A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions


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1 A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8,
2 Contents 25 Integers: Addition and Subtraction 3 26 Integers: Multiplication, Division, and Order Rational Numbers Real Numbers Functions and their Graphs Graphical Representations of Data Misleading Graphs and Statistics Measures of Central Tendency and Dispersion Probability: Some Basic Terms Probability and Counting Techniques Permutations, Combinations and Probability Odds, Expected Value, and Conditional Probability Basic Geometric Shapes and Figures Properties of Geometric Shapes Symmetry of Plane Figures Regular Polygons Three Dimensional Shapes Systems of Measurements Perimeter and Area Surface Area 229 2
3 45 Volume Congruence of Triangles Similar Triangles Solutions to Practice Problems 266 3
4 25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers other than whole numbers. In this and the next section we study the set Z of integers which consists of: the set W of whole numbers, i.e. W = {0, 1, 2, 3, }, and the negative integers denoted by N = { 1, 2, 3, }. That is, Z = {, 3, 2, 1, 0, 1, 2, 3, }. In today s society these numbers are used to record debits and credits, profits and losses, changes in the stock markets, degrees above and below zero in measuring temperature, the altitude above and below sea level, etc. As usual, we introduce the set of integers by means of pictures and diagrams with steadily increasing levels of abstraction to suggest, in turn, how these ideas might be presented to school children. Representations of Integers We will discuss two different ways for representing integers: set model and number line model. Set Model: In this model, we use signed counters. We represent +1 by and 1 by. In this case, the number +2 is represented by and the number 5 is represented by One plus counter neutralizes one minus counter. Thus, we can represent 0 by. NumberLine Model: The number line model is shown in Figure 25.1 Figure 25.1 Example 25.1 Represent 4 on a number line and as a set of signed counters. 4
5 Using signed counters we have Figure 25.2 shows the position of 4 on a numberline. Figure 25.2 Looking at Figure 25.1, we see that 2 and 2 have the same distance from 0 but on opposite sides. Such numbers are called opposite. In general, the opposite of x is x. Hence, the opposite of 5 is 5 and the opposite of 5 is ( 5) = 5. Example 25.2 Find the opposite of each of these integers. (a) 4 (b) 2 (c) 0 (a) The opposite of 4 is 4. (b) The opposite of 2 is ( 2) = 2. (c) The opposite of 0 is 0 since 0 is neither positive nor negative. Absolute Value of an Integer As an important application of the use of the number line model for integers is the concept of absolute value of an integer. We see from Figure 25.1 that the number 3 is represented by the point numbered 3 and the integer 3 is represented by the point numbered 3. Note that both 3 and 3 are three units from 0 on the number line. The absolute value of an integer x is the distance of the corresponding point on the number line from zero. We indicate the absolute value of x by x. It follows from the above discussion that 3 = 3 = 3. Example 25.3 Determine the absolute values of these integers. (a) 11 (b) 11 (c) 0 5
6 (a) 11 = 11 (b) 11 = 11 (c) 0 = 0. Practice Problems Problem 25.1 Which of the following are integers? (a) 11 (b) 0 (c) 3 4 (d) 9 3 Problem 25.2 Let N = { 1, 2, 3, }. Find (a) N W (b) N N, where N is the set of natural numbers or positive integers and W is the set of whole numbers. Problem 25.3 What number is 5 units to the left of 95? Problem 25.4 A and B are 9 units apart on the number line. A is twice as far from 0 as B. What are A and B? Problem 25.5 Represent 5 using signed counters and number line. Problem 25.6 In terms of distance, explain why 4 = 4. Problem 25.7 Find the additive inverse (i.e. opposite) of each of the following integers. (a) 2 (b) 0 (c) 5 (d) m (e) m Problem 25.8 Evaluate each of the following. (a) 5 (b) 10 (c) 4 (d) 5 Problem 25.9 Find all possible integers x such that x = 2. 6
7 Problem Let W be the set of whole numbers, Z + the set of positive integers (i.e. Z + = N), Z the set of negative integers, and Z the set of all integers. Find each of the following. (a) W Z (b) W Z (c) Z + Z (d) Z + Z (e) W Z + In the remainder of this section we discuss integer addition and subtraction. We will use the devices of signed counters and number lines to illustrate how these operations should be performed in the set of integers. Addition of Integers A football player loses 3 yards on one play and 2 yards on the next, what s the player s overall yardage on the two plays? The answer is 3+( 2) which involves addition of integers. We compute this sum using the two models discussed above. Using Signed Counters Since 3 is represented by 3 negative counters and 2 is represented by 2 negative counters then by combining these counters we obtain 5 negative counters which represent the integer 5. So 3 + ( 2) = 5. Using Number Line Move three units to the left of 0 and then 2 units to the left from 3 as shown in Figure Figure 25.3 Example 25.4 Find the following sums using (i) signed counters, and (ii) number line. (a) ( 7) + 4 (b) 5 + ( 2) (c) 2 + ( 2) 7
8 The results of the above examples are entirely typical and we state them as a theorem. Theorem 25.1 Let a and b be integers. Then the following are always true: (i) ( a) + ( b) = (a + b) (ii) If a > b then a + ( b) = a b (iii) If a < b then a + ( b) = (b a) (iv) a + ( a) = 0. Example 25.5 Compute the following sums. (a) (b) ( 6) + ( 5) (c) 7 + ( 3) (d) 4 + ( 9) (e) 6 + ( 6) (a) = 18 (b) ( 6) + ( 5) = (6 + 5) = 11 (c) 7 + ( 3) = 7 3 = 4 (d) 4 + ( 9) = (9 4) = 5 (e) 6 + ( 6) = 0 Integer addition has all the properties of whole number addition. properties are summarized in the following theorem. These 8
