1-1: Integers

Hazel McKenna; Leo Chang; Roxanne Brinkerhoff

CHAPTER 1: NUMBER SENSE

1-1: Integers

What are Integers?

Integers include all the whole numbers {0, 1, 2, 3, … } and their opposites {-1, -2, -3, …}. The scrolled letter Z is used to denote the set of all integers: $\mathbb{Z}=\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$ . On a number line positive integers lie to the right of zero while negative integers lie to the left of zero. The further right we go on the number line the larger and more positive the numbers get. On the other hand, the further left we go on the number line the smaller and more negative the numbers get.

Number line showing integers — Figure 1. Integers on a number line

The three dots … in figure 1 are called ellipsis and tell us that the pattern of numbers goes on forever in both the negative and positive directions. Notice that -3 lies to the left of -1. This tells us that $-3 < -1$ . Also -1 lies to the right of -3, so $-1 > -3$ .

We can tell how far away from zero any integer is by using absolute value. Absolute value is denoted using two vertical lines around the number and equals the positive distance from zero. For example, $|-2|=2$ and $|2|=2$ . The two integers $-2$ and $2$ have the same absolute value since they lie the same distance away from zero, just in opposite directions.

Adding and Subtracting Integers

Explore 1 – Adding and subtracting positive integers

Erica and her family are on a road trip. She stops by a gas station to get supplies every 200 miles or so. Erica prefers to use cash, but she has received too many one-dollar bills in change on this trip and she doesn’t want any more.

Her current purchase is $12. How can she pay in cash so that she can use one-dollar bills and also get a 5- or 10-dollar bill back? Find two possible ways she could pay. Write an arithmetic sentence to illustrate the calculation.

Solution

Erica could pay with twelve 1-dollar bills, but she would not get any 5- or 10-dollar bills back. Here are two possible payment methods:

Pay with 17 one-dollar bills and get one 5-dollar bill back. The arithmetic sentence is $\$17-\$12=\$5$ .
Pay with 22 one-dollar bills and get one 10-dollar bill back. The arithmetic sentence is $\$22-\$12=\$10$ .

At the previous stop, Erica paid $8 for her brother’s purchase. Now, her brother wants to return her $8, but Erica will not take any one-dollar bills. Find two possible ways for her brother to make the payment so that Erica will not receive one-dollar bills. Write an arithmetic sentence to illustrate the calculation.

Solution

Erica’s brother cannot give Erica any one-dollar bills so he must pay her more than $8 in 5- or 10-dollar bills then receive change. Here are two possible payment methods:

The brother may pay Erica $10 (one ten-dollar bill), and Erica pays her brother back two one-dollar bills. The arithmetic sentence is $\$10-\$2=\$8$ .
The brother may pay Erica $15 (one ten-dollar bill and one five-dollar bill), and Erica gives her brother back 7 one-dollar bills in change. The arithmetic sentence is $\$15-\$7=\$8$

Explore 2 – Adding negative integers

Mitya is invited to be a judge of a contest. The grading sheet has four categories with 10 points available for each category. Instead of grading using positive values, Mitya thinks it may be easier and faster to grade using negative numbers to denote the number of points below 10.

How would Mitya’s negative-number grades look like if the four category grades were 8, 5, 7, and 6?

Solution

The negative-number grades for 8, 5, 7, and 6 are found by determining the number of points each grade is below 10: 8 is 2 below 10, 5 is 5 below 10, 7 is 3 below 10, and 6 is 4 below 10. These numbers can be found by subtracting the score from 10. The negative grades are -2, -5, -3, and -4.

How would Mitya calculate the total grade based on her four negative category grades? Write the calculation in an arithmetic sentence.

Solution

The total grade will be calculated by adding the negative grades and determining how far below 40 (the total points possible across the four categories) they lie: $(-2)+(-5)+(-3)+(-4)=-14$ which is 26 below 40 ( $40+(-14)=26$ ). The arithmetic sentence is: $(-2)+(-5)+(-3)+(-4) + 40=-14+40=26$ .

