Real numbers vs integers

real numbers in algebra and real numbers natural numbers and real numbers vs natural numbers
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Prerequisites Figure 1 Credit: Andreas Kambanls It’s a cold day in Antarctica. In fact, it’s always a cold day in Antarctica. Earth’s southernmost continent, Antarctica experiences the coldest, driest, and windiest conditions known. The coldest temperature ever recorded, over one hundred degrees below zero on the Celsius scale, was recorded by remote satellite. It is no surprise then, that no native human population can survive the harsh conditions. Only explorers and scientists brave the environment for any length of time. Measuring and recording the characteristics of weather conditions in in Antarctica requires a use of die ff rent kinds of numbers. Calculating with them and using them to make predictions requires an understanding of relationships among numbers. In this chapter, we will review sets of numbers and properties of operations used to manipulate numbers. This understanding will serve as prerequisite knowledge throughout our study of algebra and trigonometry. 12 CHAPTER 1 Prere quisitse l eARnIng Obje CTIveS In this section students will: • Classify a real number as a natural, whole, integer, rational, or irrational number. • Perform calculations using order of operations. • Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity. • Evaluate algebraic expressions. • Simplify algebraic expressions. 1.1 ReAl nUmbeRS: Algeb RA eSSen TIAl S It is often said that mathematics is the language of science. If this is true, then the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization. Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts. But what if there were no cattle to trade or an entire crop of grain was lost in a o fl od? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations. Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further. Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions. Classifying a Real number e n Th umbers we use for counting, or enumerating items, are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as 1, 2, 3, . . . where the ellipsis (. . .) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: 0, 1, 2, 3, . . .. e s Th et of integers adds the opposites of the natural numbers to the set of whole numbers: . . ., −3, −2, −1, 0, 1, 2, 3, . . .. It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers. negative integers zero positive integers . . . , −3, −2, −1, 0, 1, 2, 3, . . . m _ The set of rational numbers is written as m and n are integers and n ≠ 0 . Notice from the definition that   n ∣ rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1. Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either: 15 ___ 1. a terminating decimal: = 1.875, or 8 _ 4 ___ 2. a repeating decimal: = 0.36363636 … = 0. 36 11 We use a line drawn over the repeating block of numbers instead of writing the group multiple times.SECTION 1.1 r eal Numbers: a gelb r a e sse Ntials 3 Example 1 Writing Integers as Rational Numbers Write each of the following as a rational number. a. 7 b. 0 c. −8 Solution Write a fraction with the integer in the numerator and 1 in the denominator. 7 0 8 _ _ _ a. 7 = b. 0 = c. −8 = − 1 1 1 Try It 1 Write each of the following as a rational number. a. 11 b. 3 c. − 4 Example 2 Identifying Rational Numbers Write each of the following rational numbers as either a terminating or repeating decimal. 5 15 13 _ _ _ a. − b. c. 7 5 25 Solution Write each fraction as a decimal by dividing the numerator by the denominator. _ 5 _ a. − = −0.714285 , a r epeating decimal 7 15 _ b. = 3 (or 3.0), a terminating decimal 5 13 _ c. = 0.52, a terminating decimal 25 Try It 2 Write each of the following rational numbers as either a terminating or repeating decimal. 