UNIT III STUDY GUIDE Number Theory and the Real Number System Course Learning Outcomes for Unit III Upon completion of this unit, students should be able to: 3. Perform computations involving exponents, scientific notations, and sequences within the real number system. 3.1 Identify irrational numbers. 3.2 Apply basic rules for exponents and radicals to solve problems. 3.3 Perform operations using scientific notation. 3.4 Compute future terms of arithmetic and geometric sequences. 3.5 Apply Fibonacci sequence principles to solve problems. Reading Assignment Chapter 6: Number Theory and the Real Number System: Understanding the Numbers All Around Us Section 6.4: The Real Number System, pp. 265-274 Section 6.5: Exponents and Scientific Notation, pp. 275-284 Section 6.6: Looking Deeper: Sequences, pp. 285-294 Unit Lesson Chapter 6: Number Theory and the Real Number System Unit I introduced counting numbers, integers, and rational numbers. In this section, you will learn about other types of numbers that make up the real number system. 6.4 The Real Number System Mathematicians have categorized numbers into different sets. As shown previously, some sets of numbers have different rules when performing computations. For example, integer operations depend on the sign of the number and fraction operations require that we manipulate the denominator when adding or subtracting. Irrational Numbers Recall that rational numbers are numbers that can be expressed as a fraction. For example, whole numbers, integers, and some decimals are rational numbers. Irrational numbers are numbers that are not rational and cannot be expressed as a fraction. More specifically, irrational numbers are numbers that when written as a decimal do not repeat and do not terminate. Their decimal expansion never ends and there is no pattern to the expansion. For example, 312.458967231547485693215. is an irrational number. Is the following a rational number or irrational number? 3 8 This is a rational number because it is expressed as a fraction. MAT 1301, Liberal Arts Math 1
Is the following a rational number or is it an irrational number? 1.23456789101112 This is an irrational number because the decimal does NOT terminate or repeat. Is the following a rational number or is it an irrational number? 0.101101110 This is an irrational number because the decimal does NOT terminate or repeat. Irrational numbers are identified by mathematical symbols because they are impossible to write otherwise. For example, the numbers represented by π, e, or φ are irrational. These numbers have special symbols because they reoccur in several mathematical areas such as the study of geometry. Computing Radicals The radical,, provides another way of expressing an irrational number. A radical is sometimes called a root. This symbol is found on your calculator. An example of an irrational number expressed as a radical is 2. The diagram below provides some key terms of a radical. We will need to know these terms when performing calculations with radicals. Is the following a rational number or irrational number? 81 We need to solve the radical to determine if the 81 is rational or irrational. The radical sign undoes a perfect square. Therefore, the radical can be simplified if the radicand has two factors that are the same. We can also find the 81 by using our calculator. Answer: The 81 is a rational number, because 81 = 9 and 9 is rational number. We will discuss how to multiply, divide, and add radicals. To do this, we will need to know the following property. MAT 1301, Liberal Arts Math 2
Multiplying and Dividing Radicals If a 0 and b 0, Simplify: 27 Rewrite the radicand as two factors (9 3). We know that 9 = 3 3 = 3. Therefore, we have Simplify: 189 Rewrite the radicand as two factors (9 21). We know that 9 = 3 3 = 3. Therefore, we have Perform the indicated operation, and simplify if possible: 54 6 Perform the indicated operation, and simplify if possible: 12 15 MAT 1301, Liberal Arts Math 3