9 Theorem 25.2 Let a, b, and c be integers. Closure a + b is a unique integer. Commutativity a + b = b + a. Associativity a + (b + c) = (a + b) + c. Identity Element 0 is the unique integer such that a + 0 = 0 + a = a. A useful consequence of Theorem 25.1 (iv) is the additive cancellation. Theorem 25.3 For any integers a, b, c, if a + c = b + c then a = b. Proof. a = a + 0 identity element = a + (c + ( c)) additive inverse = (a + c) + ( c) associativity = (b + c) + c given = b + (c + ( c)) associativity = b + 0 additive inverse = b identity element Example 25.6 Use the additive cancellation to show that ( a) = a. Since a + ( a) = ( ( a)) + ( a) = 0 then by the additive cancellation property we have a = ( a) Practice Problems Problem What is the opposite or additive inverse of each of the following? (a and b are integers) (a) a + b (b) a b Problem What integer addition problem is shown on the number line? 9
10 Problem (a) Explain how to compute with a number line. (b) Explain how to compute with signed counters. Problem In today s mail, you will receive a check for $86, a bill for $30, and another bill for $20. Write an integer addition equation that gives the overall gain or loss. Problem Compute the following without a calculator. (a) (b) ( 8) + ( 17) (c) ( 35) Problem Show two ways to represent the integer 3 using signed counters. Problem Illustrate each of the following addition using the signed counters. (a) 5 + ( 3) (b) (c) (d) ( 3) + ( 2) Problem Demonstrate each of the additions in the previous problem using number line model. Problem Write an addition problem that corresponds to each of the following sentences and then answer the question. (a) The temperature was 10 C and then it rose by 8 C. What is the new temperature? (b) A certain stock dropped 17 points and the following day gained 10 points. What was the net change in the stock s worth? (c) A visitor to a casino lost $200, won $100, and then lost $50. What is the change in the gambler s net worth? Problem Compute each of the following: (a) ( 9) + ( 5) (b) 7 + ( 5) (c) ( 7)
11 Problem If a is an element of { 3, 2, 1, 0, 1, 2} and b is an element of { 5, 4, 3, 2, 1, 0, 1}, find the smallest and largest values for the following expressions. (a) a + b (b) a + b Subtraction of Integers Like integer addition, integer subtraction can be illustrated with signed counters and number line. Signed Counters Take Away Approach As with the subtraction of whole numbers, one approach to integer subtraction is the notion of take away. Figure 25.4 illustrates various examples of subtraction of integers. Figure 25.4 Another important fact about subtraction of integers is illustrated in Figure Figure
12 Note that 5 3 = 5 + ( 3). In general, the following is true. Adding the Opposite Approach For all integers a and b, a b = a + ( b). Adding the opposite is perhaps the most efficient method for subtracting integers because it replaces any subtraction problem with an equivalent addition problem. Example 25.7 Find the following differences by adding the opposite. (a) 8 3 (b) 4 ( 5) (a) 8 3 = ( 8) + ( 3) = (8 + 3) = 11. (b) 4 ( 5) = 4 + [ ( 5)] = = 9. MissingAddend Approach Signed counters model can be also used to illustrate the missingaddend approach for the subtraction of integers. Figure 25.6 illustrates both equations 3 ( 2) = 5 and 3 = 5 + ( 2). Figure 25.6 Since all subtraction diagrams can be interpreted in this way, we can introduce the missingaddend approach for subtraction: If a, b, and c are any integers, then a b = c if and only if a = c + b. Number Line Model The number line model used for integer addition can also be used to model integer subtraction. In this model, the operation of subtraction corresponds of starting from the tail of the first number facing the negative direction. 12
13 Thus, from the tail of the first number we move forward if we are subtracting a positive integer and backward if we are subtracting a negative integer. Figure 25.7 illustrates some examples. Figure 25.7 In summary there are three equivalent ways to view subtraction of integers. 1. Take away 2. Adding the opposite 3. Missing addend Example 25.8 Find 4 ( 2) using all three methods of subtraction. Take away: See Figure 25.8 Figure 25.8 Adding the opposite: 4 ( 2) = 4 + [ ( 2)] = = 6. MissingAddend: 4 ( 2) = c if and only if 4 = c + ( 2). But 4 = 6 + ( 2) so that c = 6. Practice Problems 13
14 Problem Explain how to compute 5 ( 2) using a number line. Problem Tell what subtraction problem each picture illustrates. Problem Explain how to compute the following with signed counters. (a) ( 6) ( 2) (b) 2 6 (c) 2 3 (d) 2 ( 4) Problem (a) On a number line, subtracting 3 is the same as moving the. (b) On a number line, adding 3 is the same as moving. units to units to the Problem An elevetor is at an altitude of 10 feet. The elevator goes down 30 ft. (a) Write an integer equation for this situation. (b) What is the new altitude? Problem Compute each of the following using a b = a + ( b). (a) 3 ( 2) (b) 3 2 (c) 3 ( 2) Problem Use numberline model to find the following. (a) 4 ( 1) (b)
15 Problem Compute each of the following. (a) 5 11 (b) ( 4) ( 10) (c) 8 ( 3) (d) ( 8) 2 Problem Find x. (a) x + 21 = 16 (b) ( 5) + x = 7 (c) x 6 = 5 (d) x ( 8) = 17. Problem Which of the following properties hold for integer subtraction. If a property does not hold, disprove it by a counterexample. (a) Closure (b) Commutative (c) Associative (d) Identity 15