68 8 17 _ _ _ a. b. c. − 17 13 20 Irrational Numbers At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for 3 _ instance, may have found that the diagonal of a square with unit sides was not 2 or even , but was something else. 2 Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown. h  h is not a rational number Example 3 Differentiating Rational and Irrational Numbers Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal. — — 33 17 _ _ a. √25 b. c. √11 d. e. 0.3033033303333… 9 34 Solution — — — a. √25 : This can be simplified as √ 25 = 5. Therefore, √ 25 i s rational.4 CHAPTER 1 Prere quisitse 33 33 _ _ b. : Because it is a fraction, is a rational number. Next, simplify and divide. 9 9 11 _  33 33 11 _ _ _ = = = 3. 6 9  3 9 3 33 _ So, is rational and a repeating decimal. 9 — — c. √11 : This cannot be simplified any further. Therefore, √ 11 i s an irrational number. 17 17 _ _ d. : B ecause it is a fraction, i s a rational number. Simplify and divide. 34 34 1  17 17 1 _ _ _ = = = 0.5 34 2  3 4 2 17 _ So, is rational and a terminating decimal. 34 e. 0.3033033303333 … i s not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number. Try It 3 Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal. — — 7 91 _ _ a. b. √ 81 c. 4.27027002700027 … d. e. √39 77 13 Real Numbers Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or −). Zero is considered neither positive nor negative. e r Th eal numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line. e Th converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 2. 1 2 3 5 −5 −4 −2 −10 4 Figure 2 The real number line Example 4 Classifying Real Numbers Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the le ft or the right of 0 on the number line? — — 10 _ a. − b. √5 c. −√ 289 d. −6π e. 0.615384615384 … 3 Solution 10 _ a. − i s negative and rational. It lies to the left of 0 on the number line. 3 — b. √5 is positive and irrational. It lies to the right of 0. — — 2 c. −√ 289 = −√ 17 = −17 is negative and rational. It lies to the left of 0. d. −6π is negative and irrational. It lies to the left of 0. e. 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.SECTION 1.1 r eal Numbers: a gelb r a e sse Ntials 5 Try It 4 Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the le ft or the right of 0 on the number line? — — 47 √ 5 _ _ a. √ 73 b. −11.411411411 … c. d. − e. 6.210735 19 2 Sets of Numbers as Subsets Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 3. N: the set of natural numbers W: the set of whole numbers m 1, 2, 3, ... 0 ..., −3, −2, −1 , n ≠ 0 n I: the set of integers N W I Q Q: the set of rational numbers Q’: the set of irrational numbers Figure 3 Sets of numbers sets of numbers e s Th et of natural numbers includes the numbers used for counting: 1, 2, 3, .... e s Th et of whole numbers is the set of natural numbers plus zero: 0, 1, 2, 3, .... e s Th et of integers adds the negative natural numbers to the set of whole numbers: ..., −3, −2, −1, 0, 1, 2, 3, ... . m _ e s Th et of rational numbers includes fractions written as m and n are integers and n ≠ 0   n ∣ e s Th et of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: h h is not a rational number. Example 5 Differentiating the Sets of Numbers Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q'). — — 8 _ a. √36 b. c. √73 d. −6 e. 3.2121121112 … 3 Solution N W I Q Q' — × × × × a. √36 = 6 _ 8 _ b. = 2. 6   × 3 — × c. √73 d. −6 × × e. 3.2121121112... ×6 CHAPTER 1 Prere quisitse Try It 5 Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q'). — — 35 _ a. − b. 0 c. 169 d. 24 e. 4.763763763 … √ √ 7 Performing Calculations Using the Order of Operations 2 When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 = 4 ∙ 4 = 16. We can n raise any number to any power. In general, the exponential notation a m eans that the number or variable a is used as a factor n times. n factors n a = a ∙ a ∙ a ∙ … ∙ a n In this notation, a is read as the nth power of a, where a is called the base and n is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. 2 _ 2 For example, 24 + 6 ∙ − 4 is a mathematical expression. 3 To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions. Recall that in mathematics we use parentheses ( ), brackets , and braces to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols. e n Th ext step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right. Let’s take a look at the expression provided. 