2.2 Operations with Integers

Learning Objectives

You will use addition and subtraction with positive and negative integers.
You will use multiplication and division with positive and negative integers, fractions, and decimals.
You will solve problems in various contexts using inverse relationships between addition and subtraction.

Introduction

Dive Depths

Malacai and his parents are all scuba divers. Malacai learned to scuba dive two years ago when he was eleven. Kids between the ages of 11 and 14 can become certified junior divers through an organization called PADI. Since Malacai learned to dive, he has looked forward to his family’s diving vacation each year when they all take off to someplace warm to scuba dive.

One week before this year's big trip to the Caribbean, Malacai began looking through his dive book. A dive book is a book where divers keep track of their dives. They chart the depth that they went, the time they were underwater, and anything cool that they saw.

As a junior diver, Malacai is only allowed to travel to a maximum depth of 40 feet.

Here are Malacai’s dive depths from his last trip when he went diving in Jamaica.

15 feet deep

40 feet deep

25 feet deep

36 feet deep

30 feet deep

Integers can help a scuba divers in a real-world situation like this. Since Malacai traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.

To do this, you will need to know about integers. Pay attention to this lesson, and at the end of the lesson you will know how to help Malacai write out his diving depths from least to greatest.

Guided Learning

Write Integers Representing Situations of Increase/Decrease, Profit/Loss, Above/Below

Numbers can be classified in many different ways. For example, we can classify numbers as whole numbers, fractions, and decimals.

Some numbers can be classified as integers. Integers include the positive whole numbers (1, 2, 3, 4, 5, ...), their opposites(-1, -2, -3, -4, -5, ...) and zero.

This number line shows the integers from -5 to 5.

Look at the number line. The negative integers (-1, -2, -3, -4, and, -5) are to the left of 0, so their values are less than 0. The positive integers (1, 2, 3, 4, and, 5) are to the right of 0, so their values are greater than 0.

We can use numbers to describe real world situations and using integers can assist us with this as well. Let's take a look at how integers can help us describe real-world situations.

We can use integers to represent many real-world situations, such as:

Increases and decreases in temperature.
Profit and loss of money
Locations above and below sea level.

First, let's take a look at how integers can help us represent temperatures.

Example A

The temperature outside a ski lodge was $3^\circ F$ below $0^\circ F$ . Express that temperature with an integer.

To write this as an integer, we can think of a thermometer. A thermometer is just a vertical number line. Find the mark for $0^\circ F$ on this thermometer. With your finger, count $3^\circ F$ below that mark.

Your finger will point to $-3^\circ F$ . That is how the temperature $3^\circ F$ below $0^\circ F$ can be expressed as an integer. Think of the winter and the summer. When it is very cold or very hot, integers help us to understand how cold or how hot it is.

Let’s take a look at another application of negative numbers using number lines!

A fisherman is sitting 2 feet above the surface of a lake on a boat. The hook on his fishing pole is floating 6 feet below the lake's surface. Use integers to represent the position of the fisherman and his hook.

Think of a vertical number line.

The surface of the lake can be represented by the integer, 0.

The fisherman is sitting 2 feet above the surface. You can represent this as +2 or 2.

The hook is floating 6 feet below the surface. You can represent this as -6.

Wow! Working with a picture certainly helps to make it very clear!

Example B

Mr. Marsh invested in the stock market and had a loss of $45 yesterday. Mrs. Marsh also invested in the stock market. Her investment showed a gain of $20 yesterday. Represent these situations with integers.

Think of a number line from -$50 to $50. The $0 mark represents neither a gain nor a loss on an investment.

Use a negative integer to represent a loss. Mr. Marsh lost $45 on his investment. This can be represented as -$45.

Use a positive integer to represent a gain. Mrs. Marsh's investment showed a gain of $20. This can be represented as +$20 or $20, because positive integers can be written with or without a positive (+) sign.

Do you get the idea? You can look for keywords that indicate a positive or a negative number. When you look at a problem, identify any words that might tell you whether you are going to write a positive or a negative number. Think back at the last three examples and write down any keywords that you notice.

Write an integer for each example.

An increase of $200.00
Down 10%
50 feet below sea level

Compare your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Integer Addition

Knowing how to add integers can also help you solve many problems in real life. To solve a real-world problem, write an expression or an equation that can be used to solve the problem, then solve.

Example C

Molly lives in Alaska. The temperature outside her home at 6:00 A.M. one day last February was $-2^\circ F$ . Six hours later, the temperature had risen by $5^\circ F$ . What was the temperature six hours later?

