16,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2018
Use mathematical analysis in the real world Finite math takes everything you've learned in your previous math courses and brings them together into one course with a focus on organizing and analyzing information, creating mathematical models for approaching business decisions, using statistics principles to understand future states, and applying logic to data organization. Finite Math For Dummies tracks to a typical college-level course designed for business, computer science, accounting, and other non-math majors, and is the perfect supplement to help you score high! * Organize and analyze information * Apply calculation principles to real-world problems * Use models for business calculations * Supplement your coursework with step-by-step example problems If you're not a math person or just want to brush up on your skills to get a better grade, Finite Math For Dummies is your ticket to scoring higher!
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 363
Ähnliche
Finite Math For Dummies®
Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com
Copyright © 2018 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at www.wiley.com/go/permissions.
Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc. and may not be used without written permission. All other trademarks are the property of their respective owners. John Wiley & Sons, Inc. is not associated with any product or vendor mentioned in this book.
LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE. NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS. THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION. THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES. IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT. NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM. THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE. FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ.
For general information on our other products and services, please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993, or fax 317-572-4002. For technical support, please visit https://hub.wiley.com/community/support/dummies.
Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.
Library of Congress Control Number: 2018934637
ISBN 978-1-119-47636-8 (pbk); ISBN 978-1-119-47643-6 (ebk); ISBN 978-1-119-47644-3 (ebk)
Finite Math For Dummies®
To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Finite Math For Dummies Cheat Sheet” in the Search box.
Table of Contents
Cover
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting Started with Finite Math
Chapter 1: Feeling Fine with Finite Math
Getting in Line with Linear Statements
Making the Most with Matrices
Staying with the Program
Getting Set with Sets
Posing the Probability
Figuring in Financial Factors
Finding Statistical Satisfaction
Considering the Logical Side of Mathematics
Unlocking the Chains
Getting into Gaming
Chapter 2: Lining Up Linear Functions
Recognizing Equations of Lines
Graphing Lines on the Coordinate Plane
Determining Relationships between Lines
Solving for a Variable
Chapter 3: Solving Systems of Linear Equations
Solving Systems Using Elimination
Solving Systems Using Substitution
Dealing with Too Many or No Solutions
Making Linear Equations Work for You
Chapter 4: Taking on Systems of Inequalities
Ruling with Inequalities
Graphing Linear Inequalities
Part 2: Making Use of Available Methods
Chapter 5: Making Way for Matrices
Squaring Off with Matrix Basics
Investigating Row Operations
Applying Matrices and Their Operations
Chapter 6: Making Matrices Work for You
Solving Systems of Equations Using Matrices
Discovering Multiple Solutions
Applying Matrices to Tasks
Taking Advantage of Special Formulas
Chapter 7: Getting Lined Up with Linear Programming
Setting Up Linear Programming Problems
Going Three-Dimensional
Chapter 8: Simply the Simplex Method
Delineating Simplex Method Steps for Maximization
Making the Most of Minimization
Part 3: Using Finite Math to Tackle World Situations
Chapter 9: Setting Up Sets
Introducing Set Notation
Performing Basic Operations
Using Venn Diagrams for Better Views
Chapter 10: Processing the Probability
Introducing Counting Methods
Determining the Probability of an Event
Applying Probability Techniques
Chapter 11: Counting on the Mathematics of Finance
Considering Simple Interest
Compounding Things with Compound Interest
Analyzing Annuities
Amortization
Chapter 12: Telling the Truth with Statistics
Presenting Data Graphically
Measures of Central Tendency
Variance and Standard Deviation
Investigating the Normal Distribution
Chapter 13: Logic
Logically Presenting the Vocabulary
Finding the Truth with Truth Tables
Equivalent Statements
Studying the Conditional
Analyzing Arguments
Applying Logic to Circuits
Part 4: Employing the Tools of Finite Math to Expand and Investigate
Chapter 14: Markov Chains
Recognizing a Markov Chain
Coming to Terms with Markov Chains
Working with Transition Matrices
Probability Vectors
Absorbing Chains
Making Long-Term Predictions
Chapter 15: Playing Games with Game Theory
Playing Fairly
Playing by the Rules
Getting Strategic
Yielding to Domination
Determining the Moves
