Course Learning Outcomes for Unit I. Reading Assignment. Unit Lesson. UNIT I STUDY GUIDE Number Theory and the Real Number System

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1 UNIT I STUDY GUIDE Number Theory and the Real Number System Course Learning Outcomes for Unit I Upon completion of this unit, students should be able to: 2. Relate number theory, integer computation, and rational number concepts to problem solving applications. 2.1 Apply divisibility tests to determine whether numbers are prime or composite. 2.2 Determine the greatest common divisor (GCD) and least common multiple (LCM) between two sets of numbers. 2.3 Perform computations involving rational numbers. 2.4 Perform computations involving integers. Reading Assignment Chapter 6: Number Theory and the Real Number System: Understanding the Numbers All Around Us Section 6.1: Number Theory, pp Section 6.2: The Integers, pp Section 6.3: The Rational Numbers, pp Unit Lesson Chapter 6: Number Theory and the Real Number System Numbers are all around us. Your earliest memory of numbers might include learning to count as a young child. You probably learned how to count whole numbers and then tried skipping counting. Eventually, numbers were used to calculate addition, multiplication, or division problems. In this section, we will learn about natural numbers, or counting numbers. 6.1 Number Theory: Number theory is the study of natural numbers. Natural numbers are counting numbers (1, 2, 3, 4, 5, ). These numbers are whole numbers and do not include zero. For example, numbers such as 2.5, 0, and are not counting numbers. Prime Numbers Counting numbers can be written as the product of other numbers. Recall that the product is the answer to a multiplication problem. Therefore, the number 15 can be written as a product of 3 and 5. The diagram below identifies some key terms that will be used when learning about numbers. MAT 1301, Liberal Arts Math 1

2 Multiplication problems can be rewritten as division problems. For example, the UNIT problem x STUDY 3 5 = GUIDE 15 can be rewritten as the following division problems: As shown, the factors (3 and 5) are divisors of the product (15). We say that a divisor divides the product. For example, 3 and 5 both divide 15. We can also say that 3 and 5 divide 15, because 15 is divisible by 3 and 5. This means that 3 and 5 divide 15 evenly, without any remainders. Determine whether each statement is true. a) 8 divides 56. b) 27 is a multiple of 6. c) 9 is a factor of 96. d) 14 divides 42. a) True: 56 8 = 7. Therefore, 56 is divisible by 8 with a remainder of zero. b) False: 27 6 = 4.5. Therefore, 27 divided by 6 has a remainder other than zero. c) False: 96 9 = Therefore, 9 cannot be multiplied by any natural number to equal 96 and is not a factor of 96. d) True: = 3; Therefore, 42 is divisible by 14 with a remainder of zero. We will use the definition of factor, product, and divisor to help us understand prime numbers. Prime Number A natural number greater than 1 that has only itself and 1 as factors is called a prime number. Composite Number A natural number greater than 1 that is not a prime number is called a composite number. Note: All numbers are divisible by 1 and itself. Composite numbers are divisible by additional numbers. As stated above, prime numbers only have a factor of 1 and itself. This means that it is not divisible by any other number. To determine if a number is prime, we need to check to see if any other digit besides 1 can evenly be divided into the number. Determine if each number is prime or composite. (Note: if a number is not prime, it is composite.) a) 21 b) 30 c) 7 MAT 1301, Liberal Arts Math 2

3 a) Composite: 21 is composite because it is divisible by 3 and 7. b) Composite: 30 is composite because it is divisible by 2, 3, 5, 6, 10, and 15. c) Prime: 7 is prime because it is only divisible by 1 and 7. Divisibility Tests We will be asked to determine if a large number is prime or composite. To do this, we need to go through the entire set of counting numbers to determine if another number other than 1 and itself divides that number. For example, if we wanted to determine if 743 was prime, we would first ask ourselves if it was divisible by 2. If it is, then we would stop because we know that it is not prime or composite. Since 743 is not divisible by 2, we need to ask ourselves if it is divisible by 3. It is not. Next, we would ask ourselves if 743 was divisible by 4. We continue testing the counting numbers until we found at least one factor. If we could not find a factor, then the number would be prime. The process described above is very time consuming. Therefore, mathematicians have created a series of divisibility tests to determine if some numbers are divisible by others. The chart below describes tests for identifying if numbers are divisible by 2, 3, 4, 5, 6, 8, 9, and 10. This may also be found on page 236 of your textbook. Check 141,270 for divisibility by each of these numbers: 2, 3, 4, 5, 6, 8, 9, and 10. State the numbers that divide the given number. 141,270 is: Divisibility tests for some small numbers (Pirnot, 2014, p. 236.) Divisible by 2 because the last digit, 0, is divisible by 2. Divisible by 3 because the sum of the digits, 15, is divisible by 3. Not divisible by 4 because the number formed by the last two digits, 70, is not divisible by 4. Divisible by 5 because the last digit, 0, is a 0 or a 5. MAT 1301, Liberal Arts Math 3