12 15 = 12 15 Rewrite the expression so that both numbers are underneath the radical. Our answer is: 6 5. Rationalizing Denominators = 4 3 3 5 Factor 12 and 15. Notice that 12 = 4 3 and 15 = 3 5 = 4 3 3 5 We know that 4 = 2. Therefore, we separate this. = 2 9 5 Place a 2 on the outside of radical because 4 = 2. = 2 9 5 Separate the radicals. = 2 3 5 We know that 9 = 3. Move the 3 to the outside. = 6 5 Multiply the numbers that are outside the radical. The 5 is irrational and cannot be simplified. Recall that the denominator is the bottom number of a fraction and that a fraction bar represents division. In this section, we will discuss the process of rationalizing the denominator. This means that we will rewrite the quotient or fraction so that it has no radicals in the denominator. Review the following example to learn the process for rationalizing the denominator. Rationalize the denominator and simplify: 4 3 1. Multiply the numerator and denominator of the fraction by the radical that is in the denominator. 4 3 = 4 3 3 3 2. Simplify the numerator and denominator by performing the indicated operations. 3. Simplify the denominator. 4 3 3 3 = 4 3 3 3 4 3 3 3 = 4 3 9 = 4 3 3 MAT 1301, Liberal Arts Math 4
Adding Expressions Containing Radicals The radicands of a radical expression must be the same in order to add or subtract them. For example, 3 + 2 cannot be added because the radicands are different. If the radicands are the same, then we keep the radicand and add the numbers outside the radical together. For example, 3 2 + 4 2 = 7 2. If possible, combine the radicals into a single radical: 2 3 4 2 This is NOT possible. The radicands are different so we cannot simplify this expression any further. We must first simplify the radical to determine if two radicals can be added or subtracted. If the simplified radicals do not have the same radicand, then we cannot perform the operation. If possible, combine the radicals into a single radical: 20 + 6 5 Simplify each radical. The 5 cannot be simplified any further, so we will simplify 20. Properties of Real Numbers The diagram below represents the set of real numbers. As shown, there are two distinct sets of real numbers: rational and irrational numbers. Rational numbers are comprised of natural numbers, whole numbers, and integers. It is also shown that the set of whole numbers includes the set of natural numbers, and the set of integers includes the set of whole and natural numbers. Chart depicting set of real numbers (Pirnot, 2014, p. 270) MAT 1301, Liberal Arts Math 5
While different, each of the sets in the diagram follows guidelines. These guidelines UNIT x are STUDY called GUIDE the properties of real numbers. The properties are listed on page 271 of your textbook. State which property of the real numbers we are illustrating: 3(4 + 5) = 3 4 + 3 5 Distributive: The 3 is being multiplied by both numbers in the parentheses. State which property of the real numbers we are illustrating: 3 + (6 + 8) = (6 + 8) + 3 Commutative property for addition: The order in which we add changed. State which property of the real numbers we are illustrating: 3 + (6 + 8) = (3 + 6) + 8 Associative property for addition: The way we grouped the numbers in the parentheses changed. State which property of the real numbers we are illustrating: 7 + 0 = 7 Identity element for addition: Adding zero does not change the original number. State which property of the real numbers we are illustrating: 8 + ( 8) = 0 Additive inverse: We added opposite numbers together. 6.5 Exponents and Scientific Notation: Exponential and scientific notation is used to identify very small and very large numbers. For example, these notations will help us to express a number that is 100 digits long. Exponents and scientific notation will also aid in performing calculations with very small and very large numbers. MAT 1301, Liberal Arts Math 6
Exponents Exponential notation provides a way of identifying a large or small number. The diagram below represents the basic definition of an exponent operation. As shown, the exponent identifies the number of times the base is being multiplied to itself. For example, 4 6 = 4 4 4 4 4 4 and 42 2 = 42 42. We can perform the multiplication to find the answer or use the exponent key on our calculator. This key is usually represented by the ^ or x y symbol on your calculator. Evaluate: 5 3 Evaluate: 3 2 Evaluate: ( 4) 4 MAT 1301, Liberal Arts Math 7