The problem says that the temperature had risen six hours later. This means that the temperature had increased, so you should add. To find the new temperature, you can add the amount of the increase to the previous temperature.

You can find the temperature six hours later by using one of these equations.

$-2^\circ F + 5^\circ F & = ?\\\text{or} \qquad -2+5&=?$

Both integers have different signs. So, find the absolute values of both integers. Then subtract the integer with the lesser absolute value from the absolute value of the integer with the greater absolute value.

|-2|=2 and |5|=5 , so subtract: 5-2=3 .

Since 5>2 , and $5^\circ F$ has a positive sign, the temperature six hours later must be $3^\circ F$ .

Temperature is just one example of a real world problem involving integers.

Subtract Integers Using Opposites

Another strategy for subtracting integers involves using opposites. Remember, you can find the opposite of an integer by changing the sign of an integer. The opposite of any integer, , would be .

For any two integers, and , the difference of a-b can be found by adding a+(-b) . So, to subtract two integers, take the opposite of the integer being subtracted and then add that opposite to the first integer.

Write this down in your notebook and then continue with the lesson.

Example D

Find the difference of 5-(-8) .

The integer being subtracted is -8. The opposite of that integer is 8, so add 8 to 5.

5-(-8)=5+8=13 .

So, the difference of 5-(-8) is 13.

Our answer is 13.

Example E

Find the difference of -12-(-2) .

The integer being subtracted is -2. The opposite of that integer is 2, so add 2 to -12.

-12-(-2)=-12+2 .

Add as you would add any integers with different signs.

|-12|=12 and |2|=2 , so subtract the lesser absolute value from the greater absolute value.

12-2=10

Give that answer the same sign as the integer with the greater absolute value. 12>2 , so -12 has a greater absolute value than 2. Give the answer a negative sign.

So, the difference of -12-(-2) is -10.

Our answer is -10.

Example F

Find the difference of -20-3 .

The integer being subtracted is 3. The opposite of that integer is -3, so add -3 to -20.

-20-3=-20+(-3) .

Add as you would add any integers with the same sign. In this case, a negative sign.

|-20|=20 and |-3|=3 , so add their absolute values:

20+3=23

Give that answer the same sign as the two original integers, a negative sign.

So, the difference of -20-3 is -23.

Our answer is -23.

Model and Solve Real-World Problems Using Simple Equations Involving Integer Change

Knowing how to subtract integers can also help you solve many problems in real life. To solve a real-world problem, write an expression or an equation that can be used to solve the problem, then solve. Let’s practice this a bit and then return to our introductory problem.

Example G

The temperature inside a laboratory freezer was $-10^\circ$ Celsius. A scientist at the lab then lowered the temperature inside the freezer by $5^\circ$ Celsius. What was the new temperature inside the freezer?

The problem says that the temperature was lowered. This means that the temperature decreased, so you should subtract. To find the new temperature, you can subtract the amount by which the temperature was lowered from the original temperature, using one of these equations.

$-10^\circ C-5^\circ C &= ?\\\text{or} \qquad -10-5&=?$

The integer being subtracted is 5. The opposite of that integer is -5, so add -5 to -10.

-10-5=-10+(-5) .

Add as you would add any integers with the same sign––a negative sign.

|-10|=10 and |-5|=5 , so add their absolute values.

Give that answer the same sign as the two original integers, a negative sign.

10+5=15

So, the difference of -10-5 is -15.

This means that the new temperature inside the freezer must be $-15^\circ$ Celsius.

Analyze Patterns of Products of Integers with Same and Different Signs Recognizing Multiplication by Zero as Zero

Now that you have learned how to add and subtract integers, it is time to tackle multiplying them. Remember that an integer is the set of whole numbers and their opposites. There are a few vocabulary words that help us when multiplying. The first is a factor. The numbers that are multiplied are called factors. The second is product. We multiply two or more factors to get a product.

Below are some multiplication facts for 5. Notice that the products show a pattern. Suppose you did not know the product of $5 \times 0$ . How could you use the pattern shown below to determine that product?

$5 \times 4 & =20\\5 \times 3 & =15\\5 \times 2 & =10\\5 \times 1 & =5\\5 \times 0 & = ?$

Notice that each product shown is 5 less than the previous product. So, you can subtract 5 from the previous product, 5, to find the missing product. Since 5-5=0 , the product of $5 \times 0$ must be 0.

There are patterns that we can see when we multiply integers. Analyzing patterns like these can help us multiply positive and negative integers.

How do analyze these patterns?

First, we notice that the pattern for the multiplication facts of 5 could continue below zero.