2 _ 2 24 + 6 ∙ − 4 3 e Th re are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so 2 simplify 4 a s 16. 2 2 _ 24 + 6 ∙ − 4 3 2 _ 24 + 6 ∙ − 16 3 Next, perform multiplication or division, left to right. 2 _ 24 + 6 ∙ − 16 3 24 + 4 − 16 Lastly, perform addition or subtraction, left to right. 24 + 4 − 16 28 − 16 12 2 2 _ e Th refore, 24 + 6 ∙ − 4 = 12. 3 For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplie fi d before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result. order of operations Operations in mathematical expressions must be evaluated in a systematic order, which can be simplie fi d using the acronym PEMDAS: P(arentheses) E(xponents) M(ultiplication) and D(ivision) A(ddition) and S(ubtraction)SECTION 1.1 r eal Numbers: a gelb r a e sse Ntials 7 How To… Given a mathematical expression, simplify it using the order of operations. 1. Simplify any expressions within grouping symbols. 2. Simplify any expressions containing exponents or radicals. 3. Perform any multiplication and division in order, from left to right. 4. Perform any addition and subtraction in order, from left to right. Example 6 Using the Order of Operations Use the order of operations to evaluate each of the following expressions. 2 5 − 4 — 2 ______ a. (3 ∙ 2) − 4(6 + 2) b. − √11 − 2 c. 6 − 5 − 8 + 3(4 − 1) ∣ ∣ 7 14 − 3 ∙ 2 _ 2 d. e. 7(5 ∙ 3) − 2(6 − 3) − 4 + 1 2 2 ∙ 5 − 3 Solution 2 2 a. ( 3 ∙ 2) − 4(6 + 2) = (6) − 4(8) Simplify parentheses. = 36 − 4(8) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. 2 2 5 — 5 − 4 — __ ______ b. − √11 − 2 = − √9 Simplify grouping symbols (radical). 7 7 2 5 − 4 ______ = − 3 Simplify radical. 7 25 − 4 ______ = − 3 Simplify exponent. 7 21 ___ = − 3 Simplify subtraction in numerator. 7 = 3 − 3 Simplify division. = 0 Simplify subtraction. Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped. c. 6 − 5 − 8 + 3(4 − 1) = 6 − −3 + 3(3) Simplify inside grouping symbols. = 6 − 3 + 3(3) Simplify absolute value. = 6 − 3 + 9 Simplify multiplication. = 3 + 9 Simplify subtraction. = 12 Simplify addition. 14 − 3 ∙ 2 14 − 3 ∙ 2 _ _ d. = S implif y exponent. 2 2 ∙ 5 − 9 2 ∙ 5 − 3 14 − 6 _ = Simplify products. 10 − 9 8 _ = Simplify differences. 1 = 8 Simplify quotient. In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step. 2 2 e 7(5 ∙ 3) − 2(6 − 3) − 4 + 1 = 7(15) − 2(3) − 4 + 1 Simplify inside parentheses. = 7(15) − 2(3 − 16) + 1 Simplify exponent. = 7(15) − 2(−13) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.8 CHAPTER 1 Prere quisitse Try It 6 Use the order of operations to evaluate each of the following expressions. — — 7 ∙ 5 − 8 ∙ 4 2 2 2 __________ a. √ 5 − 4 + 7 (5 − 4) b. 1 + c. 1.8 − 4.3 + 0.4√ 15 + 10 9 − 6 1 1 __ 2 2 __ 2 2 d. 5 · 3 − 7 + · 9 e. (3 − 8) − 4 − (3 − 8) 2 3 Using Properties of Real numbers For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics. Commutative Properties The commutative property of addition states that numbers may be added in any order without ae ff cting the sum. a + b = b + a We can better see this relationship when using real numbers. (−2) + 7 = 5 and 7 + (−2) = 5 Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without ae ff cting the product. a ∙ b = b ∙ a Again, consider an example with real numbers. (−11) ∙ (−4) = 44 and (−4) ∙ (−11) = 44 It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 is not the same as 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. Associative Properties The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. a(bc) = (ab)c Consider this example. (3 ∙ 4) ∙ 5 = 60 and 3 ∙ (4 ∙ 5) = 60 The associative property of addition tells us that numbers may be grouped die ff rently without ae ff cting the sum. a + (b + c) = (a + b) + c Thi s property can be especially helpful when dealing with negative integers. Consider this example. 15 + (−9) + 23 = 29 and 15 + (−9) + 23 = 29 Are subtraction and division associative? Review these examples. 