Explore 3 – Subtracting negative integers

Akira has a home mortgage of $252,080. Since this is a debt, he writes the number as –252,080 in his balance book. He just made a one-time payment of $2000. Akira writes the following arithmetic sentence in his balance book.

$-252,080-(-2000) = -252,080+2000 = -250,080$

Explain why he subtracts -2000 from -252,080: $-252,080-(-2000)$

Solution

The explanation for subtracting -2,000 could be that he is removing $2,000 of debt from the mortgage. The operation subtract is to remove, and -2,000 means a debt of $2000.

Explain why we can change the sign to a + sign when subtracting a negative number: $-(-2000)$ to $+2000$ ?

Solution

The explanation for the sign change of $-(-2000)$ to $+2000$ is that removing the debt of $2,000 is equivalent to a gain of $2,000 against the total mortgage debt of -$252,080.

Examples

1. In all but 3 countries in the world^[1] (and a few US territories), temperature is measured in degrees Celsius with 0°C representing the freezing point of water at sea level and 100°C the boiling point of water at sea level. What does it mean for a temperature in °C to be negative?

Show/Hide Answer

A negative temperature in °C means that the temperature is below the freezing point of water at sea level.

2. Craig’s bank balance shows $53. If he takes his boyfriend on a date and spends $42 on dinner then $28 on movie tickets, what will his new bank balance read? Write an arithmetic sentence to show this.

Show/Hide Answer

Total spent on his date is $\$42 + \$28 = \$70$ . This is $17 more than he has in the bank. So his new bank balance will be $-\$17$ . The arithmetic sentence is $\$53-\$42-\$28=-\$17$ .

3. During 3 downs in an American football match, the team advanced the ball 2 yards by running then 5 yards in the air. On the 3rd down the quarterback was sacked and the team lost 8 yards. Explain in words where the ball will be placed for the 4th down relative to where the team started on the first down. Then write an arithmetic sentence to show this.

Show/Hide Answer

The team advanced the ball 2 + 5 = 7 yards then lost 8 yards so the ball will be placed 1 yard behind where the team started on the first down. The arithmetic sentence is $2+5+(-8)=-1$ .

Explain how to add negative numbers. Give an example.

Show/Hide Answer

Add the absolute values of the numbers then turn the answer negative. For example, $-4+(-6)+(-7)$ can be added by adding $4+6+7=17$ , then making the sum negative: $-17$ .

What does it mean to subtract a negative number? Give an example.

Show/Hide Answer

Subtracting a negative number is gaining value so subtracting a negative number is equivalent to adding the opposite of the negative number. For example, $-4-(-9)=-4+9=5$ .

Explain how to determine if the sum of two integers will turn out positive or negative. Give an example.

Show/Hide Answer

The sum will take the sign of the larger integer in absolute value. For example, $-3+6$ is positive because $6>3$ . Therefore, $-3+6=3$ . Also, $4+(-7)$ is negative because $7>4$ . Therefore, $4+(-7)=-3$ .

Explain how to subtract one integer from another. Give an example.

Show/Hide Answer

We subtract an integer by adding its opposite. For example $4-7=4+(-7)$ and $-3-(-9)=-3+9=6$ .

Now try some problems on your own. Check your answers after you have completed the problems.

Practice Exercises

Tony owes $136 on his credit card. He goes out and buys new kicks for $120. He also returns a shirt and a pair of pants whose cost of $95 is refunded to his credit card. What is his new credit card balance? Write an arithmetic sentence that shows your answer.

On a backpacking trip, Laura starts at an elevation of 5200 feet above sea level. She then climbs a 10,300 ft peak before dropping into a valley 3700 feet below to camp for the night. At what elevation is her camp site? Write an arithmetic sentence that shows your answer.

Show/Hide Answers

$-\$136+(-\$120)+\$95=-\$161$ He now owes $161 on his credit card.
$10,300-3700=6600$ The campsite is at 8800 feet above sea level.

Multiplying and Dividing Integers

Explore 4 – Multiplying positive integers

Sofia is planning a garage sale on Saturday. She needs bills of one, five, ten, and twenty dollars so she can give change.

She has decided to withdraw money from her account at the credit union to get 15 twenty-dollar bills, 15 ten-dollar bills, 25 five-dollar bills, and 50 one-dollar bills.

