CHAPTER 1: INTEGERS. Image from CHAPTER 1 CONTENTS

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1 CHAPTER 1: INTEGERS Image from CHAPTER 1 CONTENTS 1.1 Introduction to Integers 1. Absolute Value 1. Addition of Integers 1.4 Subtraction of Integers 1.5 Multiplication and Division of Integers 1.6 Exponents and Roots 1.7 Order of Operations 1

2 CCBC Math 081 Introduction to Integers Section pages 1.1 Introduction to Integers There once was a line full of numbers That never did sleep nor did slumber There are many good reasons To compute in all seasons In Fall, Winter, Spring, and the Summer Image from Microsoft Office Clip Art This little known limerick by an even lesser known mathematician-poet is convincing proof that everyone should learn mathematics. Even if your dreams of being an artist a writer, a singer, a painter, an actor are slow to materialize in the waking world, mathematical skills are valuable in the present and in the future, at home and for so many different jobs and careers. In fact, the author of the limerick pays the bills with his mathematical skills, while still having the time to explore many different avenues of artistic creativity. Images from Microsoft Office Clip Art A good number sense, as it is called, is the very foundation for understanding mathematics. However, some students feel that they are allergic to math and that working with integers (also called signed numbers) will send them into mathematical sneezing fits So, in this chapter, when we perform arithmetic with integers, we will look at a few different strategies. I am Image from Microsoft Office Clip Art confident that at least one of the strategies will make sense to you! Soon you will learn to perform operations on integers. This means that you will learn to add, subtract, multiply, and divide integers. But first, you should understand what integers are.

3 CCBC Math 081 Introduction to Integers Section pages So, let s begin by defining the integers as well as other kinds of numbers used in math. Numbers can be classified into different groups (number systems). Three number systems are described in the following chart. NUMBER SYSTEMS Natural Numbers The natural numbers are also called the counting numbers. The natural numbers are {1,,, 4, 5,... }. Look at the natural numbers on the number line below. Note: There is no greatest or last natural number. The arrow on the number line indicates that the numbers continue on endlessly. Whole Numbers The whole numbers include the set of natural numbers and the number 0. The whole numbers are {0, 1,,, 4, 5,... }. Look at the whole numbers on the number line below and notice that the number 0 is included. Note: There is no greatest or last whole number. The arrow on the number line indicates that the numbers continue on endlessly. Integers The integers are the positive and negative counting numbers and the number 0. The integers are {, -5, -4, -, -, -1, 0, 1,,, 4, 5, }. Look at the integers on the number line below. Note: There is no least or first integer and there is no greatest or last integer. The arrows on the number line indicate that the numbers continue on endlessly.

4 CCBC Math 081 Introduction to Integers Section pages Practice 1: Circle each number system that the given number belongs to, and cross out each number system that the given number does not belong to. a. 7 Natural numbers Whole numbers Integers b. 5 Natural numbers Whole numbers Integers c. Natural numbers Whole numbers Integers d. 0 Natural numbers Whole numbers Integers Answer: a. 7 Natural numbers Whole numbers Integers b. 5 Natural numbers Whole numbers Integers c. Natural numbers Whole numbers Integers d. 0 Natural numbers Whole numbers Integers (To view video links in another window, press the control key as you click the link.) 4

5 CCBC Math 081 Introduction to Integers Section pages POSITIVE AND NEGATIVE INTEGERS The set of integers can be categorized into three distinct groups (subsets): Positive Integers {1,,, 4, 5, 6, 7, 8, 9, 10, 11, } Zero { 0 } Negative Integers {, -11, -10, -9, -8, -7, -6, -5, -4, -, -, -1} Look at these subsets on the number line below. negative integers zero positive integers Imagine yourself standing at the number 0, holding a gift card that you received for your birthday. The gift card allows you to take 5 free steps, but it s up to you to decide whether to take those steps to the right or to the left. If you start at 0 and take 5 steps to the right, you will arrive at positive 5, written simply as 5. If you start at 0 and take 5 steps to the left, then you will arrive at negative 5, written as 5. 5

6 CCBC Math 081 Introduction to Integers Section pages Practice : Place each of the following numbers on the number line. (For a frame of reference, the numbers 0 and 1 are given.) Write these numbers on the number line: Answer: Practice : What is the value of each of the capital letters? (For a frame of reference, the numbers 0 and 6 are given.) 6 F A C D B E 0 A = B = C = D = E = F = Answer: A = -4 B =5 C = - D = E = 6 F =

7 CCBC Math 081 Introduction to Integers Section pages Real Numbers Before we continue our study of integers, we should point out that the number line is a densely populated line. The integers are few and far between, mathematically speaking. There are many more numbers on the number line than just the integers. For instance, there are fractions and decimals. Look at the location of some fractions and decimals on the number line below. -½ ¾.5 The fractions and decimals belong to a number system called the Rational Numbers. You will study these numbers in Chapter and Chapter 4 of this book. You will study other non-integers in future math courses. So, to help you become familiar with the various kinds of numbers used in mathematics, an overview of the number systems is presented below in diagram form. The diagram describes each number system and allows you to see the relationships among them. REAL NUMBERS All #'s on number line RATIONAL NUMBERS #'s that can be expressed as below IRRATIONAL NUMBERS decimals that do not end or repeat a pattern π INTEGERS {..., -, -, -1, 0, 1,,,...} FRACTIONS 1/ -5/8 DECIMALS decimals that end or repeat a pattern OPPOSITES OF NATURAL NUMBERS {..., -, -, -1} WHOLE NUMBERS { 0, 1,,,...} ZERO 0 NATURAL NUMBERS { 1,,,...} Strangely Beautiful Fact: If you pick any real number (integer, fraction, decimal) on the number line, there is no real number right next to it on either side. That is, there is no closest real number either to the right or the left. There are also no gaps or holes in the number line. Strange. Beautiful. Fact. This is why mathematicians say that the real number line is a continuum. Watch All: 7