Chapter 16: Applications of Matrices and Game Theory
Traffic Flow
Battle of the Bismarck Sea
The Game of Chicken
The Prisoner’s Dilemma
The Traveler’s Dilemma
Blotto’s Rules
Jack Be Nimble
Part 5: The Part of Tens
Chapter 17: Ten Financial Formulas
The Rule of 72
Leverage Ratio
Gains and Losses
Determining Depreciation
Total Return on Investments
Expected Return
Inflation-Adjusted Return
Remaining Balance
Future Value of Annuity Due
Bond Pricing Formula
Chapter 18: Ten Important Graphing Calculator Functions
Graphing Lines for Intersections
Adjusting the Window
Entering Matrices
Adding, Subtracting, and Multiplying Matrices
Powering Up Matrices
Finding Matrix Inverses
Solving Systems of Equations Using Matrices
Decimals to Fractions
Counting with Permutations and Combinations
Making Statistical Statements
Glossary
About the Author
Advertisement Page
Connect with Dummies
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
Pages
i
ii
1
2
3
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
49
50
51
52
53
54
55
56
57
58
59
60
61
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
267
268
269
270
271
272
273
274
275
276
278
277
279
280
281
282
283
284
285
286
287
289
290
291
292
293
294
295
296
297
298
299
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
319
320
321
322
323
324
325
326
Introduction
Finite math is both difficult and easy to describe. When I’m asked what finite math is, I launch into a listing of the different topics that are usually covered and then refer to all the applications that are possible to perform using the techniques. It isn’t a quick and easy explanation.
Finite math isn’t one thing, and it isn’t restricted to one area of interest or discovery. You may be a finance buff or a gambler (or both). You may want to organize your life more or organize others (or both). You may like to play games or instruct others on game-like situations (or both). Are you getting my drift? Finite math is many things to many types of people with many interests. You may find yourself loving all the topics covered here or just some of them. This is entirely your preference.
About This Book
You were brave enough to pick up this book so you could discover its secrets. Or you’re going to read the book because you want some insight into or help with some math topics that have come your way. In either case, you’re in luck.
I wrote the chapters of this book with specific goals in mind. You, the reader, may be planning on a career in business or finance. Or, if you don’t want finance as a career, perhaps you just want to be able to manage your own financial situation and not depend on others. You’ll find many examples leaning toward these topics. The mathematics presented will help your understanding and aid you in various computations that are necessary to get the right answers.
Some of the material may not be of interest to you right now, but don’t discard it yet. As you read on and discover more, you can backtrack and find the basis of premises or computation techniques. Some of the material is sequential, for instance, recognizing linear equations before you start solving systems of such equations, but most of what you find can stand alone. The topics you find here complement one another.
When I introduce a word and think it needs some explanation, I place the word in italics. Mathematics has a way of kidnapping normal, everyday words and giving them a special meaning. For example, take the word set. Are you playing tennis? Do you have dishes to put on the table? Neither works here, but set has its own meaning, as you see in Chapter 9.
So much mathematics nowadays is performed on calculators and in spreadsheets. And many of the calculations that you find in this book can and will be done using such technology. In these chapters, you find the basics behind the math that’s performed, and then you’re free to use whatever technology you have at your disposal. One chapter in the Part of Tens is devoted to some processes that you can perform using a graphing calculator.
Foolish Assumptions
You’ve picked up this book on finite math, and you open it to the chapter or page of your choice. I’m assuming that you’re interested in that particular topic or, perhaps, are just jumping into anything that pops up. This is fine. It will work.
What I do have to assume is that you have some basic understanding of algebra and its processes. Many of the topics presented assume that you understand that letters can represent numerical values and that the numerical values can be used to answer questions.
Many of the topics covered use matrices and matrix-like formats. Even if you’ve never seen matrices at work before this experience, you should be able to dive in and appreciate their value and versatility.
Icons Used in This Book
As you read this book, you’ll see icons in the margins that indicate material in support of what is being discussed. You aren’t expected to know all about these special items or formulas, but they’re presented for your reference. This section briefly describes the icons found in this book.
This icon is used to help you along the way. The material presented with this icon makes a process easier or gives more of an explanation as to why something is done.