How much money is Sofia withdrawing from the credit union? Show your work by using an arithmetic sentence.

Solution

Since she is getting 15 twenty-dollar bills, we need to multiply the value of one twenty-dollar bill by the number of bills: $15\times\$20$ . Similarly, the value of 15 ten-dollar bills will be $15\times\$10$ ; 25 five-dollar bills will be $25\times\$5$ ; and 50 one-dollar bills will be $50\times\$1$ .

She will need to take out $625 from the credit union.

The arithmetic sentence is $15\times\$20+15\times\$10+25\times\$5+50\times\$1 = \$300+\$150+\$125+\$50 = \$625$

In the arithmetic sentence what order did the multiplication and addition take place?

Solution

Multiplication was completed before the addition.

If she has a balance of $920 in her account, what will her new balance be after she withdraws the money? Show your work by using an arithmetic sentence.

Solution

If she withdraws $625 from her balance of $920, she will be left with a balance of a $295.

The arithmetic sentence is $\$920+(-\$625)=\$295$

Explore 5 – Multiplying negative integers

Temperature drops by about 5.4 degrees Fahrenheit for every 1,000 feet in elevation gain. Haoming lives in San Francisco where the temperature is currently 68^oF. She is driving with her family up to Lake Tahoe; an elevation gain of 6000 feet. This means the temperature will drop 5.4 degrees six times over the course of the journey.

What is the temperature in Lake Tahoe? Show your work in an arithmetic sentence.

Solution

Since the temperature is dropping 5.4°F every 1000 feet of elevation gain, the total drop over an elevation gain of 6000 feet is $6\times5.4\degree\text{F}=32.4\degree\text{F}$ . Since this is a drop in temperature, we have to subtract it from the temperature in San Francisco. The arithmetic sentence is $68\degree\text{F}-6\times(5.4\degree\text{F})=68\degree \text{F}-32.4\degree\text{F}=35.6\degree\text{F}$ .

Alternatively, since the temperature is dropping we can say $68\degree\text{F}+6\times(-5.4\degree\text{F})=68\degree\text{F}+(-32.4\degree\text{F})=35.6\degree\text{F}$

Is adding a negative number the same as subtracting a positive number? Explain your reasoning.

Solution

Yes. $68\degree\text{F}-32.4\degree\text{F}=35.6\degree\text{F}$ and $68\degree\text{F}+(-32.4\degree\text{F})=35.6\degree\text{F}$

In the arithmetic sentence, in what order was the multiplication and addition completed?

Solution

The multiplication was completed before the addition.

Explore 6 – Multiplying a negative integer by a negative integer

Akira has a home mortgage of $252,080. He puts –252,080 in his balance book. $500 of his monthly mortgage payment will be paid directly to the balance (the rest goes towards paying the interest on the loan). Akira writes the following arithmetic sentence to calculate the mortgage balance after 1 year (12 payments):

$\begin{aligned}-252,080+(-12)\times(-500) &= -252080+12\times500 \\&= -252080 + 6000 \\&= 246,080\end{aligned}$

Explain why he wrote $-12\times-500$ ?

Solution

A possible explanation is “to remove $500 debt 12 times from the mortgage”. The negative sign associated with 12 means “remove”. The number -500 means a debt of $500.

Explain why the signs changed from negative to positive: $-12\times-500$ to $+12\times500$ ?

Solution

The change of sign from $-12\times-500$ to $+12\times500$ happens because removing the $500 worth of debt 12 times is a gain in value of $+12\times\$500=\$6000$ to the total debt of $252,080 (i.e., the debt is reduced).

In the arithmetic sentence, in what order was the multiplication and addition completed?

Solution

The multiplication was completed before the addition.

Examples

Pearl owes $3600 on her car. If she pays $400 per month, how long will it take her to pay off her car?

Show/Hide Answer

$\$3600\div$400=9$ It will take 9 months to pay off her car.

The stock market dropped 12 points four days in a row. How far did the stock market drop over the four days?

Show/Hide Answer

Since the market dropped 12 points, the 12 points are negative. $4(-12)=-48$ The stock market dropped 48 points.