8 CCBC Math 081 Introduction to Integers Section pages 1.1 Introduction to Integers Exercises 1. Name each set of numbers shown below. {0, 1,,, 4, 5,... } {1,,, 4, 5,... } {, -5, -4, -, -, -1, 0, 1,,, 4, 5, }. Circle each number system that the given number belongs to, and cross out each number system that the given number does not belong to. (The first number is done for you.) - belongs to the natural numbers whole numbers integers 1 belongs to the natural numbers whole numbers integers 4 belongs to the natural numbers whole numbers integers 0 belongs to the natural numbers whole numbers integers - 1 belongs to the natural numbers whole numbers integers. Place each of the following numbers on the number line. The number 0 is already shown What is the value of each of the points named by the capital letters on the number line? (For a frame of reference, the number 0 is given.) 6 F A C D B E A = B = C = D = E = F = 0 8

9 CCBC Math 081 Introduction to Integers Section pages 5. Circle all the integers in the list of numbers below , 6, 0.4,, 9, 0, 1.8, 4,, Circle all the positive integers in the list of numbers below , 6., 0.4,,, 0, 8, 4.,, 7 7. Circle all the negative integers in the list of numbers below ,,,,, 0,.6, 4,,

10 CCBC Math 081 Introduction to Integers Section pages 1.1 Introduction to Integers Exercises Answers 1. Whole Numbers Natural Numbers (or Counting Numbers) Integers. - belongs to the natural numbers whole numbers integers 1 belongs to the natural numbers whole numbers integers 4 belongs to the natural numbers whole numbers integers 0 belongs to the natural numbers whole numbers integers -1 belongs to the natural numbers whole numbers integers A B 4 C 1 D E 5 F

11 CCBC Math 081 Absolute Value Section 1. 6 Pages 1. Absolute Value We will begin our discussion of absolute value with two facts. ABSOLUTE VALUE The absolute value of 0 is 0. The absolute value of every other real number is positive. For example, the absolute value of is, and the absolute value of is also. The absolute value of 17 is 17, and the absolute value of 106 is 106. The absolute value of the real number r is denoted by the symbol r. We write There are two ways to think about (and to define) absolute value. The first way is to understand absolute value as distance. The absolute value of a number is the distance from that number to 0 along the number line. Since distance is always understood to be a non-negative measurement, that is, a value greater than or equal to 0, so too is the absolute value of a number understood to be greater than or equal to 0. Example 1: Evaluate 5. 5 is 5 units away from 0. Therefore, 5 5. Practice 1: Evaluate 7. Answer: 7 Example : Evaluate. is units away from 0. Therefore,. Practice : Evaluate 4. Answer:

12 CCBC Math 081 Absolute Value Section 1. 6 Pages Example : Evaluate 0. 0 is 0 units away from 0. Therefore, 0 0. Practice : Evaluate 0. Answer: 0 The second way to think about (and to define) absolute value is in a piecewise way. Piecewise is a fancy word that means, in the context of absolute value, that the definition has two cases or two possibilities. Case 1: If the number inside the absolute value symbol is 0 or positive, then the absolute value of that number is that very same number itself. Let s evaluate 17. Since the number, 17, inside the absolute value symbol is positive, the absolute value of 17 is equal to that same number, 17. That is, Case : If the number inside the absolute value symbol is negative, then the absolute value of that number is equal to the opposite of that number. And we all know what the opposite of a negative number is, right? Yes, it s a positive number. Let s evaluate 106. Since the number, 106, inside the absolute value symbol is negative, the absolute value of 106 is the opposite of 106, which is 106. So, However, if the piecewise definition is a bit confusing, just remember the two facts you learned. ABSOLUTE VALUE The absolute value of 0 is 0. The absolute value of every other real number is positive. 1

13 CCBC Math 081 Absolute Value Section 1. 6 Pages You ve heard the phrase, Easier said than done. But absolute values are actually easier to do than to explain how to do. Using the two facts above, absolute values are Easier done than said! As you evaluate absolute value problems, keep in mind that the absolute value symbol treated as parentheses for the purpose of order of operations. Therefore, operations within the symbol must be performed before applying the definition of absolute value. is Example 4: Evaluate 1. Since the symbol that indicates absolute value order of operations, 1 absolute value symbol is applied. is treated as parentheses for the purpose of must be evaluated first, and then the negative sign in front of the 1 First, we evaluate 1. This gives 1. 1 The negative sign that was in front of the absolute value is applied by just bringing it down to the left of the number. 1 The answer is 1. Practice 4: Evaluate 4. Answer: -4 Example 5: Evaluate 57. As in Example 4, the number in the absolute value symbol must be evaluated first and then the negative sign in front of the absolute value symbol is applied. First, we evaluate 57. This gives 57. The negative sign that was in front of the absolute value is applied by just bringing it down to the left of the number. 57 The answer is 57. Practice 5: Evaluate 1. Answer:

14 CCBC Math 081 Absolute Value Section 1. 6 Pages Example 6: Evaluate 5. 5 First, perform the operation inside the absolute value. +5 gives 8. 8 Now apply the definition of absolute value. 8 The answer is 8. Practice 6: Evaluate 9 6. Answer: Example 7: Evaluate First, perform the operation inside the absolute value gives Now apply the definition of absolute value. 11 The negative sign that was in front of the absolute value is applied by bringing it down to the left of the number. 11 The answer is 11. Practice 7: Evaluate 7 5. Answer: - Watch All: 14

15 CCBC Math 081 Absolute Value Section 1. 6 Pages 1. Absolute Value Exercises Evaluate each absolute value problem

16 CCBC Math 081 Absolute Value Section 1. 6 Pages 1. Absolute Value Exercises Answers

17 CCBC Math 081 Addition of Integers Section Pages 1. Addition of Integers Recall the set of numbers {, -5, -4, -, -, -1, 0, 1,,, 4, 5, } called the integers. You will begin to use these numbers in arithmetic operations. You will learn to add, subtract, multiply, and divide with the integers. In this section, we will focus on addition of integers. PARTS OF AN ADDITION STATEMENT 4 7 Addends Sum (Answer) It is important that you become familiar with the various ways that addition problems may be written. Sometimes you will see a problem written with one or both addends in parentheses. Other times, parentheses will not be used. With or without parentheses, either way is acceptable. Look at all the ways the same problem can be written. 6 ( 6) ( ) ( 6) Now it is time to learn the procedure for adding integers. We will take the same problem and determine the answer using four different methods. Of course, all the methods will produce the same answer. The methods are listed below. (1) Number Line Method () Money Method () Signed Chip Method (4) SSS / DDD Method You may be wondering if you will need to know and use all of the methods. The answer is No. It is important that you learn the SSS / DDD Method because this method will be necessary as the problems become more complex in this and future math courses. So, why are we presenting three other methods? The reason is that the other methods provide a more visual approach to the problems that makes it easier to understand how we arrive at the answer. In other words, the three other methods may help you make sense of the SSS / DDD Method. 17

