Slide 1 / 237 Integers Integer Unit Topics Integer Definition Click on the topic to go to that section Absolute Value Comparing and Ordering Integers Integer Addition Turning Subtraction Into Addition Adding and Subtracting Integers Review Multiplying Integers Dividing Integers Powers of Integers Rules for Exponents Slide 2 / 237 Slide 3 / 237 Integer Definition Return to Table of Contents
1 Do you know what an integer is? Slide 4 / 237 Yes No Define Integer Slide 5 / 237 Definition of Integer: The set of natural numbers, their opposites and zero. Examples of Integer: {...-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...} Classify each number as an integer, or not. Slide 6 / 237 5 ½ -21-65 1 3.2 0-6.32 2.34437 x 10 3 3¾ 9 π 5 ¾ integer not an integer -6
Integers on the number line Slide 7 / 237 Negative Integers Zero Positive Integers -5-4 -3-2 -1 0 1 2 3 4 5 Numbers to the Numbers to the left of zero are Zero is neither right of zero are less than zero positive or negative greater than zero ` 2 Which of the following are examples of integers? Slide 8 / 237 A 0 B -8 C -4.5 D 7 E 1/2 3 Which of the following are examples of integers? Slide 9 / 237 A 1/2 B 6 C -4 D 0.75 E 25%
Integers can represent everyday situations Slide 10 / 237 You might hear "And the quarterback is sacked for a loss of 7 yards." This can be represented as an integer: -7 Or, "The total snow fall this year has been 9 inches more than normal." This can be represented as in integer: +9 or 9 Integers In Our World Slide 11 / 237 Write an integer to represent each situation: Slide 12 / 237 1. Spending $6 2. Gain of 11 pounds 3. Depositing $700 4. 10 degrees below zero 5. 8 strokes under par (par = 0) 6. 350 feet above sea level
4 Which of the following integers best represents the following scenario: Slide 13 / 237 The effect on your wallet when you spend 10 dollars. A -10 B 10 C 0 D +/- 10 5 Which of the following integers best represents the following scenario: Slide 14 / 237 Earning $40 shoveling snow. A -40 B 40 C 0 D +/- 40 6 Which of the following integers best represents the following scenario: Slide 15 / 237 You dive 35 feet to explore a sunken ship. A -35 B 35 C 0 D +/- 35
The numbers -4 and 4 are shown on the number line. Slide 16 / 237-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 Both numbers are 4 units from 0, but 4 is to the right of 0 and -4 is to the left of zero. The numbers -4 and 4 are opposites. Opposites are the same distance from zero. 7 What is the opposite of -7? Slide 17 / 237 8 What is the opposite of 18? Slide 18 / 237
Integers are used in game shows. Slide 19 / 237 In the game of Jeopardy you: gain points for a correct response lose points for an incorrect response and can have a positive or negative score Slide 20 / 237 When a contestant gets a $200 question correct: Score = $200 Then a $100 question incorrect: Score = $100 Then a $300 question incorrect: Score = - $200 How did the score become negative? 9 After the following 3 responses what would the contestants score be? Slide 21 / 237 $100 incorrect $200 correct $50 incorrect
10 After the following 3 responses what would the contestants score be? Slide 22 / 237 $200 correct $50 correct $300 incorrect 11 After the following 3 responses what would the contestants score be? Slide 23 / 237 $150 incorrect $50 correct $100 correct To Review Slide 24 / 237 An integer is a natural number, zero or its opposite. Number lines have negative numbers to the left of zero and positive numbers to the right. Zero is neither positive nor negative Integers can represent real life situations
Slide 25 / 237 Absolute Value Return to Table of Contents Absolute Value of Integers Slide 26 / 237 The absolute value is the distance a number is from zero on the number line, regardless of direction. Distance and absolute value are always nonnegative (positive or zero). -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 What is the distance from 0 to 5? Absolute Value of Integers The absolute value is the distance a number is from zero on the number line, regardless of direction. Slide 27 / 237 Distance and absolute value are always nonnegative. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 What is the distance from 0 to -5?