Example H

Use a pattern to find the missing products below.

$5 \times 4 & = 20\\5 \times 3 & = 15\\5 \times 2 & = 10\\5 \times 1 & = 5\\5 \times 0 & = 0\\5 \times (-1) & = ?\\5 \times (-2) & = ?$

To find the product of $5 \times (-1)$ , add 0+(-5)

|0|=0 and |-5|=5 , so subtract the lesser absolute value from the greater absolute value :

5-0=5 .

The integer with the greater absolute value is -5, so give the answer a negative sign.

0+(-5)=-5 , so $5 \times (-1)=-5$

To find the product of $5 \times (-2)$ , add -5 to the previous product, which is also -5.

In other words, add: -5+(-5)

Both integers have the same sign, so add their absolute values.

|-5|=5 , so add.

5+5=10 .

Give that answer a negative sign.

-5+(-5)=-10 , so $5 \times (-2) = -10$

Now we have our completed multiplication facts.

$5 \times 4 & = 20\\5 \times 3 & = 15\\5 \times 2 & = 10\\5 \times 1 & = 5\\5 \times 0 & = 0\\5 \times (-1) & = -5\\5 \times (-2) & = -10$

Each product is 5 less than the previous product.

You may also notice that depending on what you are multiplying the sign changes.

Here are a few rules that we can conclude from the pattern.

When 5, a positive integer, is multiplied by a positive integer, the product is positive.
When 5, a positive integer, is multiplied by zero, the product is zero.
When 5, a positive integer, is multiplied by a negative integer, the product is negative.

We can conclude the following rules.

Write these rules in your notebook.

Multiply Integers

Now that you understand the rules, we can work on applying them when actually multiplying integers.

Refer back to the rules as you work, but the key to becoming GREAT at multiplying integers is to commit these rules to memory!

(-4)(-3)

Here we have negative four times a negative three. First, we multiply the terms, remember that a set of parentheses next to another set means multiplication.

$4 \times 3 = 12$

Next, we figure out the sign.

A negative times a negative is a positive.

Our answer is 12.

Example I

$-5 \cdot 8$

Here we have a negative five times a positive eight. Remember that a dot can also mean multiplication.

$5 \times 8 = 40$

Next, we figure out the sign.

A negative times a positive is a negative.

Our answer is -40.

What about multiplying more than one term?

We can do this easily. The key is to work from left to right and remember that the sign of each product can change with each factor.

Example J

(-8)(-3)(-2)

Here we have three negative terms being multiplied. First, let’s multiply the first two terms to get a product.

$-8 \times -3 = 24$ (A negative times a negative equals a positive.)

Now we multiply that product times the factor negative two.

$24 \times -2 = -48$ (A positive times a negative equals a negative.)

Our answer is -48.

Try a few practice problems.

1. -9(-3)

2. (-3)(12)

3. (-4)(3)(-2)

Model and Solve Real-World Problems Using Simple Equations Involving Integer Multiplication

Knowing how to multiply integers can also help us represent and solve problems in real life.

To solve a real-world problem, write an expression or an equation that can be used to solve the problem, then solve.

Example K

The number of students voting in a school election has been decreasing at a rate of 15 students per year. Represent the change in the number of students voting over the next 3 years as an integer.

First, represent the decrease in the number of students voting as an integer.

The problem states that the number of students voting has been decreasing by 15 students each year. To show a decrease, use a negative integer -15.

To represent the decrease in the number of students voting over the next 3 years, multiply the number of years by the integer representing the decrease.

$3 \times (-15) = ?$

Find the product to solve the problem.

$3 \times 15 =45$ , so $3 \times (-15)=-45$

The change in the number of students voting over the next 3 years can be represented as -45.

Analyze Patterns of Quotients of Integers with Same and Different Signs, Recognizing Division by Zero as Undefined

Another important step in learning how to compute with integers is learning how to divide them. You can look for patterns in a sequence of quotients just as you looked for patterns in a sequence of products in an earlier lesson. These patterns will help you to understand the rules for dividing integers.

Let’s look at some integer patterns with division. We are looking at quotients.A quotient is the answer in a division problem.

Use a pattern to find the missing quotients below.

$6 \div 2 & = 3\\4 \div 2 & = 2\\2 \div 2 & = 1\\0 \div 2 & = 0\\-2 \div 2 & = ?\\-4 \div 2 & = ?\\$

You will see that you can subtract 1 from the previous quotient to find the next quotient. Remember, subtracting 1 is the same thing as adding its opposite, -1. Try adding -1 to the previous quotients to find the next quotients.