8 − (3 − 15) ≟ (8 − 3) − 15 64 ÷ (8 ÷ 4) ≟ (64 ÷ 8) ÷ 4 8 − ( − 12) ≟ 5 − 15 64 ÷ 2 ≟ 8 ÷ 4 20 ≠ −10 32 ≠ 2 As we can see, neither subtraction nor division is associative. Distributive Property The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. a ∙ (b + c) = a ∙ b + a ∙ cSECTION 1.1 r eal Numbers: a gelb ra e sse Ntials 9 This property combines both addition and multiplication (and is the only property to do so). Let us consider an example. 4 ∙ 12 + (−7) = 4 ∙ 12 + 4 ∙ (−7) = 48 + (−28) = 20 Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by −7, and adding the products. To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example. 6 + (3 ∙ 5) ≟ (6 + 3) ∙ (6 + 5) 6 + (15) ≟ (9) ∙ (11) 21 ≠ 99 A special case of the distributive property occurs when a sum of terms is subtracted. a − b = a + (−b) For example, consider the difference 12 − (5 + 3). We can rewrite the difference of the two terms 12 and (5 + 3) by turning the subtraction expression into addition of the opposite. So instead of subtracting (5 + 3), we add the opposite. 12 + (−1) ∙ (5 + 3) Now, distribute −1 and simplify the result. 12 − (5 + 3) = 12 + (−1) ∙ (5 + 3) = 12 + (−1) ∙ 5 + (−1) ∙ 3 = 12 + (−8) = 4 This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example. 12 − (5 + 3) = 12 + (−5 − 3) = 12 + (−8) = 4 Identity Properties e Th identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number. a + 0 = a e Th identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number. a ∙ 1 = a For example, we have (−6) + 0 = −6 and 23 ∙ 1 = 23. Th ere are no exceptions for these properties; they work for every real number, including 0 and 1. Inverse Properties The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted−a, that, when added to the original number, results in the additive identity, 0. a + (−a) = 0 For example, if a = −8, the additive inverse is 8, since (−8) + 8 = 0. The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or 1 _ reciprocal), denoted , that, when multiplied by the original number, results in the multiplicative identity, 1. a 1 _ a ∙ = 1 a 2 1 3 _ _ _ For example, if a = − , t he reciprocal, denoted , is − because a 3 210 CHAPTER 1 Prere quisitse 1 2 3 _ _ _ a ∙ = − ∙ − = 1     a 3 2 properties of real numbers e f Th ollowing properties hold for real numbers a, b, and c. Addition Multiplication Commutative Property a + b = b + a a ∙ b = b ∙ a Associative Property a + (b + c) = (a + b) + c a(bc) = (ab)c Distributive Property a ∙ (b + c) = a ∙ b + a ∙ c e Th re exists a unique real number e Th re exists a unique real number called the additive identity, 0, such called the multiplicative identity, 1, Identity Property that, for any real number a such that, for any real number a a + 0 = a a ∙ 1 = a Every real number a has an additive Every nonzero real number a has a inverse, or opposite, denoted −a, multiplicative inverse, or reciprocal, 1 _ Inverse Property such that denoted , s uch that a a + (−a) = 0 1 _ a ∙ = 1   a Example 7 Using Properties of Real Numbers Use the properties of real numbers to rewrite and simplify each expression. State which properties apply. 4 2 7 __ __ __ a. 3 ∙ 6 + 3 ∙ 4 b. (5 + 8) + (−8) c. 6 − (15 + 9) d. ∙ ∙ e. 100 ∙ 0.75 + (−2.38)   7 3 4 Solution a. 3 ∙ 6 + 3 ∙ 4 = 3 ∙ (6 + 4) Distributive property = 3 ∙ 10 Simplify. = 30 Simplify. b. (5 + 8) + (−8) = 5 + 8 + (−8) Associative property of addition = 5 + 0 Inverse property of addition = 5 Identity property of addition c. 6 − (15 + 9) = 6 + (−15) + (−9) Distributive property = 6 + (−24) Simplify. = −18 Simplify. 4 2 7 4 7 2 __ __ __ __ __ __ d. ∙ ∙ = ∙ ∙ Commutative property of multiplication     7 3 4 7 4 3 4 7 2 __ __ __ = ∙ ∙ Associative property of multiplication   7 4 3 2 __ = 1 ∙ Inverse property of multiplication 3 2 __ = Identity property of multiplication 3 e. 100 ∙ 0.75 + (−2.38) = 100 ∙ 0.75 + 100 ∙ (−2.38) Distributive property = 75 + (−238) Simplify. = −163 Simplify. Try It 7 Use the properties of real numbers to rewrite and simplify each expression. State which properties apply. 