Absolute value is symbolized by two vertical bars Slide 28 / 237 4 This is read, "the absolute value of 4" -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 What is the 4? Use the number line to find absolute value. Slide 29 / 237 9 = 9 Move to check Move to check -9 = 9-4 = 4 Move to check -10-9 -8-7 -6-5-4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 12 Find -7 Slide 30 / 237
13 Find -28 Slide 31 / 237 14 What is 56? Slide 32 / 237 15 Find -8 Slide 33 / 237
16 Find 3 Slide 34 / 237 17 What is the absolute value of the number shown in the generator? Slide 35 / 237 18 Which numbers have 15 as their absolute value? Slide 36 / 237 A -30 B -15 C 0 D 15 E 30
19 Which numbers have 100 as their absolute value? Slide 37 / 237 A -100 B -50 C 0 D 50 E 100 Slide 38 / 237 Comparing and Ordering Integers Return to Table of Contents Comparing Positive Integers Slide 39 / 237 An integer can be equal to; less than ; or greater than another integer. The symbols that we use are: Equals "=" Less than "<" Greater than ">" For example: 4 = 4 4 < 6 4 > 2 When using < or >, remember that the smaller side points at the smaller number.
20 The integer 8 is 9. Slide 40 / 237 A = B < C > 21 The integer 7 is 7. Slide 41 / 237 A = B < C > 22 The integer 3 is 5. Slide 42 / 237 A = B < C >
Use the Number Line Slide 43 / 237 To compare integers, plot points on the number line. The numbers farther to the right are greater. The numbers farther to the left are lesser. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Place the number tiles in the correct places on the number line. Slide 44 / 237-2 3 1 0-1 2-5 -3 4 5-4 Now, can you see: Slide 45 / 237 Which integer is greater? Which is lesser? -5-4 -3-2 -1 0 1 2 3 4 5 click to reveal
Put these integers on the number line. Slide 46 / 237-3 -1 0 4-2 -7 2-5 5 Now, can you see: Slide 47 / 237 Which integer is greater? Which is lesser? -7-5 -3-2 -1 0 2 4 5 click to reveal Comparing Negative Integers Slide 48 / 237 The greater the absolute value of a negative integer, the smaller the integer. That's because it is farther from zero, but in the negative direction. For example: -4 = -4-4 > -6-4 < -2-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10
Comparing Negative Integers Slide 49 / 237 One way to think of this is in terms of money. You'd rather have $20 than $10. But you'd rather owe someone $10 than $20. Owing money can be thought of as having a negative amount of money, since you need to get that much money back just to get to zero. 23 The integer -4 is -4. Slide 50 / 237 A = B < C > -10-9 -8-7 -6-5-4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 24 The integer -4 is -5. Slide 51 / 237 A = B < C > -10-9 -8-7 -6-5-4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10
25 The integer -20 is -14. Slide 52 / 237 A = B < C > 26 The integer -14 is -6. Slide 53 / 237 A = B < C > Comparing All Integers Slide 54 / 237 Any positive number is greater than zero and any negative number. Any negative number is less than zero and any positive number.
27 The integer -4 is 6. Slide 55 / 237 A = B < C > 28 The integer -3 is 0. Slide 56 / 237 A = B < C > 29 The integer 5 is 0. Slide 57 / 237 A = B < C >
30 The integer -4 is -9. Slide 58 / 237 A = B < C > 31 The integer 1 is -54. Slide 59 / 237 A = B < C > 32 The integer -480 is 0. Slide 60 / 237 A = B < C >
Drag the appropriate inequality symbol between the following pairs of integers: 1) -3 5 < > 2) -237-259 Slide 61 / 237 3) 63 36 4) -10-15 5) -6-3 6) 127 172 7) -24-17 8) -2-8 9) 8-8 10) -10-7 A thermometer can be thought of as a vertical number line. Positive numbers are above zero and negative numbers are below zero. Slide 62 / 237 33 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: Slide 63 / 237 falls 3 degrees?
34 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: Slide 64 / 237 rises 5 degrees? 35 If the temperature reading on a thermometer is 10, what will the new reading be if the temperature: Slide 65 / 237 falls 12 degrees? 36 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: Slide 66 / 237 falls 3 degrees?