4 Divisible by 6 because the number is divisible by both 2 and 3. Not divisible by 8 because the number formed by the last three digits, 270, is not divisible by 8. Not divisible by 9 because the sum of the digits, 15, is not divisible by 9. Divisible by 10 because the last digit, 0, is 0. Factoring In this section, we will learn how to factor a number into a set of prime numbers. For example, 20 = The factors (2, 2, 5) are all prime numbers. We will factor other numbers into a set of prime numbers using the fundamental theorem of arithmetic. The Fundamental Theorem of Arithmetic Every natural number greater than 1 is a unique product of prime numbers, except for the order of the factors. There are several ways to factor a number into a set of prime numbers. The next example explains how to do this using factor trees. Factor 621. We will use a factor tree to factor the number Identify one number that divides 621 evenly. 2. Divide 621 by 3 and find two factors. 621 is divisible by 3. Therefore, 3 divides 621. This means that we have found that 3 and 207 are factors of Create the factor tree by listing the two factors found in step Continue steps one through three for all composite factors in the tree: MAT 1301, Liberal Arts Math 4

5 5. Write all the prime factors that you found in step 4. These factors, when multiplied together equal 621. Therefore, the number 621 has factors 3, 3, 3, and 23. Greatest Common Divisors Recall that divisors divide a number. These are also called factors. We will learn how to find the greatest common divisor between two numbers. Greatest Common Divisor (GCD) The greatest common divisor, or GCD, is the largest divisor that divides two numbers. The GCD may be found by listing all the factors and selecting the largest factor as shown above. This method may not be realistic for larger numbers, so we will learn how to determine the GCD using prime factorization. Find the GCD of the pair of natural numbers, 60 and 72. MAT 1301, Liberal Arts Math 5

6 First, factor 60 and 72 using factor trees: Next, write the prime factors for 60 and 72 in increasing order: 60 = = Now, list the factors they have in common: (Also, list the common number of times the factor appears.) Both factors have two 2s in common. We write this as 2 2. Both factors have one 3 in common. We write this as 3 1 or 3. Lastly, multiply the common factors together: = =12 The GCD is 12. Next, we will discuss the least common multiple of two numbers. Least Common Multiple (LCM) The least common multiple or LCM of two natural numbers is the smallest natural number that is a multiple of both numbers. The LCM may also be found using prime factorization. The next example uses prime factorization to solve. Find the LCM of the pair of natural numbers, 60 and 72. MAT 1301, Liberal Arts Math 6

7 First, factor 60 and 72 using factor trees: Next, write the prime factors for 60 and 72 in increasing order: 60 = = Now, choose one number (either 60 or 72) and write down all of its factors: 60 = From here, determine what factors 72 has that 60 did not have. 72 has one more 2 and one more 3 than 60. Therefore, we will take all the factors of 60 and include one more 2 and one more 3 in the set: Multiply the factors above to get the LCM: Applying the GCD and LCM LCM = LCM = LCM = 360 It is important to know when the GCD is used and when the LCM is used. The GCD may be used when we need to take several larger objects and represent them as a collection of smaller objects. The LCM would be used when we have two objects and need to make them larger to be of equal amounts. A sporting goods store has 20 instructional DVDs on skiing and 12 DVDs on snowboarding. The owner of the store wants to display the DVDs on a shelf with stacks of the same size and each stack consisting of only one type of DVD. What is the most number of DVDs in each stack that will accomplish this? We are only allotted 20 skiing DVDs and 12 snowboarding DVDs. Therefore, we will need to find the GCD of 12 and 20. MAT 1301, Liberal Arts Math 7

8 The prime factorization of 12 and 20 is as follows: 12 = = = = Both numbers have two 2s (2 2 ) in common. So, the GCD is 2 2 = 2 2 = 4. Hence, the stacks will have a maximum of 4 videos each. 6.2 Integers: The set of integers consists of negative and positive whole numbers including zero. For example, the numbers 123, 7, -1287, and -12 are all integers. Fractions and decimals are not integers. In this section, we will learn how to add, subtract, multiply, and divide integers. Adding and Subtracting Integers It is important to pay attention to the negative (-) or positive sign (+) when adding or subtracting integers. The easiest way to understand how to use the signs to help us with the operation is by visualizing a number line. An example of a number line is illustrated below. You will see that a zero separates the negative and positive numbers. The negative numbers are to the left of the zero and the positive numbers are to the right of the zero. Calculate the sum using movements on the number line. When adding (-9) + (-4), we will start at zero and go 9 units to the left. From this position, we move 4 more units to the left. Our ending position, (-13), is our answer. This is visually shown by the number line below: MAT 1301, Liberal Arts Math 8