16 26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue is whether the result is positive or negative. This section shows how to explain the sign of an integer product or quotient using number line, patterns, and signed counters. Multiplication of Integers Using Patterns Consider the following pattern of equalities which are derived from using the repeated addition. 4 3 = = = = 0 Note that the second factors in the successive products decrease by 1 each time and that the successive results decrease by 4. If this pattern continues we obtain the following sequence of equalities. 4 3 = = = = 0 4 ( 1) = 4 4 ( 2) = 8 4 ( 3) = 12 Since these results agree with what would be obtained by repeated addition, the pattern of the product is an appropriate guide. In general, it suggests that m ( n) = (m n) where m and n are positive integers. This says that the product of a positive integer times a negative integer is always a negative integer. 16
17 Using this result we can write the following pattern of equalities 3 ( 3) = 9 2 ( 3) = 6 1 ( 3) = 3 0 ( 3) = 0 So we notice that when the first factors decrease by 1 the successive results increase by 3. Continuing the pattern we find In general, this suggests that 3 ( 3) = 9 2 ( 3) = 6 1 ( 3) = 3 0 ( 3) = 0 ( 1) ( 3) = 3 ( 2) ( 3) = 6 ( m) ( n) = m n, where m and n are positive integers. Hence the product of two negative integers is always positive. What about the product of a negative integer times a positive integer such as ( 3) 2? Using the results just derived above we see that ( 3) ( 3) = 9 ( 3) ( 2) = 6 ( 3) ( 1) = 3 ( 3) 0 = 0 Note that the second factors in the successive products increase by 1 each time and the successive results decrease by 3. Hence, continuing the pattern we obtain ( 3) ( 3) = 9 ( 3) ( 2) = 6 ( 3) ( 1) = 3 ( 3) 0 = 0 ( 3) 1 = 3 ( 3) 2 = 6 17
18 This suggests the general rule ( n) m = (n m), where m and n are positive integers. That is, a negative number times a positive number is always negative. Summarizing these results into a theorem we have Theorem 26.1 Let m and n be two positive integers. Then the following are true: (1) m( n) = mn (2) ( m)n = mn (3) ( m)( n) = mn. Multiplication of Integers Using Signed Counters The signed counters can be used to illustrate multiplication of integers, although an interpretation must be given to the sign. See Figure 26.1(a). The product 3 2 illustrates three groups each having 2 positive counters. A product such as 3 ( 2) represents three groups each having 2 negative counters as shown in Figure 26.1(b). A product like ( 3) 2 is interpreted as taking away 3 groups each having 2 positive counters as shown in Figure 26.1(c), and the product ( 3) ( 2) is interpreted as taking away three groups each consisting of two negative counters as shown in Figure 26.1(d). 18
19 Figure 26.1 Multiplication of Integers Using Number Line Since a product has a first factor and a second factor we can identify the first factor as the velocity of a moving object who starts from 0 and the second factor is time. A positive velocity means that the object is moving east and a negative velocity means that the object is moving west. Time in future is denoted by a positive integer and time in the past is denoted by a negative integer. Example 26.1 Use a numberline model to answer the following questions. (a) If you are now at 0 and travel east at a speed of 50km/hr, where will you be 3 hours from now? (b) If you are now at 0 and travel east at a speed of 50km/hr, where were you 3 hours ago? (c) If you are now at 0 and travel west at a speed of 50km/hr, where will you be 3 hours from now? (d) If you are now at 0 and travel west at a speed of 50km/hr, where were you 3 hours ago? Figure 26.2 illustrates the various products using a numberline model. 19
20 Figure 26.2 The set of integers has properties under multiplication analogous to those of the set of whole numbers. We summarize these properties in the following theorem. Theorem 26.2 Let a, b, and c be any integers. Then the following are true. Closure a b is a unique integer. Commutativity a b = b a Associativity a (b c) = (a b) c Identity Element a 1 = 1 a = a Distributivity a (b + c) = a b + a c. Zero Product a 0 = 0 a = 0. Using the previous theorem one can derive some important results that are useful in computations. Theorem 26.3 For any integer a we have ( 1) a = a. 20
21 Proof. First, note that 0 = 0 a = [1 + ( 1)] a = a + ( 1) a. But a + ( a) = 0 so that a+( a) = a+( 1)a. By the additive cancellation property we conclude that a = ( 1) a We have seen that ( a)b = ab and ( a)( b) = ab where a and b are positive integer. The following theorem shows that those two equations are true for any integer not just positive integers. Theorem 26.4 Let a and b be any integers. Then ( a)b = ab and ( a)( b) = ab. Proof. To prove the first equation, we proceed as follows. ( a)b = [( 1)a]b = ( 1)(ab) = ab. To prove the second property, we have ( a)( b) = ( 1)[a( b)] = ( 1)[( a)b] = ( 1)[( 1)(ab)] = ( 1)( 1)(ab) = ab Another important property is the socalled the multiplicative cancellation property. Theorem 26.5 Let a, b, and c be integers with c 0. If ac = bc then a = b. Proof. Suppose c 0. Then either c > 0 or c < 0. If c = c > 0 then we know from 1 Theorem 20.1 that c has a multiplicative inverse 1, i.e. c cc 1 = 1. In this case, we have a = a 1 = a(cc 1 ) = (ac)c 1 = (bc)c 1 = b(cc 1 ) = b 1 = b 21
22 If c < 0 then c > 0 and we can use the previous argument to obtain a = a 1 = a[( c)( c) 1 ] = [a( c)]( c) 1 = (ac)( c) 1 = (bc)( c) 1 = [b( c)]( c) 1 = b[( c)( c) 1 ] = b 1 = b The following result follows from the multiplicative cancellation property. Theorem 26.6 (Zero Divisor) Let a and b be integers such that ab = 0. Then either a = 0 or b = 0. Proof. Suppose that b 0. Then ab = 0 b. By the previous theorem we must have a = 0 Practice Problems Problem 26.1 Use patterns to show that ( 1)( 1) = 1. Problem 26.2 Use signed counters to show that ( 4)( 2) = 8. Problem 26.3 Use number line to show that ( 4) 2 = 8. Problem 26.4 Change 3 ( 2) into a repeated addition and then compute the answer. Problem 26.5 (a) Compute 4 ( 3) with repeated addition. (b) Compute 4 ( 3) using signed counters. (c) Compute 4 ( 3) using number line. 22