18 CCBC Math 081 Addition of Integers Section Pages Example 1a: Number Line Method The first number,, gives the point at which we should start on the number line. Since we are adding a negative number ( ), we will move units to the left. We have stopped at the number 5. Therefore, 5 SUMMARY OF THE NUMBER LINE METHOD First Number in Problem: Start at this point on the number line. Second Number in Problem: On the number line, move this many units from the starting point. If second number is negative, move left. If second number is positive, move right. Answer: The point at which you stop on the number line. Practice 1a: 5 1 Number Line Method Answer: -6 18

19 CCBC Math 081 Addition of Integers Section Pages Example 1b: Money Method The negative number ( ) indicates that you have borrowed dollars from your Uncle Larry. Thus, you are dollars in debt to him. The negative number ( ) being added indicates that you then borrowed more dollars from him. (Haven t you learned your lesson yet!) Where do you stand with Uncle Larry? (Not in his good graces, let me tell you!) You owe him 5 dollars. You re now 5 dollars in the hole and Uncle Larry is stingy even about 5 bucks. We indicate being in the hole or in debt with a negative number. Therefore, 5. By the way, Uncle Larry said he wants his money back by Tuesday! Images from Microsoft Office Clip Art SUMMARY OF THE MONEY METHOD Answer: Add the money you have or receive. Subtract the money you owe or spend. Sign of Answer: If you owe money, the answer is negative. If you have money, the answer is positive. Practice 1b: 5 1 Money Method Images from Microsoft Office Clip Art Answer: -6 19

20 CCBC Math 081 Addition of Integers Section Pages Example 1c: Signed Chip Method Draw negative chips to represent. Draw negative chips to represent. When we add, we combine all the chips. We have 5 negative chips. Therefore, 5 SUMMARY OF SIGNED CHIP METHOD Draw positive chips to represent the positive numbers in the problem. Draw negative chips to represent the negative numbers in the problem. Cross off pairs of positive and negative chips until only one type of chip remains. Each pair of positive and negative chips is equal to zero. The answer will be the number of chips left. The sign will be the sign of the remaining chips. Practice 1c: 5 1 Signed Chip Method Answer: -6 0

21 CCBC Math 081 Addition of Integers Section Pages Example 1d: Triple-S or Triple-D Method (SSS or DDD) Same: Sum: In this problem, we are adding two numbers that have the same sign: and are both negative numbers. In cases like this, we compute the sum of the absolute values of the two numbers. and First we get the absolute value of each number. 5 Next we compute their sum. This means to add the absolute values. Same: The final answer takes the same sign as the sign of the original numbers in the problem. The numbers in the original problem were and. Since those numbers were negative, our final answer is negative. Therefore, 5. SUMMARY OF THE TRIPLE-S METHOD (SSS) Same: You are adding two numbers that have the same sign. Sum: Compute the sum of the absolute values of the two numbers. In other words, add the absolute values of the numbers. Same: The final answer takes the same sign as the original numbers in the problem. Practice 1d: 5 1 Triple-S or Triple-D Method (SSS or DDD) Answer: -6 1

22 CCBC Math 081 Addition of Integers Section Pages You have now learned to add integers using four methods. Let s practice each method using a new problem. Example a: 4 ( ) Number Line Method The first number, 4, gives the point at which we should start on the number line. Since we are adding a negative number ( ), we will move units to the left. We have stopped at the number 6. Therefore, 4 ( ) 6. Practice a: ( 1) Number Line Method Answer: -4 Example b: 4 ( ) Money Method The negative number ( 4) indicates that you lost 4 dollars to your Uncle Larry during one hand of a card game. So, you are down 4 dollars. The negative number ( ) being added indicates that you then lost more dollars to Uncle Larry in the next hand of that card game. Where do you stand with Uncle Larry now? (Hopefully, at least 50 feet away!) Now you owe your uncle 6 dollars. You re now 6 dollars in the hole. We indicate being in the hole or in debt with a negative number. Therefore, 4 ( ) 6. I should have warned you about Uncle Larry. He cheats at cards! Images from Microsoft Office Clip Art Practice b: ( 1) Money Method Images from Microsoft Office Clip Art Answer: -4

23 CCBC Math 081 Addition of Integers Section Pages Example c: 4 ( ) Signed Chip Method Draw 4 negative chips to represent 4. Draw negative chips to represent. When we add, we combine all the chips. We have 6 negative chips. Therefore, 4 ( ) 6. Practice c: ( 1) Signed Chip Method Answer: -4 Example d: 4 ( ) Triple-S or Triple-D Method (SSS or DDD) Same: Sum: We are adding two numbers that have the same sign: 4 and are both negative numbers. When the signs are the same, we compute the sum of the absolute values of the two numbers. 4 4 and First we get the absolute value of each number. 4 6 Next we compute their sum. This means to add the absolute values. Same: The final answer takes the same sign as the sign of the original numbers in the problem. The numbers in the original problem were 4 and. Since those numbers were negative, our final answer is negative. Therefore, 4 ( ) 6. Practice d: ( 1) Triple-S or Triple-D Method (SSS or DDD) Answer: -4 Now we present another new problem. When you study the Signed Chip Method in Example c, notice that both positive and negative chips are used. And in Example d, notice how the Triple-S (SSS) method changes to the Triple-D (DDD) method.