37 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: Slide 67 / 237 rises 5 degrees? 38 If the temperature reading on a thermometer is -3, what will the new reading be if the temperature: Slide 68 / 237 falls 12 degrees? Slide 69 / 237 Integer Addition Return to Table of Contents
Symbols Slide 70 / 237 We will use "+" to indicate addition and "-" for subtraction. Parentheses will also be used to show things more clearly. For instance, if we want to add -3 to 4 we will write: 4 + (-3), which is clearer than 4 + -3. Or if we want to subtract -4 from -5 we will write: -5 - (-4), which is clearer than -5 - -4. Integer Addition: A walk on the number line. Slide 71 / 237 While this section is titled "Integer Addition" we're going to learn here how to both add and subtract integers using the number line. Addition and subtraction are inverse operations (they have the opposite effect). If you add a number and then subtract the same number you haven't changed anything. Addition undoes subtraction, and vice versa. Integer Addition: A walk on the number line. Slide 72 / 237 Here's how it works. 1. Start at zero 2. Walk the number of steps indicated by the first number. -Go to the right for positive numbers -Go to the left for negative numbers 3. Walk the number of steps given by the second number. 4. Look down, you're standing on the answer.
Let's do 3 + 4 on the number line. Slide 73 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Let's do 3 + 4 on the number line. 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. Slide 74 / 237-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 Go to the right for positive numbers Let's do 3 + 4 on the number line. Slide 75 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the right for positive numbers
Let's do -4 + (-5) on the number line. Slide 76 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Let's do -4 + (-5) on the number line. Slide 77 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the left for negative numbers Let's do -4 + (-5) on the number line. Slide 78 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the left for negative numbers
Let's do -4 + 9 on the number line. Slide 79 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Let's do -4 + 9 on the number line. Slide 80 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the left for negative numbers Let's do -4 + 9 on the number line. Slide 81 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the right for positive numbers
Let's do 5 + (-7) on the number line. 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. Slide 82 / 237-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 Let's do 5 + (-7) on the number line. Slide 83 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the right for positive numbers Let's do 5 + (-7) on the number line. Slide 84 / 237 1. Start at zero 2. Walk the number of steps indicated by the first number. 3. Take the number of steps given by the second number. 4. Look down, you're standing on the answer. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Go to the left for negative numbers
Integer Addition: Using Absolute Values Slide 85 / 237 You can always add using the number line. But if we study our results, we can see how to get the same answers without having to draw the number line. We'll get the same answers, but more easily. Integer Addition: Using Absolute Values Slide 86 / 237 3 + 4 = 7-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10-4 + (-5) = -9 1-10-9-8 -7-6 -5-4 -3-2 -1 0 2 3 4 5 6 7 8 9 10-4 + 9 = 5 5 + (-7) = -2-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 We can see some patterns here that allow us to create rules to get these answers without drawing. Integer Addition: Using Absolute Values Slide 87 / 237 To add integers with the same sign 1. Add the absolute value of the integers. 2. The sign stays the same. 3 + 4 = 7-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 3 + 4 = 7; both signs are positive; so 3 + 4 = 7-4 + (-5) = -9 1-10-9-8 -7-6 -5-4 -3-2 -1 0 2 3 4 5 6 7 8 9 10 4 + 5 = 9; both signs are negative; so -4 + (-5) = -9
Interpreting the Absolute Value Approach Slide 88 / 237 The reason the absolute value approach works, if the signs of the integers are the same, is: The absolute value is the distance you travel in a direction, positive or negative. If both numbers have the same sign, the distances will add together, since they're both asking you to travel in the same direction. If you walk one mile west and then two miles west, you'll be three miles west of where you started. Integer Addition: Using Absolute Values Slide 89 / 237 To add integers with different signs 1. Find the difference of the absolute values of the integers. 2. Keep the sign of the integer with the greater absolute value. -4 + 9 = 5-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 9-4 = 5; 9 > 4, and 9 is positive; so -4 + 9 = 5 5 + (-7) = -2-10-9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 7-5 = 2; 7 > 5 and 9 is negative; so 5 + (-7) = -2 Interpreting the Absolute Value Approach Slide 90 / 237 If the signs of the integers are different: For the 2 nd leg of your trip you're traveling in the opposite direction of the 1 st leg, undoing some of your original travel. The total distance you are from your starting point will be the difference between the two distances. The sign of the answer must be the same as that of the larger number, since that's the direction you traveled farther. If you walk one mile west and then two miles east, you'll be one mile east of where you started.