23 Problem 26.6 Show how ( 2) 4 can be found by extending a pattern in a wholenumber multiplication. Problem 26.7 Mike lost 3 pounds each week for 4 weeks. (a) What was the total change in his weight? (b) Write an integer equation for this situation. Problem 26.8 Compute the following without a calculator. (a) 3 ( 8) (b) ( 5) ( 8) ( 2) ( 3) Problem 26.9 Extend the following pattern by writing the next three equations. 6 3 = = = = 0 Problem Find the following products. (a) 6( 5) (b) ( 2)( 16) (c) ( 3)( 5) (d) 3( 7 6). Problem Represent the following products using signed counters and give the results. (a) 3 ( 2) (b) ( 3) ( 4) Problem In each of the following steps state the property used in the equations. a(b c) = a[b + ( c)] = ab + a( c) = ab + [ (ac)] = ab ac 23
24 Problem Extend the meaning of a whole number exponent a n = a a a a } {{ } n factors where a is any integer and n is a positive integer. Use this definition to find the following values. (a) ( 2) 4 (b) 2 4 (c) ( 3) 5 (d) 3 5 Problem Illustrate the following products on an integer number line. (a) 2 ( 5) (b) 3 ( 4) (c) 5 ( 2) Problem Expand each of the following products. (a) 6(x + 2) (b) 5(x 11) (c) (x 3)(x + 2) Problem Name the property of multiplication of integers that is used to justify each of the following equations. (a) ( 3)( 4) = ( 4)( 3) (b) ( 5)[( 2)( 7)] = [( 5)( 2)]( 7) (c) ( 5)( 7) is a unique integer (d) ( 8) 1 = 8 (e) 4 [( 8) + 7] = 4 ( 8) Problem If 3x = 0 what can you conclude about the value of x? Problem If a and b are negative integers and c is positive integer, determine whether the following are positive or negative. (a) ( a)( c) (b) ( a)(b) (c) (c b)(c a) (d) a(b c) Problem Is the following equation true? a (b c) = (a b) (a c). 24
25 Division of Integers Suppose that a town s population drops by 400 people in 5 years. What is the average population change per year? The answer is given by 400 5, where the negative sign indicates a decrease in population. Such a problem requires division of integers which we dicuss next. You recall the missingfactor approach for the division of whole numbers: we write a b, b 0 to mean a unique whole number c such that a = bc. Division on the set of integers is defined analogously: Let a and b be any integers with b 0. Then a b is the unique integer c, if it exists, such that a = bc. Example 26.2 Find the following quotients, if possible. (a) 12 ( 4) (b) ( 12) 4 (c) ( 12) ( 4) (d) 7 ( 2) (a) Let c = 12 ( 4). Then 4c = 12 and consequently, c = 3. (b) If c = ( 12) 4 then 4c = 12 and solving for c we find c = 3. (c) Letting c = ( 12) ( 4) we obtain 4c = 12 and consequently c = 3. (d) Let c = 7 ( 2). Then 2c = 7 Since there is no integer when multiplied by 2 yields 7 then we say that 7 ( 2) is undefined in the integers The previous example suggests the following rules of division of integers. (i) The quotient, if it exists, of two integers with the same sign is always positive. (ii) The quotient, if it exists, of two integers with different signs is always negative. Negative Exponents and Scientific Notation Recall that for any whole number a and n, a positive integer we have a n = } a a {{ a a}. n factors We would like to extend this definition to include negative exponents. This is done by using looking for a pattern strategy. Consider the following 25
26 sequence of equalities where a is a non zero integer. a 3 = a a a a 2 = a a a 1 = a a 0 = 1 We see that each time the exponent is decreased by 1 the result is being divided by a. If this pattern continues we will get, a 1 = 1 a, a 2 = 1 a 2, etc. In general, we have the following definition. Let a 0 be any integer and n be a positive integer then a n = 1 a n. It can be shown that the theorems on wholenumber exponents discussed in Section 10 can be extended to integer exponents. We summarize these properties in the following theorem. Theorem 26.7 Let a, b be integers and m, n be positive integers. Then (a) a m a n = a m+n (b) am = a m n a n (c) (a m ) n = a mn (d) (a b) m = a m b m (e) ( a b ) m = a m b m, assuming b 0 (f) ( a b ) m = ( b a) m. Example 26.3 Write each of the following in simplest form using positive exponents in the final answer. (a) (b) (c) ( ) 10 3 (a) = (2 4 ) 2 (2 3 ) 3 = = = 2 1 = 1 2. (b) = (5 2 2 ) = (5 2 (2 2 ) 2 ) 2 4 = ( ) 2 4 = 5 2 = 25. (c) ( ) 10 3 = = = =
27 In application problems that involve very large or very small numbers, we use scientific notation to represent these numbers. A number is said to be in scientific notation when it is expressed in the form a 10 n where 1 a < 10 is called the mantissa and n is an integer called the characteristic. For example, the diameter of Jupiter in standard notation is 143,800,000 meters. Using scientific notation this can be written in the form meters. When converting from standard notation to scientific notation and vice versa recall the following rules of multiplying by powers of 10: When you multiply by 10 n move the decimal point n positions to the right. When you multiply by 10 n move the decimal point n positions to the left. Example 26.4 Convert as indicated. (a) 38,500,000 to scientific notation (b) to standard notation (c) to standard notation (d) to scientific notation. (a) 38, 500, 000 = (b) = 413, 500, 000, 000 (c) = (d) = Order of Operations on Integers When addition, subtraction, multiplication, division, and exponentiation appear without parentheses, exponentiation is done first, then multiplication and division are done from left to right, and finally addition and subtraction from left to right. Arithmetic operations that appear inside parentheses must be done first. Example 26.5 Evaluate each of the following. (a) (2 5) (b) (c) (a) (2 5) = ( 3) = = 9. 27