24 CCBC Math 081 Addition of Integers Section Pages Example a: 5 Number Line Method The first number, 5, gives the point at which we should start on the number line. Now, since we are adding a positive number, we will move units to the right. We have stopped at the number. Therefore, 5. Practice a: 5 Number Line Method Answer: - Example b: 5 Money Method The negative number ( 5) indicates that you have borrowed 5 dollars from your Uncle Larry. Thus, you are 5 dollars in debt to him. (Not a good thing.) The positive number () being added indicates that you paid Uncle Larry back dollars. Where do you stand with your uncle? You still owe him dollars. You re still dollars in the hole. We indicate being in the hole or in debt with a negative number. Therefore, 5. By the way, Uncle Larry said that he wants his money back NOW! Images from Microsoft Office Clip Art Practice b: 5 Money Method Images from Microsoft Office Clip Art Answer: - 4

25 CCBC Math 081 Addition of Integers Section Pages Example c: 5 Signed Chip Method Draw 5 negative chips to represent 5. Draw positive chips to represent. One negative chip and one positive chip equal 0. Cross off pairs of positive and negative chips These pairs equal 0. What is left? Three negative chips. Therefore, 5. Practice c: 5 Signed Chip Method Answer: - Example d: 5 Triple-S or Triple-D Method (SSS or DDD) Different: Difference: In this problem, unlike the previous Examples 1 &, we are adding two numbers that have different signs: 5 is a negative number and is a positive number. In cases like this, we compute the difference between the absolute values of the two numbers. 5 5 and First we get the absolute value of each number. Dominant: 5 Next we compute their difference. This means to subtract: Larger number minus Smaller number. The final answer takes the sign of the dominant number. The two numbers in the original problem are 5 and. The dominant number is the one with the larger absolute value. In this problem, 5 is dominant because it has the larger absolute value. Since the dominant number is a negative number, the final answer is also a negative number. Thus, the sign of the answer to an addition problem is determined by the sign of the number with the larger absolute value. Therefore, 5. 5

26 CCBC Math 081 Addition of Integers Section Pages SUMMARY OF THE TRIPLE-D METHOD (DDD) Different: You are adding two numbers that have different signs (one positive, one negative). Difference: Compute the difference between the absolute values of the numbers. This means to subtract: Larger absolute value minus Smaller absolute value. Dominant: The final answer takes the sign of the number with the larger absolute value. Practice d: 5 Triple-S or Triple-D Method (SSS or DDD) Answer: - Now let s present our last example, again showing how to solve the problem using all four methods. Example 4a: ( ) Number Line Method The first number,, gives the point at which we should start on the number line. Since we are adding a negative number ( ), we will move units to the left. We have stopped at the number 1. Therefore, ( ) 1. Practice 4a: 6 ( 4) Number Line Method Answer: 6

27 CCBC Math 081 Addition of Integers Section Pages Example 4b: ( ) Money Method The positive number () indicates that you have dollars in your pocket. The negative number ( ) being added indicates that you then gave your Uncle Larry dollars. Where do you stand with your uncle? You are 1 dollar up or ahead. We indicate being up or ahead with a positive number. Therefore, ( ) 1. Don t you feel relieved not to owe Uncle Larry any money? Images from Microsoft Office Clip Art Practice 4b: 6 ( 4) Money Method Images from Microsoft Office Clip Art Answer: Example 4c: ( ) Signed Chip Method Draw positive chips to represent. Draw negative chips to represent. One negative chip and one positive chip equal 0. Cross off pairs of positive and negative chips These pairs equal 0. What is left? One positive chip. Therefore, ( ) 1. Practice 4c: 6 ( 4) Signed Chip Method Answer: 7

28 CCBC Math 081 Addition of Integers Section Pages Example 4d: ( ) Triple-S or Triple-D Method (SSS or DDD) Different: Difference: We are adding two numbers that have different signs: is a positive number and is a negative number. In cases like this, we compute the difference between the absolute values of the two numbers. and First we get the absolute value of each number. 1 Next we compute their difference. This means to subtract: Larger number minus Smaller number. Dominant: The final answer takes the sign of the dominant number. The two numbers in the original problem are and. The dominant number is the one with the larger absolute value. In this problem, is dominant because it has the larger absolute value. Since the dominant number is a positive number, the final answer is also a positive number. Thus, the sign of the answer to a problem is determined by the sign of the number with the larger absolute value. Therefore, ( ) 1. Practice 4d: 6 ( 4) Triple-S or Triple-D Method (SSS or DDD) Answer: When you solve the exercises that follow, remember that you should get used to using the SSS / DDD method. The other methods may be used to supplement your work and check your answers. Let s take a moment to review the four methods before you begin the exercises. Watch All: 8

29 CCBC Math 081 Addition of Integers Section Pages ADDING INTEGERS Number Line Method Money Method First number in problem start at this point on the number line. Second number in problem move this many units from the starting point If second number is negative, move left. If second number is postive, move right. Answer the point at which you stop on the number line Signed Chip Method Answer: Add the money you have or receive. Subtract the money you owe or spend. Sign of Answer: If you owe money, the answer is negative. If you have money, the answer is positive. Draw Chips: chips to represent the positive numbers chips to represent the negative numbers Cross off / pairs until only one type of chip remains Answer is the number of chips left and sign is the same as the remaining chips. Triple Method SSS Same signs Sum - add absolute values Same - answer has same sign as orignal numbers DDD Different signs Difference - subtract absolute values (larger smaller) Dominant - answer has sign of number with larger absolute value 9

30 CCBC Math 081 Addition of Integers Section Pages 1. Adding Integers Exercises Compute the answer to each addition problem ( ) ( 6) ( 8) ( 9) ( 47) ( ) ( 94) ( 1) ( 456) 0