23 5 17 4 17 _ _ _ _ _ a. − · 11 · − b. 5 · (6.2 + 0.4) c. 18 − (7 − 15) d. + + − e. 6 ⋅ (−3) + 6 ⋅ 3         5 23 18 9 18SECTION 1.1 r eal Numbers: a gelb r a e sse Ntials 11 Evaluating Algebraic Expressions So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see — 4 3 3 2 _ expressions such as x + 5, π r , or √ 2 m n . In the expression x + 5, 5 is called a constant because it does not vary 3 and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way. 5 5 ( −3) = (−3) ∙ (−3) ∙ (−3) ∙ (−3) ∙ (−3) x = x ∙ x ∙ x ∙ x ∙ x 3 3 ( 2 ∙ 7) = (2 ∙ 7) ∙ (2 ∙ 7) ∙ (2 ∙ 7) ( yz) = (yz) ∙ (yz) ∙ (yz) In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. Any variable in an algebraic expression may take on or be assigned die ff rent values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. Example 8 Describing Algebraic Expressions List the constants and variables for each algebraic expression. — 4 _ 3 3 2 a. x + 5 b. π r c. √2m n 3 Solution Constants Variables a. x + 5 5 x 4 4 _ 3 _ b. πr , π r 3 3 — 3 2 2 m, n c. √2m n Try It 8 List the constants and variables for each algebraic expression. 3 a. 2πr(r + h) b. 2(L + W) c. 4y + y Example 9 Evaluating an Algebraic Expression at Different Values Evaluate the expression 2x − 7 for each value for x. 1 _ a. x = 0 b. x = 1 c. x = d. x = −4 2 Solution a. Substitute 0 for x. 2x − 7 = 2(0) − 7 = 0 − 7 = −7 b. Substitute 1 for x. 2x − 7 = 2(1) − 7 = 2 − 7 = −512 CHAPTER 1 Prere quisitse 1 1 __ __ c. Substitute for x. 2x − 7 = 2 − 7   2 2 = 1 − 7 = −6 d. Substitute −4 for x. 2x − 7 = 2(−4) − 7 = −8 − 7 = −15 Try It 9 Evaluate the expression 11 − 3y for each value for y. 2 _ a. y = 2 b. y = 0 c. y =  d. y = −5 3 Example 10 Evaluating Algebraic Expressions Evaluate each expression for the given values. t 4 _ _ 3 a. x + 5 for x = −5 b. for t = 10 c. π r for r = 5 2t−1 3 — 3 2 d. a + ab + b for a = 11, b = −8 e. √2 m n for m = 2, n = 3 Solution a. Substitute −5 for x. x + 5 = (−5) + 5 = 0 (10) t _ _ b. Substitute 10 for t. = 2t − 1 2(10) − 1 10 _ = 20 − 1 10 _ = 19 4 4 _ 3 _ 3 c. Substitute 5 for r. πr = π(5) 3 3 4 _ = π(125) 3 500 _ = π 3 d. Substitute 11 for a and −8 for b. a + ab + b = (11) + (11)(−8) + (−8) = 11 − 88 − 8 = −85 — — 3 2 3 2 e. Substitute 2 for m and 3 for n. √2m n = √2(2) (3) — = √ 2(8)(9) — = √ 144 = 12 Try It 10 Evaluate each expression for the given values. y + 3 1 _ 2 _ a. for y = 5 b. 7 − 2t for t = −2 c. πr for r = 11 y − 3 3 2 1 2 3 _ _ d. (p q) for p = −2, q = 3 e. 4(m − n) − 5(n − m) for m = , n = 3 3SECTION 1.1 r eal Numbers: a gelb r a e sse Ntials 13 Formulas An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2x + 1 = 7 has the unique solution x = 3 because when we substitute 3 for x in the equation, we obtain the true statement 2(3) + 1 = 7. A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A of a circle in terms of the radius r of the circle: 2 2 A = πr . For any value of r, the area A can be found by evaluating the expression πr . Example 11 Using a Formula A right circular cylinder with radius r and height h has the surface area S (in square units) given by the formula S = 2πr(r + h). See Figure 4. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π. r h Figure 4 Right circular cylinder Solution Evaluate the expression 2πr(r + h) for r = 6 and h = 9. S = 2πr(r + h) = 2π(6)(6) + (9) = 2π(6)(15) = 180π Th e surface area is 180π square inches. Try It 11 A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte 2 (in square centimeters, or cm ) is found to be A = (L + 16)(W + 16) − L ∙ W. See Figure 5. Find the area of a matte for a photograph with length 32 cm and width 24 cm. 1 2 5 −5 −4 −2 −10 3 4 Figure 5 14 CHAPTER 1 Prere quisitse Simplifying Algebraic Expressions Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions. Example 12 Simplifying Algebraic Expressions Simplify each algebraic expression. 5 2 _ _ a. 3x − 2y + x − 3y − 7 b. 2r − 5(3 − r) + 4 c. 4t − s − t + 2s d. 2mn − 5m + 3mn + n     4 3 Solution a. 3x − 2y + x − 3y − 7 = 3x + x − 2y − 3y − 7 Commutative property of addition = 4x − 5y − 7 Simplify. b. 2r − 5(3 − r) + 4 = 2r − 15 + 5r + 4 Distributive property = 2r + 5r − 15 + 4 Commutative property of addition = 7r − 11 Simplify. 