The material following this icon is wonderful mathematics; it’s closely related to the topic at hand, but it’s not absolutely necessary for your understanding of the material being presented. You can take it or leave it — you’ll be fine just taking note and leaving it behind as you proceed through the section.
This icon alerts you to important information or rules needed to solve a problem or continue on with the explanation of the topic. The icon serves as a place marker so you can refer back to it as you read through the material that follows.
Beyond the Book
In addition to the material in the print or ebook you’re reading right now, this product also comes with some access-anywhere goodies on the web. No matter how well you understand the concepts of finite math, you’ll likely come across a few questions where you don’t have a clue. To get this material, simply go to www.dummies.com and search for “Finite Math For Dummies Cheat Sheet” in the Search box.
Where to Go from Here
It’s time to begin your adventure into finite math. Where should you start? Where will you end up? This is really up to you. If your algebra background is all relatively recent, then you can jump into the linear programming and maximization problems in Chapters 6 and 7. If you’re more interested in the financial part of the book, then just leap forward to Chapter 11.
But as you’re moving through any particular topic, feel free to review some of the operations and processes that are covered throughout the book, such as systems of equations in Chapter 3 or matrices in Chapter 5. You’re not expected to remember every math topic you’ve learned in the past.
Do enjoy. There’s a large enough variety to satisfy every type of reader.
Part 1
Getting Started with Finite Math
IN THIS PART …
Discover how to write and solve linear equations in two or more variables.
Get familiar with solving systems of inequalities using graphing methods.
Chapter 1
Feeling Fine with Finite Math
IN THIS CHAPTER
Lining up the lingo
Introducing multiple ways to describe mathematical situations
Looking at applications for probability
Linking logic logically
Taking on games with new vigor
What is finite mathematics? It seems that there are infinite ways to describe this subject or subjects. When applying the processes from the various topics in finite mathematics, you consider multiple applications and get to solve them in a variety of ways. Finite mathematics has become a gathering spot for many applications in business, social sciences, biological sciences, economics, finance, and so on. This gives the businessman, social scientist, biologist, economist, financial officer, and others many options for dealing with their everyday decisions.
Finite mathematics starts with the basic mathematical processes and draws in all the applications that make the processes interesting, usable, and valuable. And this is just the beginning. In addition to the basic mathematical topics and procedures, you also have all the possibilities for using modern technology to solve a particular problem or organize a situation.
Getting in Line with Linear Statements
Most of the applications in finite mathematics that involve mathematical statements are of the linear variety. A linear equation or linear inequality has only first-degree variables. You don’t find curves like parabolas or shapes like circles or ellipses in the study of linear algebra.
In Table 1-1, you find some linear statements and their descriptions. A common practice is to have the variables be letters from the end of the alphabet and the constants and coefficients come from the beginning of the alphabet.
TABLE 1-1 Linear Statements
Algebraic Statement
Description
Linear equation in two variables in standard form
Linear equation in two variables in slope-intercept form
Linear equation in three variables in standard form
Linear equation in n variables in standard form
Linear inequality, less than
Linear inequality, greater than or equal to
Compound linear inequality
Note that the power of each variable in a linear statement is equal to 1. The power isn’t showing. You don’t usually write an equation as ; the preferred format is . When there’s no exponent showing, you assume that the exponent is 1.
Making the Most with Matrices
What is a matrix? In the movie The Matrix, the characters dealt with computers, so you may find a bit of a tie-in there, because matrices provide formats that are conducive to being entered into computer programs and graphing calculators. But matrices are actually very simple structures.
A matrix is a rectangular array of numbers or other elements. By rectangular, this means that every row is the same size (making the length uniform) and every column is the same size (making the width uniform). For example, the following matrix A has four rows and two columns, so it’s a matrix.
The matrix A has eight elements, and the elements are all integers. The elements are inside brackets, and the matrix has a capital letter as its name. In Chapter 5, you find even more details about matrices and the processes that go along with them.
Most graphing calculators have built-in matrix apps so you can enter the elements in the matrix and perform operations on a matrix or multiple matrices. Excel spreadsheets also lend themselves nicely to matrix processes; and the added benefit of using computer spreadsheets is that you can easily view and print them.