31 CCBC Math 081 Addition of Integers Section Pages 1. Adding Integers Exercises Answers

32 CCBC Math 081 Subtraction of Integers Section Pages 1.4 Subtraction of Integers At all levels of mathematics, one key strategy to solving a problem is to use what you already know to build logically on ideas that you have already established. This will be our approach to subtracting integers. We will change a subtraction problem into an addition problem by adding the opposite. Then we can simply use the methods that we learned in the previous section for adding two integers. Adding the Opposite There are two types of subtraction problems to consider when subtracting integers. These are shown below along with the Add the Opposite method for changing the subtraction problems into equivalent addition problems. Look at how the first number in the subtraction problem stays the same, how the minus sign becomes a plus sign, and how we take the opposite of the second number (positive changes to negative, negative changes to positive). Subtracting a Positive a b (Same) (Becomes +) (Opposite) a + b Subtracting a Negative a ( b) (Same) (Becomes +) (Opposite) a + b Study the examples below to see how the Add the Opposite method works with integers. Subtracting a Positive -8 6 (Same) (Becomes +) (Opposite) Subtracting a Negative 9 ( 4) (Same) (Becomes +) (Opposite) Subtraction Problem: 8 6 Subtraction Problem: 9 ( 4) Addition Problem: 8 6 Addition Problem: 9 4 So, 8 6= 8 6 So, 9 ( 4) = 9 4

33 CCBC Math 081 Subtraction of Integers Section Pages Let s study each of these types more closely, beginning with the rule for Subtracting a Positive. Subtracting a Positive becomes... Adding a Negative a b a b Example 1: 1 Number Line Method First, rewrite the subtraction problem as an addition problem. The first number,, stays the same. We change the minus sign to a plus sign. The second number is 1. We take its opposite which is 1. So, our addition problem is 1. Now that we have this addition problem, we use one of the methods from the previous section. Let s go with the Number Line Method. Subtracting a Positive - 1 (Same) (Becomes +) (Opposite) The first number,, gives the point at which we should start on the number line. Since we are adding a negative number ( 1), we will move 1 unit to the left. We have stopped at the number. Therefore, 1 1. Practice 1: 4 Number Line Method Answer: -7

34 CCBC Math 081 Subtraction of Integers Section Pages Example : 5 Money Method Images from Microsoft Office Clip Art First, rewrite the subtraction problem as an addition problem. The first number,, stays the same. We change the minus sign to a plus sign. The second number is 5. We take its opposite which is 5. So, our addition problem is ( 5). Now we use an addition method from the previous section. Let s go with the Money Method. We will use a bank account to help us understand the problem. The positive number () indicates that you have dollars in your account. The negative number ( 5) being added indicates that you spent 5 dollars, either by writing a check or using your debit card. What is the balance in your account? You spent dollars more than you had, so you are overdrawn. We indicate being overdrawn with a negative number. Therefore, 5 ( 5). Subtracting a Positive 5 (Same) (Becomes +) (Opposite) + -5 Practice : 4 7 Money Method Images from Microsoft Office Clip Art Answer: - Example : 4 Triple-S or Triple-D Method (SSS or DDD) First, rewrite the subtraction problem as an addition problem. The first number, 4, stays the same. We change the minus sign to a plus sign. The second number is. We take its opposite which is. So, our addition problem is 4. Subtracting a Positive -4 (Same) (Becomes +) (Opposite) Now we will use an addition method from the previous section. We choose the SSS method. We are using this rather than DDD because the signs in the addition problem are the same. 4

35 CCBC Math 081 Subtraction of Integers Section Pages Same: Sum: Same: We are adding two numbers that have the same sign: 4 and are both negative. We compute the sum of the absolute values of the two numbers. 4 4 and First we get the absolute value of each number. 4 7 Now we compute their sum. This means to add. The final answer takes the same sign as the sign of the original numbers in the addition problem. The numbers in the addition problem were 4 and. Since those numbers were negative, our final answer is negative. Therefore, Practice : 5 Triple-S or Triple-D Method (SSS or DDD) Answer: -7 Now that you have learned how to subtract a positive, it s time to consider the second type of subtraction problem. The rule for Subtracting a Negative is given below. Subtracting a Negative becomes... Adding a Positive a ( b) a b Example 4: 4 ( 7) First, rewrite the subtraction problem as an addition problem. The first number, 4, stays the same. We change the minus sign to a plus sign. The second number is 7. We take its opposite which is 7. So, our addition problem is 4 7. Subtracting a Negative 4-7 (Same) (Becomes +) (Opposite) We know that So, 4 ( 7) Practice 4: ( ) Answer: 6 5

36 CCBC Math 081 Subtraction of Integers Section Pages Example 5: ( 6) Triple-S or Triple-D Method (SSS or DDD) First, rewrite the subtraction problem as an addition problem. The first number,, stays the same. We change the minus sign to a plus sign. The second number is 6. We take its opposite which is 6. So, our addition problem is 6. Subtracting a Negative - -6 (Same) (Becomes +) (Opposite) Now we will use an addition method from the previous section. We choose the DDD method. We are using this rather than SSS because the signs in the addition problem are different. Different: We are adding numbers that have different signs: is negative, 6 is positive. Difference: We compute the difference between the absolute values of the two numbers. and 6 6 First we get the absolute value of each number. 6 4 Next we compute their difference. This means to subtract: Larger number minus Smaller number. Dominant: The final answer takes the sign of the dominant number. Of the numbers in the addition problem, and 6, the dominant number is 6 because it has the larger absolute value. Since the dominant number is a positive number, the final answer is also a positive number. Therefore, ( 6) 6 4. Practice 5: 4 ( 7) Triple-S or Triple-D Method (SSS or DDD) Answer: 6

37 CCBC Math 081 Subtraction of Integers Section Pages Example 6: 5 ( 7) Signed Chip Method First, rewrite the subtraction problem as an addition problem. The first number, 5, stays the same. We change the minus sign to a plus sign. The second number is 7. We take its opposite which is 7. So, our addition problem is 5 7. Subtracting a Negative -5-7 (Same) (Becomes +) (Opposite) Now we will use an addition method from the previous section. For this problem we will use the signed chip method. Draw 5 negative chips to represent 5. Draw 7 positive chips to represent 7. One negative chip and one positive chip equal 0. Cross off pairs of positive and negative chips These pairs equal 0. What is left? Two positive chips. Therefore, 5 ( 7) 5 7. Practice 6: ( 5) Colored Chip Method Answer: Adding and Subtracting More Than Two Integers Sometimes we may have more than two integers to add or subtract. If this is the case, we will perform the operations in order from left to right as shown in the following two examples. 7