28 (b) = = = 18. (c) = = 4. Practice Problems Problem Perform these divisions. (a) 36 9 (b) ( 36) 9 (c) 36 ( 9) (d) ( 36) ( 9) (e) 165 ( 11) (f) Problem Write another multiplication equation and two division equations that are equivalent to the equation ( 11) ( 25, 753) = 283, 283. Problem Write two multiplication equations and another division equation that are equivalent to the equation ( 1001) 11 = 91. Problem Use the definition of division to find each quotient, if possible. If a quotient is not defined, explain why. (a) ( 40) ( 8) (b) ( 143) 11 (c) 0 ( 5) (d) ( 5) 0 (e) 0 0 Problem Find all integers x (if possible) that make each of the following true. (a) x 3 = 12 (b) x ( 3) = 2 (c) x ( x) = 1 (d) 0 x = 0 (e) x 0 = 1 Problem Write two division equations that are equivalent to 3 ( 2) = 6. Problem Explain how to compute 10 2 using signed counters. 28
29 Problem Rewrite each of the following as an equivalent multiplication problem, and give the solution. (a) ( 54) ( 6) (b) 32 ( 4) Problem A store lost $480,000 last year. (a) What was the average net change per month? (b) Write an integer equation for this situation. Problem Compute the following, using the correct rules for order of operations. (a) (b) 5 + ( 4) 2 ( 2) Problem A stock change as follows for 5 days:2,4,6,3,1. What is the average daily change in price? Problem Compute: 2 ( 2) + ( 2) ( 2) Problem For what integers a and b does a b = b a? Problem Find each quotient, if possible. (a) [144 ( 12)] ( 3) (b) 144 [ 12 ( 3)] Problem Compute the following writing the final answer in terms of positive exponents. (a) (b) (c) (3 4 ) 2 Problem Express each of the following in scientific notation. (a) (b) (c)
30 Problem Hair on the human body can grow as fast as meter per second. (a) At this rate, how much would a strand of hair grow in one month of 30 days? Express your answer in scientific notation. (b) About how long would it take for a strand of hair to grow to be 1 meter in length? Problem Compute each of these to three significant figures using scientific notation. (a) ( ) ( ) (b) ( ) ( ) Problem Convert each of the following to standard notation. (a) (b) Problem Write each of the following in scientific notation. (a) 413,682,000 (b) (c) 100,000,000 Problem Evaluate each of the following. (a) ( 2) 2 (b) (c) ( 2) Problem Evaluate each of the following, if possible. (a) ( 10 ( 2))( 2) (b) ( 10 5) 5 (c) 8 ( 8 + 8) (d) ( 6 + 6) ( 2 + 2) (e) 24 4 (3 15) Comparing and Ordering Integers In this section we extend the notion of less than to the set of all integers. We describe two equivalent ways for viewing the meaning of less than: a 30
31 number line approach and an addition (or algebraic) approach. In what follows, a and b denote any two integers. NumberLine Approach We say that a is less than b, and we write a < b, if the point representing a on the numberline is to the left of b. For example, Figure 26.3 shows that 4 < 2. Figure 26.3 Example 26.6 Order the integers 1, 0, 4, 3, 2, 2, 4, 3, 1 from smallest to largest using the number line approach. The given numbers are ordered as shown in Figure 26.4 Figure 26.4 Addition Approach Note that from Figure 26.3 we have 2 = ( 4) + 6 so we say that 2 is 6 more than 2. In general, if a < b then we can find a unique integer c such that b = a + c. Example 26.7 Use the addition approach to show that 7 < 3. Since 3 = ( 7) + 4 then 3 is 4 more than 7, that is 7 < 3 Notions similar to less than are included in the following table. 31
32 Inequality Symbol Meaning < less than > greater than less than or equal greater than or equal The following rules are valid for any of the inequality listed in the above table. Rules for Inequalities Trichotomy Law: For any integers a and b exactly one of the following is true: a < b, a > b, a = b. Transitivity: For any integers a, b, and c if a < b and b < c then a < c. Addition Property: For any integers a, b, and c if a < b then a+c < b+c. Multiplication Property: For any integers a, b, and c if a < b then ac < bc if c > 0 and ac > bc if c < 0. The first three properties are extensions of similar statements in the whole number system. Note that the fourth property asserts that an inequality symbol must be reversed when multiplying by a negative integer. This is illustrated in Figure 26.5 when c = 1. Figure 26.5 Practice Problems Problem Use the numberline approach to verify each of the following. (a) 4 < 1 (b) 4 < 2 (c) 1 > 5 Problem Order each of the following lists from smallest to largest. (a) { 4, 4, 1, 1, 0} (b) {23, 36, 45, 72, 108} 32
33 Problem Replace the blank by the appropriate symbol. (a) If x > 2 then x (b) If x < 3 then x 6 9 Problem Determine whether each of the following statements is correct. (a) 3 < 5 (b) 6 < 0 (c) 3 3 (d) 6 > 5 (e) Problem What different looking inequality means the same as a < b? Problem Use symbols of inequalities to represent at most and at least. Problem For each inequality, determine which of the numbers 5, 0, 5 satisfies the inequality. (a) x > 5 (b) 5 < x (c) 5 > x Problem Write the appropriate inequality symbol in the blank so that the two inequalities are true. (a) If x 3 then 2x 6 (b) If x + 3 > 9 then x 6 Problem How do you know when to reverse the direction of an inequality symbol? Problem Show that each of the following inequality is true by using the addition approach. (a) 4 < 2 (b) 5 < 3 (c) 17 > 23 Problem A student makes a connection between debts and negative numbers. So the number 2 represents a debt of $2. Since a debt of $10 is larger than a debt of $5 then the student writes 10 > 5. How would you convince him/her that this inequality is false? 33
34 Problem At an 8% sales tax rate, Susan paid more than $1500 sales tax when she purchased her new Camaro. Describe this situation using an inequality with p denoting the price of the car. Problem Show that if a and b are positive integers with a < b then a 2 < b 2. Does this result hold for any integers? Problem Elka is planning a rectangular garden that is twice as long as it is wide. If she can afford to buy at most 180 feet of fencing, then what are the possible values for the width? 34