39 11 + (-4) = Slide 91 / 237 40-9 + (-2) = Slide 92 / 237 41 5 + (-8) = Slide 93 / 237
42 4 + (12) = Slide 94 / 237 43-14 + 7 = Slide 95 / 237 44-4 + (-4) = Slide 96 / 237
45-5 + 10 = Slide 97 / 237 Slide 98 / 237 Turning Subtraction Into Addition Return to Table of Contents Subtracting Integers Subtracting a number is the same as adding it's opposite. Slide 99 / 237 We can see this from the number line, remembering our rules for directions. Compare these two problems: 8-5 and 8 + (-5). For "8-5" we move 8 steps to the right and then move 5 steps to the left, since the negative sign tells us to move in the opposite direction that we would for +5. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 For "8 + (-5)" we move 8 steps to the right, and then 5 steps to the left since we're adding -5. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Either way, we end up at +3.
Subtracting Negative Integers Compare these two problems: 8 - (-2) and 8 + 2. Slide 100 / 237 For "8 - (-2)" we move 8 steps to the right and then move 2 steps to the right, since the negative sign tells us to move in the opposite direction that we would for -2. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 For "8 + 2" we move 8 steps to the right, and then 2 steps to the right since we're adding 2. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 Either way, we end up at +10. Subtracting Integers Slide 101 / 237 Any subtraction can be turned into addition by: Changing the subtraction sign to addition. Changing the integer after the subtraction sign to its opposite. EXAMPLES: 5 - (-3) is the same as 5 + 3-12 - 17 is the same as -12 + (-17) 46 Convert the subtraction problem into an addition problem. Slide 102 / 237 8 4 A -8 + 4 B 8 + (-4) C -8 + (-4) D 8 + 4
47 Convert the subtraction problem into an addition problem. -3 (-10) Slide 103 / 237 A -3 + 10 B 3 + (-10) C -3 + (-10) D 3 + 10 48 Convert the subtraction problem into an addition problem. -9 3 Slide 104 / 237 A -9 + 3 B 9 + (-3) C -9 + (-3) D 9 + 3 49 Convert the subtraction problem into an addition problem. 6 (-2) Slide 105 / 237 A -6 + 2 B 6 + (-2) C -6 + (-2) D 6 + 2
50 Convert the subtraction problem into an addition problem. Slide 106 / 237 A -1 + 9 1-9 B 1 + (-9) C -1 + (-9) D 1 + 9 Slide 107 / 237 Adding and Subtracting Integers Review Return to Table of Contents 51 Calculate the sum or difference. -6 2 Slide 108 / 237
52 Calculate the sum or difference. -10 + 6 Slide 109 / 237 53 Calculate the sum or difference. 7 (-4) Slide 110 / 237 54 Calculate the sum or difference. 4 7 Slide 111 / 237
55 Calculate the sum or difference. 5 + (-5) Slide 112 / 237 56 Calculate the sum or difference. 9 + (-8) Slide 113 / 237 57 Calculate the sum or difference. -4 + (-5) Slide 114 / 237
58 Calculate the sum or difference. -2 (-3) Slide 115 / 237 59 Calculate the sum or difference. -2 + 4 + (-12) Slide 116 / 237 60 Calculate the sum or difference. 5-6 + (-7) Slide 117 / 237
61 Calculate the sum or difference. -8 - (-3) + 5 Slide 118 / 237 62 Calculate the sum or difference. 16 - (-9) - 21 Slide 119 / 237 63 Calculate the sum or difference. 19 + (-12) - 11 Slide 120 / 237
Slide 121 / 237 Multiplying Integers Return to Table of Contents Symbols Slide 122 / 237 In the past, you may have used "x" to indicate multiplication. For example "3 times 4" would have been written as 3 x 4. However, that will be a problem in the future since the letter "x" is used in Algebra as a variable. There are two ways we will indicate multiplication: 3 times 4 will be written as either 3 4 or 3(4). Other commonly used symbols for multiplication include: * [ ] { } ( ) For example "3 times 4" could be written as: 3*4 3[4] 3{4} 3(4) Parentheses Slide 123 / 237 The second method of showing multiplication, 3(4), is to put the second number in parentheses. Parentheses have also been used for other purposes. When we want to add -3 to 4 we will write that as 4 + (-3), which is clearer than 4 + -3. Also, whatever operation is in parentheses is done first. The way to write that we want to subtract 4 from 6 and then divide by 2 would be (6-4) / 2 = 1. Removing the parentheses would yield 6-4/2 = 4, since we work from left to right.