38 CCBC Math 081 Subtraction of Integers Section Pages Example 7: ( ) ( 4) Let s perform the additions in order from left to right. ( ) ( 4) First we will add the numbers and 4. We use the SSS method because the numbers have the same sign. We compute the sum of the absolute values: 4 7. Then we keep the same sign as the numbers in the problem. Therefore, the answer is Practice 7: 4 ( ) ( 8) But we are not finished. Now we have another addition problem. We use the DDD method because the numbers have different signs. We compute the difference of the absolute values: 7 5 The dominant number is 7. It is negative, so the sign of the answer is negative. Therefore, the answer is 5. Answer: -7 Example 8: 7 98 Let s perform the operations in order from left to right First we will add the numbers 7 and 9. We use the DDD method because the numbers have different signs. We compute the difference of the absolute values: 9 7. The dominant number is 9. It is positive, so the sign of the answer is positive. Therefore, the answer is. 8 ( 8) But we are not finished. Now we have a subtraction problem. First we must change the problem to addition. We use the Add the Opposite method. The problem 8 becomes ( 8). 6 To add, we use the DDD method because the numbers have different signs. We compute the difference of the absolute values: 8 6 The dominant number is 8. It is negative, so the sign of the answer is negative. Therefore, the answer is 6. Practice 8: Answer: 0 Watch All: 8

39 CCBC Math 081 Subtraction of Integers Section Pages 1.4 Subtracting Integers Exercises Change each subtraction problem to an addition problem, then use any method from the previous section to compute the answer ( 7) ( 7) 6. 1 ( ) ( ) 9. 1 ( 1) Mixed Practice Exercises Now, we ll mix it up a little bit. In some problems, you ll be adding integers, and in others, you ll be subtracting integers. Remember to change all subtraction problems to addition problems first ( 8) ( 5) ( 4) ( ) ( 6) 4. 8 ( 9) 9

40 CCBC Math 081 Subtraction of Integers Section Pages 1.4 Subtracting Integers Exercises Answers = ( 4) =. ( 7) + ( 7) = ( 10) = = = ( ) = = = ( 48) = ( 6) = = Mixed Practice Exercises Answers

41 CCBC Math 081 Mid-Chapter Review Section CHAPTER 1 Mid-Chapter Review 1. Circle each number system that the given number belongs to: a. 9 Natural Numbers Whole Numbers Integers b. 55 Natural Numbers Whole Numbers Integers c. 0 Natural Numbers Whole Numbers Integers. Circle all the integers in the list of numbers: Circle all the positive integers in the list of numbers: 4. What is the value of each of the points named by the capital letters on the number line? 5. Place the following numbers on the number line: B A ( 6) ( ) ( 7) ( 4) ( 4). 16

42 CCBC Math 081 Mid-Chapter Review Section M i d - C h a p t e r 1 R e v i e w A n s w e r s 1. a. Natural Numbers Whole Numbers Integers b. Natural Numbers Whole Numbers Integers c. Natural Numbers Whole Numbers Integers A = B =

43 CCBC Math 081 Multiplication and Division of Integers Section Pages 1.5 Multiplication and Division of Integers In the previous two sections, you learned two operations on integers: addition and subtraction. In this section, you will learn the operations of multiplication and division of integers. Multiplying Signed Numbers PARTS OF A MULTIPLICATION STATEMENT 4 1 Factors Product (Answer) RULES FOR MULTIPLYING REAL NUMBERS (INCLUDING INTEGERS) If two non-zero numbers have the same sign, the product (answer) is positive. positive positive = positive negative negative = positive If two non-zero numbers have different signs, the product (answer) is negative. positive negative = negative negative positive = negative NOTATION TO REPRESENT MULTIPLICATION For this course and for future math courses, you should be familiar with all three ways to represent multiplication. Review all the different notation used to write the problem below. They all express the same multiplication problem. 6 6 ( 6) ( 6) () ( 6) ()( 6) () ( 6) Times Sign 4 Raised Dot 4 Parentheses ()(4) 4

44 CCBC Math 081 Multiplication and Division of Integers Section Pages Example 1: 5 ( 8) The number 5 is positive, and the number 8 is negative. These numbers have different signs. Therefore, the product is negative. 5 ( 8) 40 Practice 1: 4 ( 6) Answer: -4 Example : ( )( 5) The number is negative, and the number 5 is negative. These numbers have the same sign. Therefore, the product is positive. ( ) ( 5) 10 Practice : ( 5)( 7) Answer: 5 Example : 9 6 The number 9 is positive, and the number 6 is positive. These numbers have the same sign. Therefore, the product is positive Practice : 5 Answer: 15 Example 4: ( 7) 4 The number 7 is negative, and the number 4 is positive. These numbers have different signs. Therefore, the product is negative. ( 7) 4 8 Practice 4: ( 6) Answer:

45 CCBC Math 081 Multiplication and Division of Integers Section Pages ZERO PROPERTY OF MULTIPLICATION WORDS SYMBOLS EXAMPLE The product of a number and 0 is 0. a a Example 5: The product of any number and 0 is 0. Therefore, the answer is 0. Practice 5: Answer: 0 Example 6: 0( 7) 0 The product of any number and 0 is 0. Therefore, the answer is 0. Practice 6: 0( 9) Answer: 0 Example 7: 8 ( 6) 0 0 The product of any number and 0 is 0. Therefore, the answer is 0. Practice 7: 0 ( 7) Answer: 0 Multiplying More Than Two Integers When you multiply two or more than two integers, you can multiply in any order that you want. You might find it easier to multiply from left to right, but keep in mind that the order in which you multiply does not matter. 45