35 27 Rational Numbers Integers such as 5 were important when solving the equation x+5 = 0. In a similar way, fractions are important for solving equations like 2x = 1. What about equations like 2x + 1 = 0? Equations of this type require numbers like 1. In general, numbers of the form a where a and b are integers with b 0 2 b are solutions to the equation bx = a. The set of all such numbers is the set of rational numbers, denoted by Q : Q = { a b : a, b Z, b 0}. That is, the set of rational numbers consists of all fractions with their opposites. In the notation a we call a the numerator and b the denominator. b Note that every fraction is a rational number. Also, every integer is a rational number for if a is an integer then we can write a = a. Thus, Z Q. 1 Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. The relationship is shown in Figure 27.1 Figure 27.1 All properties that hold for fractions apply as well for rational numbers. Equality of Rational Numbers: Let a and c be any two rational numbers. b d Then a = c if and only if ad = bc (Crossmultiplication). b d 35
36 Example 27.2 Determine if the following pairs are equal. (a) 3 12 (b) and and. 215 (a) Since 3(144) = ( 12)( 36) then 3 = (b) Since ( 21)(215) (86)( 51) then The Fundamental Law of Fractions: Let a b and n be a nonzero integer then a b = an bn = a n b n. be any rational number As an important application of the Fundamental Law of Fractions we have a b = ( 1)a ( 1)( b) = a b. We also use the notation a for either a a or. b b b Example 27.3 Write three rational numbers equal to 2. 5 By the Fundamental Law of Fractions we have 2 5 = 4 10 = 6 15 = 8 20 Rational Numbers in Simplest Form: A rational number a is in simplest form if a and b have no common factors greater than 1. b The methods of reducing fractions into simplest form apply as well with rational numbers. Example 27.4 Find the simplest form of the rational number Using the prime factorizations of 294 and 84 we find = ( 2) = 7 2 =
37 Practice Problems Problem 27.1 Show that each of the following numbers is a rational number. (a) 3 (b) (c) 5.6 (d) 25% Problem 27.2 Which of the following are equal to 3? 3 1, 3 1, 3 1, 3 1, 3 1, 3 1, 3 1. Problem 27.3 Determine which of the following pairs of rational numbers are equal. (a) 3 5 (b) and and. 60 Problem 27.4 Rewrite each of the following rational numbers in simplest form. (a) 5 (b) 21 (c) 8 (d) Problem 27.5 How many different rational numbers are given in the following list? 2 4, 3, 5 10, 39 13, 7 4 Problem 27.6 Find the value of x to make the statement a true one. (a) 7 = x (b) 18 = x Problem 27.7 Find the prime factorizations of the numerator and the denominator and use them to express the fraction 247 in simplest form. 77 Problem 27.8 (a) If a = a, what must be true? b c (b) If a = b, what must be true? c c 37
38 Addition of Rational Numbers The definition of adding fractions extends to rational numbers. a b + c b = a + c b Example 27.5 Find each of the following sums. (a) (b) (c) a b + c d = ad + bc bd (a) = ( 5) = = ( 10)+3 = 7 ( 3) ( 5) (b) = ( 2) ( 5) ( 5) ( 7) = = ( 14)+( 20) 35 = (c) = 3+( 5) = Rational numbers have the following properties for addition. Theorem 27.1 Closure: a + c is a unique rational number. b d Commutative: a + c = c + a b d d b Associative: ( a + ) ( ) c b d + e = a + c + e f b d f Identity Element: a + 0 = a b b Additive inverse: a + ( ) a b b = 0 Example 27.6 Find the additive inverse for each of the following: (a) 3 (b) 5 (c) 2 (d) (a) 3 5 = 3 5 = 3 5 (b) 5 11 (c) 2 3 (d)
39 Subtraction of Rational Numbers Subtraction of rational numbers like subtraction of fractions can be defined in terms of addition as follows. a b c d = a b + ( c d ). Using the above result we obtain the following: ) Example 27.7 Compute Practice Problems a c = a + ( c b d b d a = + c b d = ad+b( c) bd ad bc = bd = = 206 ( 105) 48 = = Problem 27.9 Use number line model to illustrate each of the following sums. (a) (b) (c) Problem Perform the following additions. Express your answer in simplest form. (a) 6 25 (b) Problem Perform the following subtractions. Express your answer in simplest form. (a) (b) Problem Compute the following differences. (a) 2 9 (b) ( )
40 Multiplication of Rational Numbers The multiplication of fractions is extended to rational numbers. That is, if are any two rational numbers then a b and c d a b c d = ac bd. Multiplication of rational numbers has properties analogous to the properties of multiplication of fractions. These properties are summarized in the following theorem. Theorem 27.2 Let a, c, and e be any rational numbers. Then we have the following: b d f Closure: The product of two rational numbers is a unique rational number. Commutativity: a c b ( = c a d ). d b Associativity: a c e = ( a ) c b d f b d e. f Identity: a 1 = a = 1 a. b b b a Inverse: b = 1. We call b the reciprocal of a or the multiplicative b a a b inverse of a. b ( ) Distributivity: a c + e = a c + a e. b d f b d b f Example 27.8 Perform each of the following multiplications. Express your answer in simplest form. (a) 5 7 (b) (a) We have (b) = ( 5) = = = ( 1)( 5) 2(9) = 5 18 Example 27.9 Use the properties of multiplication of rational numbers to compute the following. 40
41 (a) 3 ( 11 ) (b) 2 ( 3 + ) (c) (a) 3 ( 11 ) = = 3 55 = 1 11 = (b) 2 ( 3 + ) = 2 31 = 1 31 = (c) = = ( 5 + ) = Division of Rational Numbers We define the division of rational numbers as an extension of the division of fractions. Let a and c be any rational numbers with c 0. Then b d d a b c d = a b d c. Using words, to find a b c d multiply a b by the reciprocal of c d. By the above definition one gets the following two results. and a b c b = a c a b c d = a c b d. Remark 27.1 After inverting, it is often simplest to cancel before doing the multiplication. Cancelling is dividing one factor of the numerator and one factor of the 2 denominator by the same number. For example: 3 = 2 12 = 2 12 = = Remark 27.2 Exponents and their properties are extended to rational numbers in a natural way. For example, if a is any rational number and n is a positive integer then a n = a a a } {{ } n factors and a n = 1 a n 41
