - 2 b - 8

when* * *

a = −6

and $b = −3$

−56

7 m - 4 n - 2

when* * *

m = −4

and $n = −9$

3 x^{2} - 4 x y + 2 y^{2}

when* * *

x = −2, y = −3

6

4 x^{2} - x y + 3 y^{2}

when* * *

x = −3, y = −2

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

the sum of 3 and $−15,$

increased by 7

(3 + (−15)) + 7; - 5

the sum of $−8$

and $−9,$

increased by $23$

ⓐ the difference of $10$

and $−18$

ⓑ subtract $11$

from $−25$

ⓐ $10 - (−18); 28$

ⓑ $−25 - 11; - 36$

ⓐ the difference of $−5$

and $−30$

ⓑ subtract $−6$

from $−13$

the quotient of $−6$

and the sum of $a$

and $b$

\frac{−6}{a + b}

the product of $−13$

and the difference of $c$

and $d$

Use Integers in Applications

In the following exercises, solve.

Temperature On January 15, the high temperature in Anaheim, California, was 84°. That same day, the high temperature in Embarrass, Minnesota, was $− 12 ° .$

What was the difference between the temperature in Anaheim and the temperature in Embarrass?

96 °

Temperature On January 21, the high temperature in Palm Springs, California, was $89 °,$

and the high temperature in Whitefield, New Hampshire, was $− 31 ° .$

What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Football On the first down, the Chargers had the ball on their 25-yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down?

Football On the first down, the Steelers had the ball on their 30-yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down?

Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?

− $ 28

Checking Account Reymonte has a balance of $− $ 49$

in his checking account. He deposits $281 to the account. What is the new balance?

Writing Exercises

Explain why the sum of $−8$

and 2 is negative, but the sum of 8 and $−2$

is positive.

Answers will vary.

Give an example from your life experience of adding two negative numbers.

In your own words, state the rules for multiplying and dividing integers.

Answers will vary.

Why is $− 4^{3} = {(−4)}^{3} ?$

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?


{: valign=”top”}

Simplify Expressions with Absolute Value

Add and Subtract Integers

Multiply and Divide Integers

Simplify Expressions with Integers

Evaluate Variable Expressions with Integers

Translate Phrases to Expressions with Integers

Use Integers in Applications

Key Concepts

Practice Makes Perfect

Writing Exercises

Self Check

Glossary