46 CCBC Math 081 Multiplication and Division of Integers Section Pages Example 8: ( ) ( 4) Let s multiply from left to right. ( ) ( 4) 1 4 First multiply the numbers and 4. Both numbers are negative. They have the same sign. Therefore, the product of these two numbers is positive 1. But we are not finished. Now multiply 1 by the third integer. Both of these numbers are positive. They have the same sign. Therefore, the product is positive 4. Practice 8: 5 ( ) ( ) Answer: 0 Example 9: ( 1) ( ) () ( 8) Let s multiply from left to right. ( 1) ( ) () ( 8) First multiply the numbers 1 and. Both numbers are negative. They have the same sign. Therefore, the product is positive. Next multiply with. Both of these numbers are positive. They have the same sign. Therefore, the product is positive 6. 6 ( 8) 48 Finally, multiply 6 with 8. One of these numbers is positive and the other is negative. These numbers have different signs. Therefore, the product is negative 48. Practice 9: ( 1) (4) ( ) ( ) Answer:

47 CCBC Math 081 Multiplication and Division of Integers Section Pages Dividing Signed Numbers PARTS OF A DIVISION STATEMENT 8 4 Dividend Divisor Quotient 8 4 Dividend Quotient Quotient Divisor Divisor 8 Dividend 4 RULES FOR DIVIDING REAL NUMBERS (INCLUDING INTEGERS) If two non-zero numbers have the same sign, the quotient (answer) is positive. negative negative = positive positive positive = positive If two non-zero numbers have different signs, the quotient (answer) is negative. negative positive = negative positive negative = negative Notice that the Division Rules are the same as the Multiplication Rules! Example 10: 56 ( 8) The number 56 is positive and the number 8 is negative. These numbers have different signs. Therefore, the quotient (answer) is negative. 56 ( 8) 7 Practice 10: 6 ( 6) Answer: -6 Example 11: ( 7) ( 6) The numbers 7 and 6 are both negative. These numbers have the same sign. Therefore, the quotient is positive. ( 7) ( 6) 1 Practice 11: ( 49) ( 7) Answer:

48 CCBC Math 081 Multiplication and Division of Integers Section Pages Example 1: ( 14) 7 The number 14 is negative and the number 7 is positive. These numbers have different signs. Therefore, the quotient is negative. ( 14) 7 Practice 1: ( 18) Answer: -6 Example 1: 8 The numbers 8 and are both positive. These numbers have the same sign. Therefore, the quotient is positive Practice 1: 4 Answer: 8 There are two special division rules that need to be stated before we conclude this lesson. They pertain just to division problems that involve the number 0. RULES FOR DIVIDING WHEN 0 IS IN THE PROBLEM If 0 is divided by any number (except 0), the answer is 0. 0n n If any number is divided by 0, the answer is undefined. In other words, there is no answer. n n 0 undefined undefined 0 Example 14: ( 7) 0 undefined Since 0 is the divisor, the answer is undefined. Practice 14: ( 4) 0 Answer: Undefined Example 15: 06 0 Since 0 is the dividend, the answer is 0. Practice 15: 0 7 Answer:

49 CCBC Math 081 Multiplication and Division of Integers Section Pages I know that you are very interested in why these two rules work as they do. So, let s explain. We start with a division problem that does not involve ? 1 We can verify our answer to the division problem by using a multiplication problem. Think of it this way: times what number gives 1? 4 1 And of course, that number is 4, the same number we got as the answer in the division problem. Now let s look at a division problem that has 0 as the dividend. 0 0? 0 We can verify our answer to the division problem by using a multiplication problem. Think of it this way: times what number gives 0? 00 And of course, that number can only be 0, the same number we got as the answer in the division problem. Last, let s look at a division problem that has 0 as the divisor. 0 undefined 0? We can verify our answer to the division problem by using a multiplication problem. Watch All: Think of it this way: 0 times what number gives? 0? But there is no number we can multiply 0 by to get. That s why the answer to the division problem is undefined. 49

50 CCBC Math 081 Multiplication and Division of Integers Section Pages 1.5 Integer Multiplication and Division Exercises Perform the indicated operation(s). Remember that multiplication may be represented with parentheses, or symbols. 1. ( 5) 9. ( 11)( ) ( ) 16. ( 11) ( 4) 4. ( 4) ( 10) 6 5. ( 15) ( ) 18. ( 87) ( 9) ( 8) 8. ( 44) ( 7) ( 57) 11. ( 4) ( 6) 1. 17( 8) ( 1) ( 9) 0. ( 9) ( 6). ( ) ( 4) ( 7). ( 1)( 1)( 1)( 1) 4. ( ) ( ) ( 8) 6 ( 1) 6. ( )()( )( ) 50

51 CCBC Math 081 Multiplication and Division of Integers Section Pages 1.5 Integer Multiplication and Division Exercises Answers Undefined

52 CCBC Math 081 Exponents and Roots Section Pages 1.6 Exponents and Roots You have learned to add, subtract, multiply, and divide integers. There are two other mathematical operations to consider in this chapter. They are exponents and roots. Exponents Do you know where the popular search engine Google got its name? It was named for the 100 number 10 which is called a googol! The number is equal to 1 followed by 100 zeros, but it is 100 so much easier to represent the number using the notation 10. This is the type of notation you will learn in this section. The notation uses exponents and is very useful in math and science applications to represent very small or very large numbers. The box below reviews some basic vocabulary and explains the meaning of the notation. EXPONENTIAL NOTATION Base n Exponent a An exponent indicates how many times the base should be multiplied with itself. This means that a will be multiplied n times. Example 1: Evaluate 4 This is read, Two to the fourth power. 4 The base is and the exponent is 4. Multiply with itself four times. = 16 The answer is 16. Practice 1: Evaluate 4 Answer: 81 Example : Evaluate 4 This is read, Four to the third power or Four cubed 4 The base is 4 and the exponent is. 4 4 Multiply 4 with itself three times. = 64 The answer is 64. Practice : Evaluate Answer: 8 5