Multiplying Integers Slide 124 / 237 Multiplication is just a quick way of writing multiple additions. These are all equivalent: 3 4 3 +3 + 3 + 3 4 + 4 + 4 they all equal 12. Multiplying Integers Slide 125 / 237 We know how to add with a number line. Let's just do the same thing with multiplication by just doing repeated addition. To do that, we'll start at zero and then just keep adding: either 3+3+3+3 or 4+4+4. We should get the same result either way, 12. Let's do 4 x 3 on the number line. We'll do it as 3+3+3+3 and as 4+4+4 Slide 126 / 237-3 -2-1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17
Try 5 x 2 on the number line. Slide 127 / 237 Try it as 5+5 and as 2+2+2+2+2-3 -2-1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 Multiplying Negative Integers Slide 128 / 237 Let's use the same approach to determine rules for multiplying negative integers. If we have 4 x (-3) we know we can think of that as (-3) added to itself 4 times. But we don't know how to think of adding 4 to itself -3 times, so let's just get our answer this way: 4 x (-3) = (-3)+(-3)+(-3)+(-3) 4 x (-3) On the Number Line Slide 129 / 237 4 x (-3) = (-3)+(-3)+(-3)+(-3) -17-16-15-14-13-12-11-10 -9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 So we can see that 4 x (-3) = -12
Sign Rules for Multiplying Integers Slide 130 / 237 4 3 4 + 4 + 4 12 Multiplying Positive Integers has a positive value. 4(-3) (-3) + (-3) + (-3) -12 Multiplying a negative integer and a positive integer has a negative value.? What about multiplying together two negative integers: what is the sign of (-4)(-3) Multiplying Negative Integers Slide 131 / 237 We can't add something to itself a negative number of times; we don't know what that means. But we can think of our rule from earlier, where a (-) sign tells us to reverse direction. -5 - (-2) = -5 + 2-17-16-15-14-13-12-11-10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 Multiplying Negative Integers Slide 132 / 237 So if we think of (-4)(-3) as -(4)(-3) we can then see that the answer will be the opposite of (-12): +12 Each negative sign makes us reverse direction once, so two multiplied together gets us back to the positive direction.
Sign Rules for Multiplying Integers Slide 133 / 237 4 3 4 + 4 + 4 12 4(-3) (-4) + (-4) + (-4) -12 Multiplying Positive Integers yields a positive result. Multiplying a negative integer and a positive integer yields a negative result. (-4)(-3) -((-4) + (-4) + (-4)) -(-12) 12 Multiplying two negative integers together yields a positive result. Multiplying Integers Slide 134 / 237 Every time you multiply by a negative number you change the sign. Multiplying with one negative number makes the answer negative. Multiplying with a second negative change the answer back to positive. 1(-3) = -3-3(-4) = 12 Multiplying Integers Slide 135 / 237 When multiplying two numbers with the same sign (+ or -), the product is positive. When multiplying two numbers with different signs, the product is negative.