42 Example Compute the following and express the answers in simplest form. (a) 7 2 (b) 13 4 (c) (a) = = ( 7)(3) (4)(2) = (b) = = ( 4) = (c) = = = 3 Practice Problems Problem Multiply the following rational numbers. form. Write your answers in simplest (a) 3 10 (b) 6 33 (c) Problem Find the following quotients. Write your answers in simplest form. (a) 8 2 (b) 12 4 (c) Problem State the property that justifies each statement. (a) ( 5 ) = 5 ( 7 ) ( (b) ) = Problem Compute the following and write your answers in simplest form. (a) (b) 21 3 (c) Problem Find the reciprocals of the following rational numbers. (a) 4 (b) 0 (c) 3 (d)
43 Problem Compute: ( 4 ) Problem If a b 4 7 = 2 3 what is a b? Problem Compute Problem Compute Problem Compute each of the following: (a) ( 3 2 4) (b) ( ) 3 2 (c) ( 3 2 ( 4 4) 3 ) 7 4 Comparing and Ordering Rational Numbers In this section we extend the notion of less than to the set of all rationals. We describe two equivalent ways for viewing the meaning of less than: a number line approach and an addition (or algebraic) approach. In what follows, a and c denote any two rationals. b d NumberLine Approach We say that a is less than c, and we write a < c, if the point representing b d b d a on the numberline is to the left of c. For example, Figure 27.2 shows that b d 1 < Figure 27.2 Example Use the number line approach to order the pair of numbers 3 7 and 5 2. When the two numbers have unlike denominators then we find the least common denominator and then we order the numbers. Thus, 3 = 6 and 5 =
44 Hence, on a number line, 3 7 is to the left of 5 2. Addition Approach As in the case of ordering integers, we say that a b fraction e such that a + e = c. f b f d Example Use the addition approach to show that 3 7 < 5 2. Since 5 = then 3 7 < 5 2 < c d if there is a unique Notions similar to less than are included in the following table. Inequality Symbol Meaning < less than > greater than less than or equal greater than or equal The following rules are valid for any of the inequality listed in the above table. Rules for Inequalities Trichotomy Law: For any rationals a and c exactly one of the following b d is true: a b < c d, a b > c d, a b = c d. Transitivity: For any rationals a, c, and e if a < c and c < e then a < e. b d f b d d f b f Addition Property: For any rationals a, c, and e if a < c then a + e < b d f b d b f c + e. d f Multiplication Property: For any rationals a, c, and e if a < c then b d f b d a e < c e if e > 0 and a e > c e if e < 0 b f d f f b f d f f Density Property: For any rationals a and c, if a < c then b d b d Practice Problems a b < 1 ( a 2 b + c ) < c d d. Problem True or false: (a) 2 3 < 3 7 (b)
45 Problem Show that 3 < Problem Show that 3 < using the addition approach. by using a number line. Problem Put the appropriate symbol, <, =, > between each pair of numbers to make a true statement. (a) (b) 1 3 (c) (d) Problem Find three rational numbers between 1 4 and 2 5. Problem The properties of rational numbers are used to solve inequalities. For example, x < 5 10 x ( 3 5 5) < 7 + ( ) Solve the inequality x < x > 1. Problem Solve each of the following inequalities. (a) x 6 5 < 12 7 (b) 2 5 x < 7 8 (c) 3 7 x >
46 Problem Verify the following inequalities. (a) 4 < 3 (b) 1 < 1 (c) 19 > Problem Use the numberline approach to arrange the following rational numbers in increasing order: (a) 4, 1, (b) 7, 2, Problem Find a rational number between 5 12 and 3 8. Problem Complete the following, and name the property that is used as a justification. (a) If 2 3 < 3 4 and 3 4 < 7 5 then (b) If 3 < 6 then ( ) ( ) ( 6 ) ( ) 3 (c) If 4 7 < 7 4 then < (d) If 3 < 11 then ( ) ( ) ( ) ( 5 ) 7 (e) There is a rational number any two unequal rational numbers. 46
47 28 Real Numbers In the previous section we introduced the set of rational numbers. We have seen that integers and fractions are rational numbers. Now, what about decimal numbers? To answer this question we first mention that a decimal number can be terminating, nonterminating and repeating, nonterminating and nonrepeating. If the number is terminating then we can use properties of powers of 10 to write the number as a rational number. For example, = 123 and = Thus, every terminating decimal number is a rational number. 100 Now, suppose that the decimal number is nonterminating and repeating. For the sake of argument let s take the number where 341 repeats indefinitely. Then = Let x = Then 1000x = = x. Thus, 999x = 341 and therefore x = 341. It follows that = = Hence, every nonterminating repeating decimal is a rational number. Next, what about nonterminating and non repeating decimals. Such numbers are not rationals and the collection of all such numbers is called the set of irrational numbers. As an example of irrational numbers, let s consider finding the hypotenuse c of a right triangle where each leg has length 1. Then by the Pythagorean formula we have c 2 = = 2. We will show that c is irrational. Suppose the contrary. That is, suppose that c is rational so that it can be written as a b = c where a and b 0 are integers. Squaring both sides we find a2 = c 2 = 2 or b 2 a 2 = 2 b 2. If a has an even number of prime factors then a 2 has an even number of prime factors. If a has an odd number of prime factors then a 2 has an even number of prime factors. So, a 2 and b 2 have both even number of prime factors. But 2 b 2 has an odd number of prime factors. So we have that a 2 has an even number and an odd number of prime factors. This 47