53 CCBC Math 081 Exponents and Roots Section Pages Example : Evaluate 10 This is read, Ten to the third power or Ten cubed. The base is 10 and the exponent is. (10) (10) (10) Multiply 10 with itself three times. = 1000 The answer is Practice : Evaluate 4 10 Answer: 10,000 Example 4: Evaluate The base is 1 and the exponent is 5. (1) (1) (1) (1) (1) Multiply 1 with itself five times. = 1 The answer is 1. Practice 4: Evaluate 4 1 Answer: 1 Example 5: Practice 5: Evaluate This is read, The opposite of to the second power or The opposite of squared. This problem is different than the previous ones because it has a negative sign in front. However, we do the problem the same way as the previous problems. We just have to remember to keep the negative sign out in front. The base is and the exponent is. ( ) Multiply with itself two times to get 9. (9) Take the opposite of 9. 9 The answer is 9. Evaluate 4 Answer:

54 CCBC Math 081 Exponents and Roots Section Pages Example 6: Evaluate ( ) This is read, Negative to the second power or Negative squared. This problem is different than the last because it has a number in parentheses. The number in parentheses is the base. The exponent is outside the parentheses. ( ) The base is since it is in parentheses. The exponent is. ( )( ) Multiply with itself two times. = 9 The answer is 9. Practice 6: Evaluate ( 4) Answer: 16 Example 7: Evaluate 5 This is read, The opposite of 5 cubed. 5 The base is 5 and the exponent is. (5 5 5) Multiply 5 with itself three times to get 15. (15) Take the opposite of The answer is 15. Practice 7: Evaluate Answer: -8 Example 8: Evaluate ( 5) This is read, Negative 5 to the third power or Negative 5 cubed. ( 5) The base is 5 since it is in parentheses. The exponent is. ( 5)( 5)( 5) Multiply 5 with itself three times. = 15 The answer is 15. Practice 8: Evaluate ( ) Answer:

55 CCBC Math 081 Exponents and Roots Section Pages Example 9: Evaluate 6 This is read, The opposite of to the sixth power. 6 The base is and the exponent is 6. ( ) Multiply with itself six times to get 64. (64) Take the opposite of The answer is 64. Practice 9: Evaluate 4 Answer: Example 10: Evaluate 6 ( ) This is read, Negative to the sixth power. 6 ( ) The base is since it is in parentheses. The exponent is 6. ( )( )( )( )( )( ) Multiply with itself six times to get 64. = 64 The answer is 64. Practice 10: Evaluate 4 ( ) Answer: 16 Example 11: Evaluate The base is 1 and the exponent is 1. 1 Since the exponent is 1, there is only one 1. Practice 11: Evaluate 1 45 Answer: 45 Example 1: Evaluate The base is 0 and the exponent is Multiply 0 with itself five times. = 0 Remember that the product of any number and 0 is 0. Practice 1: Evaluate 4 0 Answer:

56 CCBC Math 081 Exponents and Roots Section Pages PROPERTY OF THE ZERO POWER a0 1 Any number (except 0) raised to the 0 power is 1. You will learn the proof of this in a future math course. Example 1: Evaluate The base is 6 and the exponent is 0. = 1 Any number (except for 0) raised to the 0 power is 1. Practice 1: Evaluate 0 5 Answer: 1 Roots An operation related to exponents is taking the root of a number. You will be working with expressions that look like 81 or 7. The box below reviews some basic vocabulary and explains the meaning of the notation. ROOT NOTATION Radical Sign Index n a Radicand n a means the number that when raised to the power n, gives a. In other notation,? n a Note: When there is no index shown, it is understood to be. 56

57 CCBC Math 081 Exponents and Roots Section Pages You should try to become familiar with the perfect squares, cubes, and fourths that are listed in the chart below. This will help you to simplify roots. Perfect Squares Perfect Cubes Perfect Fourths Example 14: Evaluate 9 9 This is read, The square root of 9. We need to know what number can be raised to the nd power to get 9. Or, in symbols, The answer is because So, What number would you put in the box? 9. Practice 14: Evaluate 16 Answer: 4 Example 15: Evaluate 15 This is read, The cube root of 15. We need to know what number can be raised to the rd power to get 15. Or, in symbols, The answer is 5 because So, What number would you put in the box? Practice 15: Evaluate 7 Answer: 57

58 CCBC Math 081 Exponents and Roots Section Pages Example 16: Evaluate 169 Remember that since there is no index shown, it is understood to be. So, 169 means 169 which is read The square root of 169. We need to know what number can be raised to the nd power to get 169. Or, in symbols, The answer is 1 because So, What number would you put in the box? 1 (1)(1) 169. Practice 16: Evaluate 5 Answer: 15 Example 17: Evaluate 4 81 This is read, The fourth root of 81. We need to know what number can be raised to the 4 th power to get 81. Or, in symbols, The answer is because So, What number would you put in the box? Practice 17: Evaluate 4 56 Answer: 4 Watch All: 58

59 CCBC Math 081 Exponents and Roots Section Pages 1.6 Exponent Exercises Evaluate each of the following ( ) ( 4) ( 8) ( 1) ( )

60 CCBC Math 081 Exponents and Roots Section Pages 1.6 Exponent Exercises Answers ,

61 CCBC Math 081 Order of Operations Section Pages 1.7 Order of Operations Now you know how to perform all the operations addition, subtraction, multiplication, division, exponents, and roots. But what if we have a problem that contains more than one operation? For instance, what if we have the problem 5 4? What operation should we perform first the addition, the exponent, or the multiplication? It is important that everyone do the problem the same way in order to get the same answer. For this reason, mathematicians developed a set of rules for evaluating problems that involve more than one arithmetic operation. The rules, called the Order of Operations, specify the order in which the computations should be performed. The Order of Operations is given below. It is important to follow these rules, one step at a time, in the order in which they are presented. ORDER OF OPERATIONS Step 1: Parentheses If there are any operations in parentheses, those computations should be performed first. Step : Exponents and Roots Simplify any numbers being raised to a power and any numbers under the symbol. Step : Multiplication and Division Do these two operations in the order in which they appear, working from left to right. Step 4: Addition and Subtraction Do these two operations in the order in which they appear, working from left to right. To help remember the Order of Operations, try using the phrase in the box on the left below. Please Excuse My Dear Parentheses Exponents and Roots Multiplication and Division... working from left to right Aunt Sally Addition and Subtraction... working from left to right 61