Rules for Exponents There are several rules that can be applied to exponents. These rules will help us when performing mathematical operations. First, we will discuss the product rule for exponents. Product Rule for Exponents If x is a real number and m and n are natural numbers, then x m x n = x m+n Note: We can only apply this rule if both bases are the same. Evaluate: 3 2 3 4 Each exponential expression has a base of 3. Therefore, we can apply the product rule for exponents. To do this, keep the base and add the exponents together. Then, expand the exponential problem to find the answer: 3 2 3 4 = 3 2+4 = 3 6 = 3 3 3 3 3 3 = 729 The power is applied when an exponential expression is raised to a power or exponent. In this case, we will multiply the exponents together and keep the base. Power Rule for Exponents If x is a real number and m and n are natural numbers, then (x m ) n = x m n Evaluate: (7 2 ) 3 Apply the power rule for exponents to solve. To do this, multiply the exponents together and keep the base. Then, expand the exponential expression to find the answer: (7 2 ) 3 = 7 2 3 = 7 6 = 7 7 7 7 7 7 = 117, 649 Another rule we will discuss is the quotient rule for exponents. This rule is applied when we are dividing by two exponential expressions. Evaluate: (7 1 ) 3 MAT 1301, Liberal Arts Math 8
Apply the power rule to solve. To do this, multiply the exponents together and keep the base. Then, expand the exponential expression. Note: A negative number times a negative number is a positive number. (7 1 ) 3 = 7 1( 3) = 7 3 = 7 7 7 = 343 Quotient Rule for Exponents If x is a nonzero real number and both m and n are natural numbers, then x m = xm n xn Note: We can only apply this rule if both bases are the same. Evaluate: 59 5 7 Apply the quotient rule for exponents to solve. To do this, subtract the exponent attached to the denominator from the exponent attached to the numerator and keep the base. Then, expand the exponential problem to find the answer: Special Exponent Definitions 5 9 5 7 = 59 7 = 5 2 = 5 5 = 25 Exponential expressions have special definitions when they are raised to a zero or negative power. Evaluate: 3 0 Definition 1 above states that any number raised to the zero power is 1. So, 3 0 = 1 Evaluate: 5 2 MAT 1301, Liberal Arts Math 9
Definition 2 above states to make the exponent positive by placing a 1 over the exponential expression. So, 5 2 = 1 5 2 = 1 5 5 = 1 25 Next, we will combine rules and definitions to solve more complex exponential problems in the following examples. Evaluate: 5 4 5 6 First, apply the product rule for exponents. (Keep the base and add the exponents.) 5 4 5 6 = 5 4+( 6) = 5 2 The first step resulted in a negative exponent. Therefore, apply the definition for negative exponents. (Make the exponent positive and place a 1 over the exponential expression.) = 1 5 2 = 1 5 5 = 1 25 Evaluate: ( 3)6 ( 3) 9 First, apply the quotient rule for exponents. (Subtract the exponent attached to the denominator from the exponent attached to the numerator and keep the base.) ( 3) 6 ( 3) 9 = ( 3)6 9 = ( 3) 3 The first step resulted in a negative exponent. Therefore, apply the definition for negative exponents. (Make the exponent positive and place a 1 over the exponential expression.) Any number raised to the zero power is 1. So, Scientific Notation = 1 ( 3) 3 = 1 3 3 3 = 1 27 = 1 27 Evaluate: (3 4 ) 0 (3 4 ) 0 = 3 4(0) = 3 0 = 1 Scientific notation is used when representing very small or very large numbers. The form for scientific notation is listed below. Notice that this form includes an exponential expression whose base is always 10. MAT 1301, Liberal Arts Math 10
Scientific notation A number is written in scientific notation if it is in the form a 10 n Where 1 a 10, and n is any integer. The following chart is listed on page 280 of your textbook. It reviews the steps for converting a number into scientific notation. Rewrite 3,200,000,000 in scientific notation. This number is very large, so we will move the decimal to the left until there is only one number in front of the decimal. To do this, we will move the decimal nine places to the left. Therefore, our exponent is 9. 3,200,000,000 = 3. 2 10 9 Rewrite 0.000258 in scientific notation. This number is very small, so we will move the decimal to the right until there is one number in front of the decimal. To do this, we will move the decimal four places to the right. Therefore, our exponent is 4. 0.000258 = 2. 58 10 4 The next examples practice converting scientific notation to a decimal. Chart containing rules for converting a decimal number to scientific notation (Pirnot, 2014, p. 280) Rewrite 1.78 10 3 in standard notation. MAT 1301, Liberal Arts Math 11