5 2 5 2 _ _ _ _ c. 4 t − 4 t − s − t + 2s = 4t − s − t − 2s Distributive property     4 3 4 3 2 5 _ _ = 4t − t − s − 2s Commutative property of addition 3 4 10 13 _ _ = t − s Simplify. 3 4 d. mn − 5m + 3mn + n = 2mn + 3mn − 5m + n Commutative property of addition = 5mn − 5m + n Simplify. Try It 12 Simplify each algebraic expression. 2 4 5 3 _ _ _ _ a. y − 2 y + z b. − 2 − + 1 c. 4p(q − 1) + q(1 − p) d. 9r − (s + 2r) + (6 − s)   3 3 t t Example 13 Simplifying a Formula A rectangle with length L and width W has a perimeter P given by P = L + W + L + W. Simplify this expression. Solution P = L + W + L + W P = L + L + W + W Commutative property of addition P = 2L + 2W Simplify. P = 2(L + W) Distributive property Try It 13 If the amount P is deposited into an account paying simple interest r for time t, the total value of the deposit A is given by A = P + Prt. Simplify the expression. (This formula will be explored in more detail later in the course.) Access these online resources for additional instruction and practice with real numbers. • Simplify an expression (http://openstaxcollege.org/l/simexpress) • evaluate an expression1 (http://openstaxcollege.org/l/ordofoper1) • evaluate an expression2 (http://openstaxcollege.org/l/ordofoper2)SECTION 1.1 s ectio N e xercises 15 1.1 SeCTIOn exeRCISeS veRbAl — 2. What is the order of operations? What acronym is 1. Is √ 2 an example of a rational terminating, rational used to describe the order of operations, and what repeating, or irrational number? Tell why it fits that does it stand for? category. 3. What do the Associative Properties allow us to do when following the order of operations? Explain your answer. nUmeRIC For the following exercises, simplify the given expression. 2 3 2 2 4. 10 + 2 · (5 − 3) 5. 6 ÷ 2 − (81 ÷ 3 ) 6. 18 + (6 − 8) 7. −2 · 16 ÷ (8 − 4) 8. 4 − 6 + 2 · 7 9. 3(5 − 8) 10. 4 + 6 − 10 ÷ 2 11. 12 ÷ (36 ÷ 9) + 6 2 12. (4 + 5) ÷ 3 13. 3 − 12 · 2 + 19 14. 2 + 8 · 7 ÷ 4 15. 5 + (6 + 4) − 11 2 16. 9 − 18 ÷ 3 17. 14 · 3 ÷ 7 − 6 18. 9 − (3 + 11) · 2 19. 6 + 2 · 2 − 1 2 2 2 20. 64 ÷ (8 + 4 · 2) 21. 9 + 4(2 ) 22. (12 ÷ 3 · 3) 23. 25 ÷ 5 − 7 1 2 _ 24. (15 − 7) · (3 − 7) 25. 2 · 4 − 9(−1) 26. 4 − 25 · 27. 12(3 − 1) ÷ 6 5 Algeb RAIC For the following exercises, solve for the variable. 28. 8(x + 3) = 64 29. 4y + 8 = 2y 30. (11a + 3) − 18a = −4 31. 4z − 2z(1 + 4) = 36 2 2 32. 4y(7 − 2) = −200 33. −(2x) + 1 = −3 34. 8(2 + 4) − 15b = b 35. 2(11c − 4) = 36 1 _ 2 36. 4(3 − 1)x = 4 37. (8 w − 4 ) = 0 4 For the following exercises, simplify the expression. a 2 __ 38. 4x + x(13 − 7) 39. 2y − (4) y − 11 40. (64) − 12a ÷ 6 41. 8b − 4b(3) + 1 3 2 2 42. 5l ÷ 3l · (9 − 6) 43. 7z − 3 + z · 6 44. 4 · 3 + 18x ÷ 9 − 12 45. 9(y + 8) − 27 2 9 4 _ _ 47. 6 + 12b − 3 · 6b 48. 18y − 2(1 + 7y) 46. t − 4 2 49. · 27x     6 9 2 50. 8(3 − m) + 1(−8) 51. 9x + 4x(2 + 3) − 4(2x + 3x) 52. 5 − 4(3x)CHAPTER 1 Prerequisites 16 ReAl-W ORld A PPl ICATIOnS For the following exercises, consider this scenario: Fred earns 40 mowing lawns. He spends 10 on mp3s, puts half of what is left in a savings account, and gets another 5 for washing his neighbor’s car. 53. Write the expression that represents the number of 54. How much money does Fred keep? dollars Fred keeps (and does not put in his savings account). Remember the order of operations. For the following exercises, solve the given problem. 55. According to the U.S. Mint, the diameter of a 56. Jessica and her roommate, Adriana, have decided quarter is 0.955 inches. The circumference of the to share a change jar for joint expenses. Jessica put quarter would be the diameter multiplied by π. Is her loose change in the jar first, and then Adriana the circumference of a quarter a whole number, a put her change in the jar. We know that it does not rational number, or an irrational number? matter in which order the change was added to the jar. What property of addition describes this fact? For the following exercises, consider this scenario: There is a mound of g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel. 57. Write the equation that describes the situation. 58. Solve for g. For the following exercise, solve the given problem. 59. Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got 2.5 million for the annual marketing budget. He must spend the budget such that 2,500,000 − x = 0. What property of addition tells us what the value of x must be? TeChn Ol Ogy For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth. 3 2 _ 2 60. 0.5(12.3) − 48x = 61. (0.25 − 0.75) x − 7.2 = 9.9 5 exTen SIOnS 62. If a whole number is not a natural number, what 63. Determine whether the statement is true or false: must the number be? e m Th ultiplicative inverse of a rational number is also rational. 64. Determine whether the statement is true or false: 65. Determine whether the simplified expression is — e p Th roduct of a rational and irrational number is rational or irrational: √ −18 − 4(5)( −1) . always irrational. 66. Determine whether the simplified expression is 67. e di Th vision of two whole numbers will always result — in what type of number? rational or irrational: √ −16 + 4(5) + 5 . 68. What property of real numbers would simplify the following expression: 4 + 7(x − 1)?SECTION 1.2 e x Po Ne Nts a N d s cie Nti icf Ntioato N 17 l eARnIng Obje CTIveS In this section students will: • Use the product rule of exponents. • Use the quotient rule of exponents. • Use the power rule of exponents. • Use the zero exponent rule of exponents. • Use the negative rule of exponents. • Find the power of a product and a quotient. • Simplify exponential expressions. • Use scientific notation. 1.2 exPOnen TS And S CIen TIFIC nOTATIOn Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number. Using a calculator, we enter 2,048 · 1,536 · 48 · 24 · 3,600 and press ENTER. The calculator displays 1.304596316E13. What does this mean? The “ E13” portion of the result represents the exponent 13 of ten, so there are a maximum of 13 approximately 1.3 · 10 bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. Using the Product Rule of exponents 3 4 Consider the product x ∙ x . Both terms have the same base, x, but they are raised to die ff rent exponents. Expand each expression, and then rewrite the resulting expression. 3 factors 4 factors 3 4 x · x = x · x · x · x · x · x · x 7 factors = x · x · x · x · x · x · x 7 = x 3 4 3 + 4 7 e r Th esult is that x ∙ x = x = x . Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents. m n m + n a · a = a Now consider an example with real numbers. 3 4 3 + 4 7 2 · 2 = 2 = 2 3 4 7 We can always check that this is true by simplifying each exponential expression. We find that 2 is 8, 2 is 16, and 2 is 128. The product 8 ∙ 16 equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but die ff rent exponents. the product rule of exponents For any real number a and natural numbers m and n, the product rule of exponents states that m n m + n a · a = a18 CHAPTER 1 Prere quisitse Example 1 Using the Product Rule Write each of the following products with a single base. Do not simplify further. 5 3 5 2 5 3 a. t ∙ t b. (−3) ∙ (−3) c. x ∙ x ∙ x Solution Use the product rule to simplify each expression. 5 3 5 + 3 8 a. t ∙ t = t = t 5 5 1 5 + 1 6 b. (−3) ∙ (−3) = (−3) ∙ (−3) = (−3) = (−3) 2 5 3 c. x ∙ x ∙ x At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two. 2 5 3 2 5 3 2 + 5 3 7 3 7 + 3 10 x · x · x = (x · x ) · x = (x )· x = x · x = x = x Notice we get the same result by adding the three exponents in one step. 2 5 3 2 + 5 + 3 10 x · x · x = x = x Try It 1 Write each of the following products with a single base. Do not simplify further. 4 2 2 6 9 _ _ 3 6 5 a. k · k b. · c. t · t · t     y y Using the Quotient Rule of exponents The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but m y ___ different exponents. In a similar way to the product rule, we can simplify an expression such as , where m n. n y 9 y _ Consider the example . P erform the division by canceling common factors. 5 y 9 y · y · y · y · y · y · y · y · y y ___ ___ = 5 y · y · y · y · y y      y · y · y · y · y · y · y · y · y ___ =      y · y · y · y · y y · y · y · y __________ = 1 4 = y Notice that the exponent of the quotient is the die ff rence between the exponents of the divisor and dividend. m a ___ m − n = a n a In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. 