A Course in Quantitative Literacy
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A Course in Quantitative Literacy Mark Beintema and Azar Khosravani A Course in Quantitative Literacy by Mark Beintema and Azar Khosravani College of Lake County Columbia College Chicago Preface Why study Quantitative Literacy? Most students sign up for this course to fulfill a general education mathematics requirement. And this text is certainly aimed at that general audience. But by the time the course is completed, the authors hope that you will have developed some appreciation for the usefulness and elegance of the subject. Without doubt, some level of competency and comfort in working with numerical data is needed to navigate the modern world; and we have tried to cover topics that can be used in day to day life. In this book, we will focus on problem solving and critical thinking skills. Our goal is not to prepare you just for the next math class, but to equip you with the necessary tools so that you can apply basic mathematical reasoning to a wide variety of commonly encountered problems. Along the way, we will learn basic logic, how to work with percentages and units, the basics of consumer finance, and how to use and interpret basic statistical data. TABLE OF CONTENTS Chapter 1: Problem Solving 1.1: Inductive and Deductive Reasoning . 1 1.2: General Principles of Problem Solving. 4 1.3: Estimation Methods . 11 Exercises . .15 Chapter 2: Logic 2.1: Statements and Truth Tables . .19 2.2: The Conditional and Related Statements . 29 2.3: Deductive Arguments . .38 2.4: Euler Diagrams . .51 2.5: Fallacies . .62 Chapter 3: Measurement and Error 3.1: Working with Numbers . .67 3.2: Working with Percents . .70 3.3: Working with Units . .85 3.4: Estimation and Measurement Error . .99 Chapter 4: Mathematical Modeling 4.1: Linear Models . 110 4.2: Quadratic Models . 122 4.3: Exponential Growth and Decay . 132 4.4: Other Models . 151 Chapter 5: Mathematics of Finance 5.1: Simple Interest . 159 5.2: Compound Interest . 164 5.3: Consumer Loans . .182 5.4: Annuities . .197 Chapter 6: Statistics 6.1: Sampling, Surveys and Experiments . .208 6.2: Statistical Graphs . 218 6.3: Descriptive Statistics . .231 6.4: Normal Distributions . .248 6.5: Confidence Intervals . .261 6.6: Linear Regression and Correlation . .271 CHAPTER 1: PROBLEM SOLVING In this chapter we will introduce and review some general problem-solving techniques that will be used throughout the text. 1.1 Inductive and Deductive Reasoning. In order to discuss Quantitative Reasoning, we must first distinguish between two different styles of reasoning: Inductive and Deductive. Inductive reasoning is the process of generalizing from experience and/or observation to reach a general conclusion. This type of reasoning is widely used, but can be unreliable as there is no guarantee that the conclusions reached by “inductive reasoning” are correct. For example, consider the following argument: All of the squirrels in this city are grey. Therefore, all squirrels are grey. Even if the premise is true, the conclusion is not. In fact, there are also red squirrels and black squirrels (as a quick internet search or visit to the zoo will verify). So this argument is flawed, and a few minutes reflection reveals why: the conclusion was based on observation from only one specific locale, which would not be inhabited by every possible species of Eurasian red squirrel (Sciurus vulgaris) . squirrel. This argument also illustrates a key concept: To prove an assertion is false, we only need to find one counterexample. For example, in order to disprove the conclusion above, we would only need to observe a single squirrel that was not grey. Thus, inductive reasoning can provide evidence in favor of an assertion, but cannot provide proof. Nonetheless, inductive reasoning is quite useful and even necessary; in the natural and social sciences, information gathered from observation is often used to formulate general hypotheses. Similarly, in mathematics inductive reasoning is often used to find patterns and formulate conjectures. 