9 Calculate the sum using movements on the number line. When adding (+4) + (6), we will start at zero and go 4 units to the right. From this position, we move 6 more units to the right. Our ending position, (10), is our answer. This is visually shown by the number line below: Note: An integer is positive if it does not have a (-) sign next to it. Therefore, (+6) and (6) are the same number. Calculate the sum using movements on the number line. When adding (- 7) + (3), we will start at zero and go 7 units to the left. From this position, we move 3 more units to the right. Our ending position, (- 4), is our answer. This is visually shown by the number line below: Not all integer calculations can be done on a number line. Read the next example to learn how to add integers without a number line. Calculate the sum. It is not practical to draw a number line when calculating with such large numbers; however, you can still imagine movements along the number line. If you visualize moving 57 units to the right and then 38 units to the left, you can see that the net effect is that you moved = 19 units to the right. Therefore: MAT 1301, Liberal Arts Math 9

10 Next, we will learn how to subtract integers. We can change any subtraction problem UNIT x STUDY into an addition GUIDE by using the following mathematical theorem: Definition If a and b are integers, then a b = a + (- b). For example, 9 10 = 9 + (- 10) = - 1. Notice that we changed the subtraction sign to an addition sign and then took the opposite of 10, which is (-10) to write the problem. Rewrite the following subtraction problem as an addition problem and then find the answer: Rewrite the following subtraction problem as an addition problem and then find the answer: Rewrite the following subtraction problem as an addition problem and then find the answer: MAT 1301, Liberal Arts Math 10

11 Rewrite the following subtraction problem as an addition problem and then find the answer: Multiplying and Dividing Integers In section 6.1 above, we saw that multiplication problems can be written as division and vice versa. This means that multiplication and division are related. Therefore, the same rules apply for both operations when multiplying and dividing integers. Rules for Multiplying Integers If a and b are integers, then: a) If a and b have the same sign, then a b is positive. 4 6 = 24 AND (-4) (-6) = 24 b) If a and b have opposite signs, then a b is negative. (-4) 6 = (-24) AND 4 (-6) = (-2) Rules for Dividing Integers If a and b are integers and b 0, then: a) If a and b have the same sign, then a is positive. b 24 4 = 6 AND 24 6 = 4 b) If a and b have opposite signs, then a is negative. b (-24) 4 = (-6) AND 24 (-4) = (-6) MAT 1301, Liberal Arts Math 11

12 Find the products: a) (+5)(+3) b) (-3)(-9) c) (-6)(+7) a) Two positive numbers multiplied together gives a positive result. b) Two negative numbers multiplied together gives a positive result. c) A negative number multiplied by a positive number gives a negative result. Divide: a) b) c) a) A positive number divided by a positive number gives a positive result. b) A positive number divided by a negative number gives a negative result. c) A negative number divided by a negative number gives a positive result. We will use what we have learned about integers to perform more complex operations. Perform the following calculation: MAT 1301, Liberal Arts Math 12

13 To solve, we will multiply each set of integers and then add them together. Simplify the expression: To solve, simplify the numerator and denominator of the fraction and then multiply by (-4). Simplify the expression: To solve, simplify the numerator and denominator of each fraction. Then multiply the simplified fractions. MAT 1301, Liberal Arts Math 13

$6.3 The Rational Numbers: In this section, we will learn about another set of UNIT numbers x STUDY called GUIDE rational numbers. A rational number is any number that can be expressed as a fraction.$

14 6.3 The Rational Numbers: In this section, we will learn about another set of UNIT numbers x STUDY called GUIDE rational numbers. A rational number is any number that can be expressed as a fraction. The diagram below identifies some key terms about fractions and rational numbers. Numbers such as are rational numbers. The set of integers (negative and positive whole numbers) are also rational numbers because they can be expressed as fractions. For example, 0, 119, and - 67 can all be expressed as a fraction by placing a one as their denominator. That is, they can be rewritten in the form shown below: Equality of Rational Numbers Imagine that you have two whole pies. One pie is cut to form two equal slices, and the other pie is cut to form four equal slices. Suppose that we take one slice from the first pie and two slices from the second pie. The illustration below shows the fraction that represents this scenario. We can see that the same amount of pie was taken from pie one and pie two; however, the representing fractions are different. Rational numbers sometimes look different, but are equal. We say that two rational numbers are equal if their cross product is equal. Equality of Rational Numbers This is known as cross multiplication. MAT 1301, Liberal Arts Math 14

15 Use cross multiplication to determine if: We will cross multiply to determine if the rational numbers are equal. First, multiply 7 and 24. Then, multiply 8 and 21. The rational numbers are equal if the products are the same. Determine if the pair of rational numbers is equal: We will cross multiply to determine if the rational numbers are equal. The two rational numbers are NOT equal. Reducing Rational Numbers Rational numbers can be reduced or expressed in lowest terms when the numerator and denominator have a factor in common. Please see the example below. MAT 1301, Liberal Arts Math 15