Multiplying Integers Slide 136 / 237 We can also see these rules when we look at the patterns below: 3(3) = 9-5(3) = -15 3(2) = 6-5(2) = -10 3(1) = 3-5(1) = -5 3(0) = 0-5(0) = 0 3(-1) = -3-5(-1) = 5 3(-2) = -6-5(-2) = 10 3(-3) = -9-5(-3) = 15 64 What is the value of 3 7? Slide 137 / 237 65 What is the value of 5(-4)? Slide 138 / 237
66 What is the value of -3(-6)? Slide 139 / 237 67 What is the value of (-3)(-9)? Slide 140 / 237 68 What is the value of -8 7? Slide 141 / 237
69 What is the value of -5(-9)? Slide 142 / 237 70 What is the value of 4(-2)(5)? Slide 143 / 237 71 What is the value of -2(-7)(-4)? Slide 144 / 237
72 What is the value of 6(3)(-8)? Slide 145 / 237 Slide 146 / 237 Dividing Integers Return to Table of Contents Division Symbols Slide 147 / 237 You may have mostly used the " " symbol to show division. Other common symbols include "/" and " " We will also represent division as a fraction. Remember: 9 9 3 = 3 = 3 3 are both ways to show division.
Dividing Integers Slide 148 / 237 Division is the inverse operation of multiplication, just like subtraction is the inverse of addition. When you divide an integer, by a number, you are finding out how many of that second number would have to add together to get the first number. For instance, since 5 2 = 10, that means that I could divide 10 into 5 2's, or 2 5's. This is just what we did on the number line for multiplication, but backwards. Let's try 10 2 Try 10 2 on the number line Slide 149 / 237 This means how many lengths of 2 would be needed to add up to 10. -3-2 -1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 The answer is 5: the number of red arrows of length 2 that end to end give a total length of 10. Try 10 5 on the number line Slide 150 / 237 This means how many lengths of 5 would be needed to add up to 10. -3-2 -1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 The answer is 2: the number of green arrows of length 5 that, end to end, give a total length of 10.
-12 3 On the Number Line Slide 151 / 237 This can be read as what would each step have to be if 3 of them was to take you to -12. -17-16-15-14-13-12-11-10 -9-8 -7-6 -5-4 -3-2 -1 0 1 2 3 Each red arrow represents a step of 3, so we can see that -12 3 = -4 (The answer is negative because the steps are to the left.) Dividing Integers -15 3 = -5 Slide 152 / 237-15 3 = -5 We know that -5(3) = -15, so it makes sense that -15 3 = -5. We also know 3(-5) = -15. So, what is the value of -15-5 The value must be positive 3, because 3(-5) = -15 Dividing Integers Slide 153 / 237 The quotient of two positive integers is positive The quotient of a positive and negative integer is negative. The quotient of two negative integers is positive.
73 Find the value of 32 4 Slide 154 / 237 74 Find the value of -48 12 Slide 155 / 237 75 Find the value of -27-9 Slide 156 / 237
76 Find the value of -25 5 Slide 157 / 237 77 Find the value of 0-13 Slide 158 / 237 Slide 159 / 237 Powers of Integers Return to Table of Contents
Powers of Integers Slide 160 / 237 Powers are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition. These are all equivalent: 2 4 2 2 2 2 16 In this example 2 is raised to the 4 th power. That means that 2 is multiplied by itself 4 times. Powers of Integers Slide 161 / 237 Bases and Exponents When "raising a number to a power", The number we start with is called the base, the number we raise it to is called the exponent. The entire expression is called a power. 2 4 When a number is written as a power, it is written in exponential form. If you multiply the base and simplify the answer, the number is now written in standard form. Slide 162 / 237 EXAMPLE: 3 5 = 3(3)(3)(3)(3) = 243 Power Expanded Notation Standard Form TRY THESE: 1. Write 5 3 in standard form. 2. Write 7(7)(7)(7)(7)(7)(7) as a power.