Let’s reflect on what we just learned.

What does it mean to multiply a negative number by a positive number? Give an example.

Show/Hide Answer

Multiplication is repeated addition, so multiplying a negative number means adding the negative number to itself over and over. For example, $5 \times -3$ means to add -3 to itself 5 times. $5 \times -3=-3+(-3)+(-3)+(-3)+(-3)=-15$

What does it mean to multiply a negative number by another negative number? Give an example.

Show/Hide Answer

The meaning of multiplying a negative number by another negative number is to repeatedly add the opposite of the negative. When we multiply a negative number times a negative number, we get less negative.

For example, $-5(-3)=-(-3)-(-3)-(-3)-(-3)-(-3)=3+3+3+3+3=15$

If there is both addition and multiplication in an arithmetic sentence, in what order are the operations of addition and multiplication completed? Give an example to explain your answer.

Show/Hide Answer

Multiplication is completed before addition. For example, in $12+2(-5)$ 2 is multiplied by -5 first:

$12+2(-5)=12+(-10)=2$

Now try a couple of problems on your own. Try the problem before checking the answer.

Practice Exercises

Andrew is building a deck. He bought twenty-four pieces of lumber that cost $18 each to build the deck’s surface. This turned out to be 5 pieces more than he needed so he returned the extra. However, the lumber yard had a restocking fee of $1 per piece of lumber. How much did the surface of the deck cost him to build? Write an arithmetic sentence to show your work.

Show/Hide Answer

The deck surface cost $347. The arithmetic sentence is: $24(-\$18)+5(\$18)+5(-\$1)=-\$432+\$90+(-\$5)=-\$347$

A bow and arrow shooting target has both positive and negative score values.

Target with values by color: red = 5 points; blue = 3 points; white = 1 point; yellow = –4 points

Ema shot six arrows. Two arrows hit the red bulls eye for 5 points each. One arrow hit the blue area for 3 points. One arrow hit the white area for 1 point. Two arrows hit the yellow area for -4 points each.

What was Ema’s score for this round? Write an arithmetic sentence to show your work.

Show/Hide Answer

Ema’s score was 6.

Score = Red score + Blue score + White score + Yellow score

= $2(5)+1(3)+1(1)+2(-4)=10+3+1+(-8)=6$

In this section, we will take what we have learned and apply the concepts to new situations.

Perspectives

Salt Lake City’s record highest temperature is 42°C set on July 13, 2002. In contrast, Salt Lake City’s record lowest temperature is -34°C set on February 9, 1933. What is the difference from the record highest temperature to the record lowest temperature? Write an arithmetic sentence to show your work.

Time across the globe is decided by adding or subtracting hours from Coordinated Universal Time (UTC). Orem, Utah’s time is -7 UTC, while Auckland, New Zealand is +12 UTC. How many hours separate the time in Aukland from the time in Orem? Write an arithmetic sentence to show your work.

On a recent round of golf, Sally scored 4 birdies, 8 pars, 3 bogies, and 3 double bogies. Par is what a scratch golfer is expected to shoot on each hole. Each birdie scores 1 under par, each bogie scores 1 over par, and each double bogie scores 2 over par. How many points did Sally score with respect to par? If par for the course is 72, what score did Sally shoot?

Show/Hide Answers

$42-(-34)=76$ . Difference is 76°C.
$+12-(-7)=12+7=19$ Auckland’s time is 19 hours ahead of Orem’s time.
4 birdies scored $4(-1)=-4$ points; 3 bogies scored $3(+1)=+3$ points; 3 double bogies scored $3(+2)=+6$ points. Sally scored $-4+3+6=+5$ points with respect to par. Since par is 72, Sally shot a round of $72+5=77$ .

Skills Icon In this section, we will use what we have learned so far to practice skill problems.

Skills Practice

Calculator Tips Icon There are different ways to input a negative number in a calculator. Some calculators simply put a minus sign before the number. Others put in the positive number then change the sign using a $\boxed{\pm}$ or a $\boxed{(-)}$ button. Find out how to input a negative number on the calculator you are using.