The exponent is negative. This means that our number will be small. Therefore, we will move the decimal 3 places to the left. Multiplying and Dividing Numbers in Scientific Notation 1.78 10 3 = 0.00178 We can multiply and divide numbers that are written in scientific notation by following the rules for exponents. The examples shown below discuss how to do this. Use scientific notation to perform the following operation. Leave your answer in scientific notation form. (4 10 3 )(2 10 4 ) First, separate the numbers and exponential expressions. (4 10 3 )(2 10 4 ) = (4 2) (10 3 10 4 ) Next, multiply each set. Use the product rule for exponents to multiply the exponential expressions. = 8.0 10 3+4 = 8. 0 10 7 Use scientific notation to perform the following operation. Leave your answer in scientific notation form. First, write the division problem as a fraction: (8 10 2 ) (2 10 3 ) (8 10 2 ) (2 10 3 ) = 8 10 2 2 10 3 Next, separate the numbers and exponential expressions: = 8 2 10 2 10 3 Divide each set. Use the quotient rule for exponents to divide the exponential expressions. = 4 10 2 3 = 4 10 2+( 3) = 4 10 5 Use scientific notation to perform the following operation. Leave your answer in scientific notation form. (9.6368 10 3 )(4.15 10 6 ) 1.52 10 4 MAT 1301, Liberal Arts Math 12
First, separate the numbers and exponential expressions. (9.6368 10 3 )(4.15 10 6 ) 1.52 10 4 = Next, multiply the numbers or expressions in the numerator. = 39.99272 103+( 6) 1.52 10 4 9.6368 4.15 103 10 6 1.52 10 4 Divide both sets. Use the quotient rule for exponents to divide the exponential expressions. = 26.311 10 3 10 4 = 26.311 10 3 4 = 26.311 10 3+( 4) = 26.311 10 7 Move the decimal one more place to the left. This is our answer. Applications of Scientific Notation = 2. 6311 10 6 Scientific notation is used in real-world applications. The next problem provides an example. In 1977, scientists sent the spaceship Voyager II to Neptune, which is about 2.8 billion miles away. If the spacecraft averaged about 25000 miles per hour, how long did the journey take? The numbers presented in this problem are very large. Therefore, we will use scientific notation to solve. In scientific notation, 2.8 billion miles = 2.8 10 9 In scientific notation, 25,000 miles per hour = 2.5 10 4 Next, divide the numbers of miles by the number of miles per hour to find how long the journey took. 2.8 10 9 miles 2.5 10 4 miles = 2.8 2.5 109 10 4 = 1.12 109 4 = 1. 12 10 5 hours hour For more information regarding scientific notation and its use, please view the following interactive presentation. https://media.pearsoncmg.com/pcp/pls/pls_mycoursetools/fufillment/mct_1256689785_csu/trigonometry/redir ect_math_custom_index_20.html 6.6 Sequences: As a child, we were taught to recognize patterns. We might have been given three shapes and were asked to provide the fourth shape based upon a recognizable pattern. As adults, we can recognize patterns or trends in things such as utility bills or gas prices. For example, we may expect an increase in our cable bill if that bill increased in previous years. In this section, we will learn how to recognize patterns in number sequences and predict the next number in a given sequence. MAT 1301, Liberal Arts Math 13
Sequence - A sequence is a list of numbers that follows some rule or pattern. UNIT The numbers x STUDY in GUIDE the list are called terms of the sequence. A sequence is in the form: We will discuss three sequences: arithmetic, geometric, and the Fibonacci sequence. Arithmetic Sequence An arithmetic sequence is a sequence in which the future terms are found by subtracting or adding a fixed constant to a previous term. The fixed constant is referred to as the common difference. For example, the sequence 2, 8, 14, 20. Is an arithmetic sequence because each term is found by adding 6 to the previous term. The common difference for this example is 6. List the next two terms of the arithmetic sequence: 11, 7, 3, -1, The first term of the sequence is 11. To proceed, we need to determine what operation was done to result in a 7. We know that 11 4 = 7. Let s see if this rule works for the other numbers in the sequence: 11 4 = 7 7 4 = 3 3 4 = -1 Subtracting 4 from the previous terms in the sequence results in the next consecutive term. Therefore, the common difference is 4. We will find the next two terms by following this pattern: The next two terms are -5 and -9. -1 4 = -5-5 4 = -9 We will be given certain criteria about an arithmetic sequence and will be asked to find the n th term of that sequence. To do this, we need to know the common difference (d), the first term (a 1 ), and the number of terms in the sequence (n). The n th Term of an Arithmetic Sequence The n th term of an arithmetic sequence with first term a 1 and common difference d is a n = a 1 + (n 1)d. MAT 1301, Liberal Arts Math 14