1 EXAMPLE 1.1.1: Consider the partial list of numbers: 4, 12, 20, 28, 36, … a) Identify and describe a pattern in this list of numbers. b) Use the pattern to find the next number. c) Use the pattern to find the 12th number in the list d) If possible, find a formula for the n-th number in the list. Solution: A quick examination of the list reveals that the numbers are evenly spaced; that is, the difference between any two consecutive numbers is 8. Thus, we can describe the pattern: a) Each number in the list is obtained by adding 8 to its predecessor. b) Thus, we would expect the next number in the list to be 36 + 8 = 44. c) To find the 12th number, we just continue listing numbers according to the rule in part a: 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92 For part d, we look at the pattern more carefully. The first number is 4 + 0*8 The second number is 4 + 1*8 The third number is 4 + 2*8 … The 12th number is 4 + 11*8 This suggests that for the n-th number in the list, we will have added 8 a total of n – 1 times. So we can generalized the pattern by saying that the n-th number is 4 + (n – 1)*8. Example (DIY) Repeat the previous example for the list 5, 15, 45, 135, … Again, inductive reasoning cannot guarantee that the conclusion is correct. For this reason, inductive thinking has been treated skeptically from the time of the Roman empire to more modern philosophers (e.g. Hume and Russell). But inductive arguments run the gamut from very weak to very strong. For example, our squirrel example was a fairly weak inductive argument, as it was based on very limited observation. On the other hand, Charles Darwin’s inductive argument in support of the theory of natural selection was based on careful observation of hundreds of different species from locations all around the world, and he wrote several detailed books in support of his theory. This constituted an extraordinarily strong argument. As another example, the modern study of Statistics provides a systematic 2 way of using observed sample data to infer conclusions about a large population which is in essence strong inductive reasoning. And carefully designed statistical studies can provide very strong evidence in support of a hypothesis. (We will explore some of these ideas in a later chapter). Deductive reasoning involves deriving specific conclusions from general statements that are known or assumed to be true. For example, suppose that a college catalog states that all new freshmen must enroll in an English class. Suppose further that Mary is a freshman; then it follows that Mary must enroll in an English class. Most mathematical studies involve a mix of inductive and deductive reasoning. When faced with a new problem, we usually start by working some basic examples, and try to recognize a pattern in the results. That is, we use inductive reasoning to develop a conjecture about the situation – then we use deductive reasoning to prove that our approach and answer are correct. This is not unlike the scientific method – scientists usually formulate an initial hypothesis based on observation (either in the natural world or in the lab), and then test that hypothesis experimentally. We will discuss deductive reasoning at length in Chapter 2, but for now let’s look at an example involving both types of reasoning: EXAMPLE 1.1.2: Select a number, and perform the following steps: Multiply the number by 6. Add 8 to the product. Divide this sum by 2. Subtract 4 from the quotient. a) Repeat this procedure for at least four different numbers. Write a conjecture that relates the result of this process to the original number selected. b) Represent the original number as n, and use deductive reasoning to prove the conjecture in part (a). Solution: a) Let’s carry the steps out for the numbers 1, 5, 7 and 12. It will help to organize the results into a table: Starting number x 6 + 8 ÷ 2 – 4 1 6 14 7 3 5 30 38 19 15 7 42 50 25 21 12 72 80 40 36 In each case, the ending value is three times the starting number, so we conjecture that: No matter what number we start with, the ending number will be triple the starting number. 3 But we should understand that these limited calculations do not constitute a proof. In fact, it is not even a particularly strong inductive argument. First, using just four numbers does provide a lot of evidence. And for simplicity we used four positive whole numbers – but maybe the conjecture would fail if we used negative numbers, or if we used fractions, or even if we used anything besides these four values! To answer part b, we let n be any number. Let’s perform all of the steps and see what we end up with: Select a number: n Multiply the number by 6: 6n Add 8 to the product: 6n + 8 Divide the sum by 2: 3n + 4 Subtract 4 from the quotient: 3n This shows that when we start with n, we end with 3n; and this will be true no matter what number we start with. Thus, the conjecture is true. 1.2 General Problem-Solving Techniques. Our goal in this chapter (and throughout the text) is to develop problem-solving and critical thinking skills. And no discussion of problem solving is complete without a review of Pólya’s acclaimed “Four-Step Process”. George Pólya was a Hungarian mathematician who made significant contributions to multiple areas of Mathematics.