48 cannot happen by the Fundamental Theorem of Arithmetic which says that every positive integer has a unique number of prime factors. In conclusion, c cannot be written in the form a so it is not rational. Hence, its decimal b form is nonterminating and nonrepeating. Remark 28.1 It follows from the above discussion that the equation c 2 = 2 has no answers in the set of rationals. This is true for the equation c 2 = p where p is a prime number. Since there are infinite numbers of primes then the set of irrational numbers is infinite. We define the set of real numbers to be the union of the set of rationals and the set of irrationals. We denote it by the letter R. The relationships among the sets of all numbers discussed in this book is summarized in Figure 28.1 Figure 28.1 Now, in the set of real numbers, the equation c 2 = 2 has a solution denoted by 2. Thus, ( 2) 2 = 2. In general, we say that a positive number b is the square root of a positive number a if b 2 = a. We write a = b. Representing Irrational Numbers on a Number Line Figure 28.2 illustrates how we can accurately plot the length 2 on the number line. Figure
49 Practice Problems Problem 28.1 Given a decimal number, how can you tell whether the number is rational or irrational? Problem 28.2 Write each of the following repeating decimal numbers as a fraction. (a) 0.8 (b) 0.37 (c) Problem 28.3 Which of the following describe 2.6? (a) A whole number (b) An integer (c) A rational number (d) An irrational number (e) A real number Problem 28.4 Classify the following numbers as rational or irrational. (a) 11 (b) 3 7 (c) π (d) 16 Problem 28.5 Classify the following numbers as rational or irrational. (a) (b) 0.26 (c) Problem 28.6 Match each word in column A with a word in column B. A Terminating Repeating Infinite nonrepeating B Rational Irrational Problem 28.7 (a) How many whole numbers are there between 3 and 3 (not including 3 and 3)? (b) How many integers are there between 3 and 3? (c) How many real numbers are there between 3 and 3? 49
50 Problem 28.8 Prove that 3 is irrational. Problem 28.9 Show that is irrational. Problem Find an irrational number between 0.37 and 0.38 Problem Write an irrational number whose digits are twos and threes. Problem Classify each of the following statement as true or false. counter example. If false, give a (a) The sum of any rational number and any irrational number is a rational number. (b) The sum of any two irrational numbers is an irrational number. (c) The product of any two irrational numbers is an irrational number. (d) The difference of any two irrational numbers is an irrational number. Arithmetic of Real Numbers Addition, subtraction, multiplication, division, and less than are defined on the set of real numbers in such a way that all the properties on the rationals still hold. These properties are summarized next. Closure: For any real numbers a and b, a + b and ab are unique real numbers. Commutative: For any real numbers a and b, a + b = b + a and ab = ba. Associative: For any real numbers, a, b, and c we have a+(b+c) = (a+b)+c and a(bc) = (ab)c. Identity element: For any real numbers a we have a + 0 = a and a 1 = a. Inverse Elements: For any real numbers a and b 0 we have a + ( a) = 0 and b 1 = 1. b Distributive: For any real numbers a, b, and c we have a(b + c) = ab + ac. Transitivity: If a < b and b < c then a < c. Addition Property: If a < b then a + c < b + c. Multiplication Property: If a < b then ac < bc if c > 0 and ac > bc if 50
51 c < 0. Density: If a < b then a < a+b 2 < b. Practice Problems Problem Construct the lengths 2, 3, 4, 5, as follows. (a) First construct a right triangle with both legs of length 1. What is the length of the hypotenuse? (b) This hypotenuse is the leg of the next right triangle. The other leg has length 1. What is the length of the hypotenuse of this triangle? (c) Continue drawing right triangles, using the hypotenuse of the proceeding triangle as a leg of the next triangle until you have constructed one with length 6. Problem Arrange the following real numbers in increasing order. 0.56, 0.56, 0.566, , Problem Which property of real numbers justify the following statement Problem Find x : x = = (2 + 5) 3 = 7 3 Problem Solve the following equation: 2x 3 6 = 3x + 6 Problem Solve the inequality: 3 2 x 2 < 5 6 x Problem Determine for what real number x, if any, each of the following is true: (a) x = 8 (b) x = 8 (c) x = 8 (d) x = 8 Radical and Rational Exponents By rational exponents we mean exponents of the form a m n where a is any real number and m and n are integers. First we consider the simple case a 1 n (where n is a positive integer) which is defined as follows 51
52 a 1 n = b is equivalent to b n = a. We call b the nth root of a and we write b = n a. We call a the radicand and n the index. The symbol n is called a radical. It follows that a 1 n = n a. Note that the above definition requires a 0 for n even since b n is always greater than or equal to 0. Example 28.1 Compute each of the following. (a) (b) ( 8) 1 3 (c) ( 16) 1 4 (d) is un (a) Since 5 2 = 25 then = 5. (b) Since ( 2) 3 = 8 then ( 8) 1 3 = 2. (c) Since the radicand is negative and the index is even then ( 16) 1 4 defined. (d) Since 2 4 = 16 then = 2 As the last two parts of the previous example have once again shown, we really need to be careful with parentheses. In this case parenthesis makes the difference between being able to get an answer or not. For a negative exponent we define a 1 n = 1 Now, for a general rational exponent we define That is, a 1 n a m n = (a 1 n ) m. a m n = ( n a) m. Now, if b = (a 1 n ) m then b n = ((a 1 n ) m ) n = (a 1 n ) mn = ((a 1 n ) n ) m = (a n n ) m = a m. This shows that b = (a m ) 1 n. Hence, we have established that. or (a 1 n ) m = (a m ) 1 n ( n a) m = n a m. 52
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