We can find the sum of first n terms of an arithmetic sequence once the n th term( UNIT a n x ) of STUDY the sequence GUIDE is found. The Sum of the First n Terms of Arithmetic Sequence The sum of the first n terms of an arithmetic sequence is given by n(a 1 + a n ) 2 For the arithmetic sequence, find the specified term a n and then find the sum of the terms from a 1 to a n, inclusive. 11, 17, 23, 29 Find a 9 Use the following formula to solve for a 9 : a n = a 1 + (n 1)d First, we need to identify a 1, n, and d of the sequence. 11, 17, 23, 29 a 1 = first term = 11 Next, plug in a 1 = 11, d = 6, and n = 9 into the formula: Therefore, a 9 = 59. a n = a 1 + (n 1)d a 9 = 11 + (9 1) 6 = 11 + 8 6 = 11 + 48 = 59 Now we will find the sum of the terms from a 1 to a n. To do this, we will use the value we found for a 9 and solve the following formula: Recall that a 1 = 11, a 9 = 59, and n = 9. Therefore, n(a 1 + a n ) 2 = n(a 1 + a n ) 2 9(11 + 59) 2 = 9 70 2 = 9 35 = 315 MAT 1301, Liberal Arts Math 15
The sum of the terms from a 1 to a 9 is 315. Geometric Sequence A geometric sequence is a sequence in which the future terms are found by multiplying a fixed constant to a previous term. The fixed constant is referred to as a common ratio. For example, the sequence 3, 9, 27, 81. Is a geometric sequence because each term is found by multiplying 3 to the previous term. The common ratio for this example is 3. List the next two terms of the sequence: 8, 24, 72, 216, The first term of the sequence is 8. To proceed, we need to determine what operation was done to result in a 24. We know that 8 3 = 24. Let s see if this rule works for the other numbers in the sequence: 8 3 = 24 24 3 = 72 72 3 = 216 Multiplying 3 by the previous terms in the sequence results in the next consecutive term. Therefore, the common ratio is 3. We will find the next two terms by following this pattern: The next two terms are 648 and 1,944. 216 3 = 648 648 3 = 1,944 We will be given certain criteria about a geometric sequence and will be asked to find the n th term of that sequence. To do this, we need to know the common ratio, the first term (a 1 ), and the number of terms in the sequence (n). The n th Term of a Geometric Sequence The n th term of a geometric sequence with a common ratio r is For the geometric sequence for the indicated term: We will use the following formula to solve for a 10 : a n = a 1 r n 1. 2, -4, 8, -16, Find a 10. a n = a 1 r n 1 MAT 1301, Liberal Arts Math 16
First, identify a 1, n, and r of the sequence: Next, plug in a 1 = 2, r = -2, and n = 10 into the formula: Therefore, a 10 = 1,024. The Fibonacci Sequence a n = a 1 r n 1 a 10 = 2 ( 2) 10 1 = 2 ( 2) 9 = 2 ( 512) = 1, 024 The Fibonacci Sequence is a special sequence that was discovered in 1202 to aid in numeric calculations. The Fibonacci Sequence A sequence whose future terms is the sum of the two previous terms in the sequence: 1,1,2,3,5,8,13,21,34,55,89,. Note: we label the terms of the sequence F1, F2, F3, F4 Two terms are given in the Fibonacci sequence. Find the specified term. F11 = 89 and F13 = 233. Find F12. A term of the Fibonacci sequence is found by adding the two previous terms together. Assume that we are given the 11 th, 12 th, and 13 th term of the sequence. This means that the 13 th term would be found by adding the 11 th and 12 th terms together. This is represented by the formula F 13 = F 11 + F 12 MAT 1301, Liberal Arts Math 17
We will plug in the values for F 11 and F 13 and solve for F 12. For more information about sequencing, please view the following interactive presentation: https://media.pearsoncmg.com/pcp/pls/pls_mycoursetools/fufillment/mct_1256689785_csu/basic_math/redire ct_math_custom_index_64.html Reference Pirnot, T. L. (2014). Mathematics all around (5th ed.). Boston, MA: Pearson. MAT 1301, Liberal Arts Math 18