16 Reduce the fraction: Reduce the fraction: MAT 1301, Liberal Arts Math 16

17 Adding and Subtracting Rational Numbers Two numbers must have the same denominator when adding or subtracting. For example, A common denominator must be found in order to add or subtract fractions. Adding and Subtracting Rational Numbers (Fractions) Find: Express your answer in reduced form (lowest terms). Use the following formula: We can also add fractions by finding the LCM, or least common multiple. This concept was introduced in section 6.1. We will use the LCM when adding and subtracting fractions in the examples that follow. Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. First, find the least common multiple (LCM) of the denominators. MAT 1301, Liberal Arts Math 17

18 The multiples of 6 and 2 are: 6: 6, 12, 18, 24, 2: 2, 4, 6, 8, 10, The LCM is 6. Next, rewrite the fractions so that each has a denominator of 6. Now, we will replace the fractions with those found above and subtract. Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. MAT 1301, Liberal Arts Math 18

19 First, subtract the first two fractions in the problem. The LCM of 9 and 27 is 27. Therefore, Next, add the fraction we just found, 4 and 1 together. The LCM of 27 and 4 is 108. Therefore, 27 4 Multiplying and Dividing Rational Numbers We do not need to find a common denominator when multiplying and dividing fractions. To multiply a fraction, multiply the numerators together and then multiply the denominators together. Reduce the fraction to find your final answer. Multiplying Rational Numbers Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. We can solve this two ways. The first way we will solve this problem is by multiplying the numerators and denominators together and reducing our answer. MAT 1301, Liberal Arts Math 19

20 The second way we can solve this problem is by canceling factors before we multiply UNIT x STUDY the numerator GUIDE and denominators together. We divide two fractions by changing the division sign to a multiplication sign and flipping the second fraction. Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. When dividing by a number, we can choose to multiply by the reciprocal of that number instead. The following examples combine multiplication and division operations. Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. MAT 1301, Liberal Arts Math 20

21 Perform the following operation. Express your answer as a positive or negative quotient of two integers in reduced form. Mixed Numbers An improper fraction is a fraction whose denominator is smaller than its numerator. Improper fractions may be changed into mixed numbers by dividing the denominator into the numerator. A mixed number consists of a whole number and a fraction. Numbers such as the ones below are mixed numbers: Convert the improper fraction to a mixed number: First, we will perform long division as follows: The answer will then be written as an improper fraction: MAT 1301, Liberal Arts Math 21

22 Reduce the fractional part of the mixed number to get the final answer: Mixed numbers may be converted to improper fractions by multiplying the denominator and whole number together and then adding the numerator. Converting a Mixed Number to an Improper Fraction The mixed number q r b Equals the improper fraction q b+r b. Convert the mixed number to an improper fraction: Multiply the whole number by the denominator, add that result to the numerator, and place over the denominator. Repeating Decimals Decimals can either be described as terminating or repeating. A terminating decimal is a decimal that ends. For example, or are terminating decimals. A repeating decimal never ends and repeats digits. For example, is a repeating decimal. We signify that a decimal is repeating by placing a bar over the numbers that repeat. So, = Write the rational number as a terminating or repeating decimal: MAT 1301, Liberal Arts Math 22

23 We will need to perform long division as follows: The result is a terminating decimal, Write the rational number as a terminating or repeating decimal: We will need to perform long division as follows: If we continued the long division process, we would see that 81 repeats continuously. Therefore, the answer is a repeating decimal: Next, we will earn how to convert a decimal into a fraction. To do this, it is important to review the place values of a decimal. MAT 1301, Liberal Arts Math 23

24 Write the decimal as a quotient of two integers in reduced form: This number is read as twelve and two tenths. We can write this as: This can now be converted from a mixed number to an improper fraction, then simplified. Write the decimal as a quotient of two integers in reduced form: x = 0.38 This is a repeating decimal. We will use a technique of creating another number that has the same tail as x. From there we will subtract the two numbers to obtain a number that does not have a repeating infinite tail. First, multiply x by 10 as follows: MAT 1301, Liberal Arts Math 24

25 Next, subtract the original value from this, solve for x, and reduce if necessary: Reference Pirnot, T. L. (2014). Mathematics all around (5th ed.). Boston, MA: Pearson. MAT 1301, Liberal Arts Math 25

COMPETENCY 1.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY

COMPETENCY 1.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY SUBAREA I. NUMBERS AND OPERATIONS COMPETENCY.0 UNDERSTAND THE STRUCTURE OF THE BASE TEN NUMERATION SYSTEM AND NUMBER THEORY Skill. Analyze the structure of the base ten number system (e.g., decimal and