To find the quotient of $-2 \div 2$ , add 0+(-1)

|0|=0 and |-1|=1 , so subtract the lesser absolute value from the greater absolute value.

1-0=1

The integer with the greater absolute value is -1, so give the answer a negative sign.

0+(-1)=-1 , so $-2 \div 2=-1$

To find the quotient of $-4 \div 2$ , add -1+(-1)

Both integers have the same sign, so add their absolute values.

|-1|=1 , so add

1+1=2

Give that answer a negative sign.

-1+(-1)=-2 , so $-4 \div 2 =-2$ .

This shows the completed division facts.

$6 \div 2 & = 3\\4 \div 2 & = 2\\2 \div 2 & = 1\\0 \div 2 & = 0\\-2 \div 2 & = -1\\-4 \div 2 & = -2\\$

Each quotient is still 1 less than the previous quotient.

Write these rules in your notebook.

Based on the patterns, here are the rules for dividing integers.

Divide Integers

Now we can use these rules to divide integers. Just like with the rules for multiplying, becoming great at dividing integers will require that you memorize these rules.

Now, let’s apply these rules to dividing integers.

Example L

Find the quotient $(-33) \div (-3)$

To find this quotient, we need to divide two negative integers.

Divide the integers without paying attention to their signs. The quotient will be positive.

$(-33) \div (-3) = 33 \div 3 = 11$

The quotient is 11.

Example M

Find the quotient $(-20) \div 5$ .

To find this quotient, we need to divide two integers with different signs.

Divide the integers without paying attention to their signs. Give the quotient a negative sign because the signs are different, indicating a negative answer.

$20 \div 5 =4$ , so $(-20) \div 5 = -4$ .

The quotient is -4.

These problems used a division sign, but remember we can also show division using a fraction bar where the numerator is divided by the denominator.

Now, it’s time for you to practice applying these rules to figuring out quotients.

1. $-12 \div -3$

2. $\frac{18}{-3}$

3. $-24 \div 8$

Check your answers with a peer.

Model and Solve Real-World Problems Using Simple Equations Involving Integer Division

We can apply these rules for dividing integers to real-world problems.

To solve a real-world problem, write an expression or an equation that can be used to solve the problem. Then solve.

Example N

On 3 consecutive plays, a football team lost a total of 30 yards. The team lost the same number of yards on each play. Represent the number of yards lost on each play as a negative integer.

First, represent the total number of yards lost as an integer.

Since the integer shows a loss of 30 yards, use a negative integer -30.

To represent the loss for each of the 3 plays, divide the integer representing the total number of yards lost by 3.

We write this equation and then fill in the given values

Total yards lost $\div$ number of plays = yds lost on each play

$-30 \div 3 =?$

To find this quotient, we need to divide two integers with different signs.

Divide the integers without paying attention to their signs. Give the quotient a negative sign.

$30 \div 3 = 10$ , so $(-30) \div 3 = -10$ .

The integer -10 represents the number of yards lost on each play.

Now let’s apply what we have learned to the problem from the introduction.

Real Life Example Completed

Dive Depths

As a junior diver, Malacai is only allowed to travel to a maximum depth of 40 feet.

Here are Malacai’s dive depths from his last trip when he went diving in Jamaica.

15 feet deep

40 feet deep

25 feet deep

36 feet deep

30 feet deep

Integers can help scuba divers in a real-world situation like this. Since Malacai traveled below the surface, we can use integers to write each of his depths. Then we can write them in order from least to greatest.

First, let’s write each of Malacai’s depths as an integer. Depth is a word that tells us that we are going below the surface of the water. If the surface is zero, then anything below the surface would be represented by a negative number. Malacai’s depths are all negative numbers.

-25

-30

-15

-36

-40

We can order these integers from least to greatest by thinking of the deepest dive as the least and the dive closest to the surface, zero, as the greatest.

-40, -36, -30, -25, -15

Review

Remember that a number line will support you with problems in various contexts with adding and subtracting negative numbers.
Make sure to check the sign on your answer when you have finished calculating with multiplication and division.

Negative Number

A negative number is any number less than zero.

Product

A product is the answer in a multiplication problem.

Quotient

A quotient is the answer in a division problem.

Video Resources

Khan Academy Negative Numbers Introduction

James Sousa Multiplying Integers, The Basics

James Sousa Example of Multiplying Integers

Khan Academy Multiplying and Dividing Negative Numbers

James Sousa, Division of Integers: The Basics

James Sousa, Example of Dividing Integers

James Sousa, Multiplying and Dividing Involving Zero