Contemporary Mathematics

# 3.2The Integers

Contemporary Mathematics3.2 The Integers

Figure 3.11 A ledger comparing assets to debts, resulting in net wealth. (credit: modification of work “Reviewing Financial Statements” by Mary Cullen/Flickr, CC BY 2.0)

After completing this section, you should be able to:

1. Define and identify numbers that are integers.
2. Graph integers on a number line.
3. Compare integers.
4. Compute the absolute value of an integer.
6. Multiply and divide integers.

Positive net wealth is when the total value of a person’s assets, such as their home, their 401(k), their car, and savings account balance, exceed that of their debts, such as car loans, mortgages, or credit card debt. However, when the total value of debt exceeds the total value of assets, then the person has negative net wealth. Expressing the negative net wealth as a negative number allows people to work with the positive net values and negative net values with the same mathematical processes, and in the same applications. This section introduces the integers and operations with integers.

### Defining and Identifying Integers

Extending the counting numbers to include negative numbers and zero forms the integers. Any other number that cannot be written as ${ …−3,−2,−1,0,1,2,3,… }{ …−3,−2,−1,0,1,2,3,… }$ is not an integer.

### Example 3.26

#### Identifying Integers

Which of the following are integers and which are not?

 −3 Is an integer, as it is the negative of a counting number $2424$ This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is 4.899 (rounded off). Since this is not an integer, then $2424$ is not an integer. 36/4 Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer. 45 Is an integer, as it is a counting number 63.9 Is not an integer, because it is not a counting number and not the negative of a counting number. 2/7 Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. So, 2/7 is not an integer. −16.0 Is an integer, since the decimal part is 0

Which of the following are integers?
1.
−214
2.
38/11
3.
$\sqrt {90}$
4.
$\sqrt {121}$
5.
420/35

### Graphing Integers on a Number Line

Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the negative direction, while going to the right is going forward, or in the positive direction. A number line (Figure 3.12) helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number line.

Figure 3.12

### Example 3.27

#### Graphing Numbers on the Number Line

Graph the following on the number line:

1. 1
2. −4
3. 3

Graph the following numbers on the number line:
1.
−10
2.
4
3.
0

### Comparing Integers

When determining if one quantity or size is larger than another, we know it means there is more of whatever is being discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. This idea applies to negative integers also. This means that $aa$ is greater than $bb$ when $aa$ is to the right of $bb$ on the number line. We write $a>ba>b$. When $aa$ is greater than $bb$, we can also say that $bb$ is less than $aa$. On the number line, $bb$ would be to the left of $aa$. We write $b.

We need to recognize that $a>ba>b$ means the same thing as $b. This can be seen on the number line in Figure 3.16. On this number line, $aa$ is to the right of $bb$, so $a>ba>b$. But this means $bb$ is to the left of $aa$, so $b.

Figure 3.16

### Example 3.28

#### Comparing Integers Using a Number Line

Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less than notations.

1.
Determine which of −38 and 27 is larger using a number line, and express that using both the greater than and the less than notations.

### Example 3.29

#### Comparing Negative Integers

Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations.

### Checkpoint

Warning: People often ignore the negative signs, and think than since 6 is greater than 2, −6 is greater than −2. To avoid that error, remember that the greater number is to the right on the number line.

1.
Determine which of −63 and −213 is larger, and express that using both the greater than and the less than notations.

### Example 3.30

#### Comparing Integers by Quantity

Determine which of 27 and 410 is larger, and express that using both the greater than and the less than notations.

1.
Determine which of 101 and 98 is larger, and express that using both the greater than and the less than notations.

### The Absolute Value of an Integer

When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away from 0. The absolute value of an integer $nn$ is the distance from $nn$ to 0, regardless of the direction. The notation for absolute value of the integer $nn$ is $|n||n|$.

If we think of an integer as both direction and magnitude (size), absolute value is the magnitude part.

Calculating the absolute value of an integer is very straightforward. If the integer is positive, then the absolute value of the integer is just the integer itself. If the integer is negative, then to compute the absolute value of the integer, simply remove the negative sign. Keeping in mind the number line as a path, when you’ve gone 10 steps to the left of 0, you have still taken 10 steps, and the direction does not matter.

Calculate |19|.

1.
Calculate |38|.

### Example 3.32

#### Calculating the Absolute Value of a Negative Integer

Calculate |−435|.

1.
Calculate |−81|.

You may recall having approached adding and subtracting integers using the number line from earlier in your academic life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a negative integer results in a move to the right.

Rule 1: Subtracting a negative is the same as adding a positive.

Rule 2: Adding two negative integers always results in a negative integer.

Rule 3: Adding two positive integers always results in a positive integer.

Rule 4: The sign when adding integers with opposite signs is the same as the integer with the larger absolute value.

These rules are good to keep in the back of your mind, as they can serve as a quick error check when you use a calculator.

### Example 3.33

Use your calculator to calculate 4 + (−7). Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate (−18) + 11. Explain how the answer agrees with what was expected.

### Example 3.34

#### Subtracting Positive Integers

Use your calculator to calculate 18 − 9. Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate 38 − 100. Explain how the answer agrees with what was expected.

### Example 3.35

#### Subtracting with Negative Integers

Use your calculator to calculate 27 − (−13). Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate 45 − (−26). Explain how the answer agrees with what was expected.

### Example 3.36

#### Adding Integers with Opposite Signs

Use your calculator to calculate (−13) + 90. Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate 19 + (−36). Explain how the answer agrees with what was expected.