9 y __ 9 − 5 4 = y = y 5 y For the time being, we must be aware of the condition m n. Otherwise, the difference m − n could be zero or negative. o Th se possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers. the quotient rule of exponents For any real number a and natural numbers m and n, such that m n, the quotient rule of exponents states that m a ___ m − n = a n a Example 2 Using the Quotient Rule Write each of the following products with a single base. Do not simplify further. 5 — 14 23   (−2) t z 2 √ ______ __ ________ a. b. c. — 9 15 (−2) t z √2 SECTION 1.2 e x Po Ne Nts a N d s cie Nti icf Ntioato N 19 Solution Use the quotient rule to simplify each expression. 14 (−2) ______ 14 − 9 5 a. = (−2) = (−2) 9 (−2) 23 t __ 23 − 15 8 b. = t = t 15 t — 5   — 5 − 1 — 4 z √2 ________     c. = z √2 = z√ 2 — z√ 2 Try It 2 Write each of the following products with a single base. Do not simplify further. 2 5 75 6 (ef ) s (−3) __ _____ _ a. b. c. 2 3 68 (ef ) s −3 Using the Power Rule of exponents Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power 2 3 rule of exponents. Consider the expression (x ) . The expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3. 3 factors 2 3 2 2 2 ( x ) = (x ) · (x ) · (x ) 3 factors 2 factors 2 factors 2 factors = · ·       x · x x · x x · x = x · x · x · x · x · x 6 = x 2 3 2 ∙ 3 6 The exponent of the answer is the product of the exponents: (x ) = x = x . In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents. m n m ∙ n (a ) = a Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, die ff rent terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents. Product Rule Power Rule 3 4 3 + 4 7 3 4 3 ∙ 4 12 5 ∙ 5 = 5 = 5 but (5 ) = 5 = 5 5 2 5 + 2 7 5 2 5 ∙ 2 10 x ∙ x = x = x but (x ) = x = x 7 10 7 + 10 17 7 10 7 ∙ 10 70 (3a) ∙ (3a) = (3a) = (3a) but ((3a) ) = (3a) = (3a) the power rule of exponents For any real number a and positive integers m and n, the power rule of exponents states that m n m ∙ n (a ) = a Example 3 Using the Power Rule Write each of the following products with a single base. Do not simplify further. 2 7 5 3 5 11 a. (x ) b. ((2t) ) c. ( (−3) ) Solution Use the power rule to simplify each expression. 2 7 2 ∙ 7 14 a. ( x ) = x = x 5 3 5 ∙ 3 15 b. ((2t) ) = (2t) = (2t) 5 11 5 ∙ 11 55 c. ((−3) ) = (−3) = (−3)20 CHAPTER 1 Prere quisitse Try It 3 Write each of the following products with a single base. Do not simplify further. 8 3 5 7 4 4 a. ((3y) ) b. (t ) c. ((−g) ) Using the Zero exponent Rule of exponents Return to the quotient rule. We made the condition that m n so that the die ff rence m − n would never be zero or negative. What would happen if m = n? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example. 8 8  t t __ __ = = 1 8 8  t t If we were to simplify the original expression using the quotient rule, we would have 8 t __ 8 − 8 0 = t = t 8 t 0 If we equate the two answers, the result is t = 1. This is true for any nonzero real number, or any variable representing a real number. 0 a = 1 0 e s Th ole exception is the expression 0 . This appears later in more advanced courses, but for now, we will consider the value to be undefined. the zero exponent rule of exponents For any nonzero real number a, the zero exponent rule of exponents states that 0 a = 1 Example 4 Using the Zero Exponent Rule Simplify each expression using the zero exponent rule of exponents. 2 4 3 5 2 2 (j k) 5(rs ) c −3x __ _____ _ _ a. b. c. d. 2 2 3 2 2 3 5 (j k) ∙ (j k) (rs ) c x Solution Use the zero exponent and other rules to simplify each expression. 3 c __ 3 − 3 a. = c 3 c 0 = c 5 5 −3x x _____ __ b. = −3 ∙ 5 5 x x 5 − 5 = −3 ∙ x 0 = −3 ∙ x = −3 ∙ 1 = −3 2 4 2 4 (j k) (j k) _ _ c. = Use the product rule in the denominator. 2 2 3 2 1 + 3 (j k) ∙ (j k) (j k) 2 4 (j k) _ = Simplify. 2 4 (j k) 2 4 − 4 = (j k) Use the quotient rule. 2 0 = (j k) Simplify. = 1 2 2 5(rs ) ______ 2 2 − 2 d. = 5(rs ) Use the quotient rule. 2 2 (rs ) 2 0 = 5(rs ) Simplify. = 5 ∙ 1 Use the zero exponent rule. = 5 Simplify.