78 What is the base in this expression? 3 2 Slide 163 / 237 79 What is the exponent in this expression? 3 2 Slide 164 / 237 80 What is the base in this expression? 7 3 Slide 165 / 237
81 What is the exponent in this expression? 4 3 Slide 166 / 237 82 What is the base in this expression? 9 4 Slide 167 / 237 Squares Squares - Raising a number to the power of 2 is called squaring it. Slide 168 / 237 2 2 is two squared, and 4 is the square of 2 3 2 is three squared, and 9 is the square of 3 4 2 is four squared, and 16 is the square of 4 2 2 Area 2 x 2 = 4 units 2 Area = 3 3 x 3 = 4 9 units 2 3 Area = 4 x 4 = 16 units 2 4
Slide 169 / 237 This comes from the fact that the area of a square whose sides have length 3 is 3x3 or 3 2 = 9; The area of a square whose sides have length 5 is 5x5 or 5 2 = 25; What would the area of a square with side lengths of 6 be? Cubes Cubes - Raising a number to the power of 3 is called cubing it. Slide 170 / 237 2 3 is two cubed, and 8 is the cube of 2 3 3 is three cubed, and 27 is the cube of 3 4 3 is four cubed, and 64 is the cube of 4 That comes from the fact that the volume of a cube whose sides have length 3 is 3x3x3 or 3 3 = 27; The volume of a cube whose sides have length 5 is 5x5x5 or 5 3 = 125; etc. 2 2 = 2 x 2 Let's do 2 2 on the number line. Slide 171 / 237 Travel a distance of 2, twice -3-2 -1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17
Let's do 3 2 on the number line. 3 2 = 3 x 3 = 3 + 3 + 3 = 9 Slide 172 / 237 Travel a distance of 3, three times -3-2 -1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 Let's do 2 3 on the number line. 2 3 = 2 x 2 x 2 2 3 = (2 x 2) x 2 First, travel a distance of 2, twice: 4 2 3 = 4 x 2 = 8 Then, travel a distance of 4, twice: 8 Slide 173 / 237-3 -2-1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 Let's do 3 2 on the number line. 4 2 = 4 x 4 = 4+4+4+4 = 16 Slide 174 / 237 Travel a distance of 4, four times -3-2 -1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17
Let's do 2 4 on the number line. 2 4 = 2 x 2 x 2 x 2 2 4 = 2 x 2 x 2 x 2 First, travel a distance of 2, twice: 4 2 4 = 4 x 2 = 8 x 2 Then, travel a distance of 4, twice: 8 2 4 = 8 x 2 = 16 Then, travel a distance of 8, twice: 16 Slide 175 / 237-3 -2-1 0 1 2 3 4 5 6 7 8 9 1011 1213 14 1516 17 83 Evaluate 3 2. Slide 176 / 237 84 Evaluate 5 2. Slide 177 / 237
85 Evaluate 8 2. Slide 178 / 237 86 Evaluate 5 3. Slide 179 / 237 87 Evaluate 7 2. Slide 180 / 237
88 Evaluate 2 2. Slide 181 / 237 These are all equivalent: Slide 182 / 237 3 4 3 3 3 3 81 If we now take (-3) 4 (-3)(-3)(-3)(-3) 81 In this example we still get 81. Why? Slide 183 / 237 If we take (-3) 3 It is equivalent to (-3)(-3)(-3) Which equals -27 Why is this example negative?
89 Evaluate (-5) 2. Slide 184 / 237 90 Evalute (-2) 3. Slide 185 / 237 91 Evaluate (-4) 4. Slide 186 / 237
92 Evaluate (-7) 2. Slide 187 / 237 93 Evaluate (-3) 5. Slide 188 / 237 Remember...these are all equivalent: 3 4 (-3) 4 3 3 3 3 (-3)(-3)(-3)(-3) 81 81 In both examples we get 81. Slide 189 / 237 If we take - 3 4 We get -81. This answer is negative because - 3 4 means the opposite of 3 4. Note: When there are no parentheses, we take the exponent first and then apply the negative sign!