One use of negative numbers is determining net worth, which is all the weath someone owns less all that someone owes. Sometimes net worth is positive (which is good), and sometimes net worth is negative (which can be stressful).

### Example 3.37

#### Calculating Net Worth

Jennifer is owed $50 from her friend Janice, but owes her friend Pat$87. What is Jennifer’s net worth?

1.
Christian is owed $180 from his friend Chanel, but owes his friend Jeff$91. What is Christian’s net worth?

### Multiplying and Dividing Integers

Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules.

Rule 1: When multiplying or dividing two integers with the same sign, the result is positive.

Rule 2: When multiplying or dividing two integers with opposite signs, the result is negative.

Just as before, these rules can serve as a quick error check when using a calculator.

### Example 3.38

#### Multiplying Positive Integers

Use your calculator to calculate 4 × 8. Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate 81 × 26. Explain how the answer agrees with what was expected.

### Example 3.39

#### Multiplying Integers with Different Signs

Use your calculator to calculate 9 × (−10). Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate (−18) × 13. Explain how the answer agrees with what was expected.

### Example 3.40

#### Dividing Integers with Different Signs

Use your calculator to calculate 400/(−25). Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate (−116)/4. Explain how the answer agrees with what was expected.

### Example 3.41

#### Dividing Negative Integers

Use your calculator to calculate −750/(−3). Explain how the answer agrees with what was expected.

1.
Use your calculator to calculate (−77)/(−11). Explain how the answer agrees with what was expected.

At the end of a season, a team may wish to buy their coach an end-of season gift. It makes sense to share the cost equally among the members. To do so, the team would need to find the average (or mean) cost per member. The average (or mean) of a set of numbers is the sum of the numbers divided by the number values that are being averaged.

### Example 3.42

#### Finding the Average of a Set of Numbers

The daily low temperatures in Barrie, Ontario, for the week of February 14, 2021, were −20°, −12°, −15°, −23°, −17°, −13°, and −19° degrees Celsius. What was the average daily temperature for the week of February 14, 2021, in Barrie?

1.
Banks and credit cards often base their interest on the average daily balance of an account, which is the average of the balance from each day of the period. The account balance of Jada’s checking account on each day of the week of December 13, 2020, was $1,250,$673, −$1,500,$1,000, $785,$785, and $710. What was Jada’s average daily balance for the week of December 13, 2020? Assume Jada pays no fees for a negative balance. ### Check Your Understanding 6. Identify which of the following numbers are integers: −4, 15.2, $\sqrt 2$, $\frac{3}{{20}}$, 430 7. Graph the following integers on the number line: 4, −2, 7. 8. Place these integers in increasing order: 4, −2, −7, 10, −13. 9. Calculate |−7|. 10. Calculate 4 − (−9). 11. Calculate $(-3)\times(-12)$. ### Section 3.2 Exercises 1 . Identify all the integers in the following list: 4, −17, 8, 0.5, $\sqrt 7$, $\frac{1}{9}$, −300. 2 . Identify all integers in the following list: −9.2, 13, −1, $\sqrt {47}$, $\frac{{\sqrt 3 }}{2}$, 567, −300. For the following exercises, plot the integers on the same number line. 3 . 4, −2, 10, 0 4 . −6, −3, 10, 1, 4 5 . −3, −10, 7, 2 6 . 2, 4, 8, −2, −5 For the following exercises, determine if the comparison is true or false. 7 . −3 < −10 8 . −10 > −3 9 . 7 > −6 10 . −6 < 7 11 . 18 < 20 12 . 20 < 18 13 . What are two numbers with an absolute value of 76? 14 . What are two numbers with an absolute value 87? 15 . Determine |−67|. 16 . Determine |98|. 17 . Determine |61|. 18 . Determine |−903|. 19 . What are two numbers that are 23 away from 0? 20 . What are two numbers 13 away from 0? For the following exercises, complete the indicated calculation. 21 . 47 + 200 22 . 67 + (−86) 23 . (−86) + 104 24 . 13 – (−54) 25 . (−45) – (−26) 26 . (−13) + (−102) 27 . $38 \times 12$ 28 . $(-12)\times(-10)$ 29 . $14\times(-5)$ 30 . $421\times(-13)$ 31 . $(-412)\times504$ 32 . $(-8,919)\times(-3,401)$ 33 . $(-130)\div(-65)$ 34 . $450 \div 9$ 35 . $2000 \div (-40)$ 36 . $(-910) \div 14$ 37 . The daily low temperatures, in degrees centigrade, in Fargo for the week of January 17, 2021, were −9, −17, −18, −14, −17, −19, and −11. What was the average low temperature in Fargo that week? 38 . Riley collects checks for a fundraiser supporting the homeless in town. Through Venmo, they collect the following amounts:$20, $20,$50, $75,$250, $10,$15, $65,$30, $15. What was the average donation that Riley collected? 39 . Heath has$495 in an account. They will collect two paychecks this week, one for $150 and the other for$250. Heath also will pay three bills, one for $50, one for$110, and one for $300. After all those transactions, how much will Heath have in their account? 40 . Five diners decide to split the check evenly. The total bill comes to$475. How much does each diner owe?
41 .
There are many people who are single-issue voters, which means that they will vote for (or against!) a candidate based on one issue and one issue only. Suppose a politician wants to earn votes based on single issues: Issue 1, Issue 2, and Issue 3. By publicly supporting issue 1, the politician gains 127 voter but loses 154. By publicly supporting issue 2, the politician gains 350 voters but loses 83. By publicly denouncing Issue 3, the politician gains 306 voters but loses 158. By publicly taking those stances, what is the politician’s net gain or loss in number of voters?
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