94 Evaluate -5 2. Slide 190 / 237 95 Evalute -2 3. Slide 191 / 237 96 Evaluate -4 4. Slide 192 / 237
97 Evaluate -7 2. Slide 193 / 237 98 Evaluate -3 5. Slide 194 / 237 Slide 195 / 237 Rules for Exponents Return to Table of Contents
Exponent Rules - Multiplying Slide 196 / 237 You can only directly multiply numbers which have exponents if the bases are the same. If the bases are the same, just add the exponents. (4 3 )(4 2 ) = 4 (3+2) = 4 5 Here's why: (4 3 )(4 2 ) = (4x4x4)(4x4) = 4x4x4x4x4 = 4 5 This will be true for any base, not just 4, so this rule always works. 99 (5 4 )(5 3 ) = 5? Slide 197 / 237 100 (4 3 )(4 2 ) = 4? Slide 198 / 237
101 (3 7 )(3 1 ) = 3? Slide 199 / 237 102 (2 4 )(2-2 ) = 2? Slide 200 / 237 103 (15 8 )(15 3 ) = 15? Slide 201 / 237
104 (6 4 )(6-8 ) = 6? Slide 202 / 237 105 (4 3 )(4 2 )(4 5 ) = 4? Slide 203 / 237 106 (2 4 )(2 5 )(2 7 ) = 2? Slide 204 / 237
107 (6 4 )(6? ) = 6 10 Slide 205 / 237 108 (5 7 )(5? ) = 5 3 Slide 206 / 237 Exponent Rules - Dividing You can only directly divide numbers which have exponents if the bases are the same. Slide 207 / 237 If the bases are the same, just subtract the exponent of the denominator from that of the numerator. 7 5 7 3 = 7 (5-3) = 7 2 7 5 = 7x7x7x7x7 = 7 2 7 3 = 7x7x7 This will be true for any base, not just 7, this rule always works.
109 5 4 5 3 = 5? Slide 208 / 237 110 7 4 7 3 = 7? Slide 209 / 237 111 9 3 9 6 = 9? Slide 210 / 237
112 3 4 3 8 = 3? Slide 211 / 237 113 8 14 8 6 = 8? Slide 212 / 237 114 8 4 8 4 = 8? Slide 213 / 237
115 (5 5 )(5 3 ) 5 2 = 5? Slide 214 / 237 116 9 8 9? = 9 5 Slide 215 / 237 117 6 6 6? = 6 13 Slide 216 / 237
118 4? 4 8 = 4 3 Slide 217 / 237 Exponent Rules - Exponent of zero Slide 218 / 237 ANY non-zero base raised to the zero power is equal to one. Consider the following pattern of powers: 2 3 = 8 2 2 = 4 2 1 = 2 2 0 = 1 _ 2 _ 2 _ 2 119 8 0 =? Slide 219 / 237
120 4 0 =? Slide 220 / 237 121 (-5) 0 =? Slide 221 / 237 122 25 0 =? Slide 222 / 237
Exponent Rules - Power of a power Slide 223 / 237 What happens when a power is raised to a power? Let's see... (3 4 ) 2 = 3 4 (3 4 ) = [3(3)(3)(3)][3(3)(3)(3)] = 3 8 (a 3 ) 4 = (a 3 )(a 3 )(a 3 )(a 3 ) = a(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a) = a 12 What do you notice? Simplify: (5 2 ) 6 Move the box to see the rule... (x m ) n = x mn 123 (3 8 ) 3 = 3? Slide 224 / 237 124 (13 9 ) 12 = 13? Slide 225 / 237
125 (2 8 ) 5 = 2? Slide 226 / 237 126 (11 4 ) 8 = 11? Slide 227 / 237 Exponent Rules - Negative Exponents Slide 228 / 237 Consider the following pattern of powers: 2 3 = 8 2 2 = 4 2 1 = 2 2 0 = 1 2-1 = 2-2 = 2-3 = _ 2 _ 2 _ 2 _ 2 _ 2 _ 2
Negative Exponents Considering the pattern of powers for any nonzero number: Slide 229 / 237 5-2 = Then, = 5 2 = 1 _ = 25 127 2-1 =? Slide 230 / 237 Give your answer as a fraction 128 1 4-1 =? Slide 231 / 237
129 10-2 =? Slide 232 / 237 130 5-1 =? Slide 233 / 237 131 1 3-3 =? Slide 234 / 237
132 2-5 =? Slide 235 / 237 133 1 5-2 =? Slide 236 / 237 134 1 9-2 =? Slide 237 / 237