You can solve systems of linear equations by the tried-and-true methods from algebra: substitution and elimination. But matrix mathematics also includes methods that you can use to solve systems of linear equations. Matrices also help by changing the format of mathematical statements to make them more usable and understandable. The results are easily read after performing matrix computations. You just have to follow steps provided in Chapter 6.
Staying with the Program
Finite mathematics involves quite a bit of linear programming, in one form or another. Basically, this means that the topics covered take applications that involve linear statements and find a solution. Typically, the solution is in the form of finding the maximum or minimum value possible.
For example, say that you’re trying to take care of some dietary problems and don’t want to spend too much money while doing this. You’re trying to minimize the cost. You need to add just so much vitamin A, some vitamin D, some iron, and some potassium to your diet. Pill I has certain amounts of each, Pill II has three out of four of those elements, and Pill III has a different three out of four. And, of course, they each cost a different amount of money.
A linear programming process associated with this situation has you write statements that represent the amounts of the vitamins, iron, and potassium and their relative cost. Then you write inequalities expressing that you want at least the minimum of each added to your diet. Finally, you write the statement that you want to minimize — the total cost.
Yes, this may seem very complicated, but all this becomes clear in Chapters 7 and 8. The steps are spelled out and the options for solving the problem presented.
Getting Set with Sets
A set can be many things, and it can be used in many ways. In mathematics, a set is a grouping or collection of objects. Yes, the objects are usually numbers, but they really can be anything.
When you describe a set in mathematics, you usually name the set with a capital letter, and you list the objects or elements of the set in braces, with the elements separated by commas.
The set of states starting with the letter i can be described with . This set has four elements. And this isn’t the only way to describe the set. You can also say that . The order in which you list the elements doesn’t matter.
If the set is very large and you don’t want to list all the elements, then you can use a rule or an ellipsis. For example, if the set H contains all the positive integers smaller than 100, then you can use one of the following formats:
Each description of the set H means the same thing — that is, creates the same elements. The positive integers smaller than 100 are 1, 2, 3, 4, …, 98, 99. You don’t want to list all those numbers, so you can use an alternate form for the set of numbers.
How many elements are there in the set H? You answer that question with the notation . This says that set H has 99 elements. And, again, they don’t have to be listed in order, if you choose to list all the elements.
You can accomplish many operations and other calculations using sets. One of the most popular processes involves Venn diagrams. A Venn diagram usually involves a geometric figure (most often, a circle) that represents a set and its elements, and it shows where the set intersects (shares) with another set or two. Figure 1-1 shows you a Venn diagram illustrating the relationship between sets M and F.
FIGURE 1-1: States starting with M and states with five or fewer letters.
From both the figure and the set listings, you see that and . The intersection of the two sets is what they share, and that contains one element. The union of the two sets is the combination of the two sets put together. There are elements in the union, because you don’t count Maine twice.
Sets provide a great way of organizing information and making conclusions about how they relate to one another.
Posing the Probability
What is the probability that it will rain tomorrow? What is the probability that you’ll land on Park Place in the game Monopoly? Each of these answers or predictions is based on the numbers 0 through 100. If something has 0% probability, then it isn’t supposed to happen, and 100% probability is a sure thing. If you’re four spaces away from Park Place, then the probability is about 11% that you’ll land on that spot with its hotel!
You write probability amounts as percentages, decimals, or fractions. Each has an equivalence to the other two, and the use of one or another form is usually just a preference or whatever works best in the situation.
To change a fraction to a percentage, you first change the fraction to its equivalent decimal form and then that decimal to a percent. For example, the fraction . Changing the decimal to a percentage, you move the decimal point two places to the right and get 62.5%.
What is the big advantage of using percentages? They’re much easier to compare to one another. If you wanted to know which is the greater probability, or , you get a better idea by comparing their percentages. The fraction is equal to 62.5%, and the fraction or 56%, so represents the greater probability.
What do you do about decimals that don’t end? Some don’t even repeat! The short answer is to shorten them or round to a certain number of decimal places. If you want the decimal equivalent of , you divide 12 into 11 and get 0.9166666 … with the digit 6 repeating forever. Choosing to round the percentage to the nearer hundredth, you first change the decimal to a percent, getting 91.6666 … % and then round to the nearest hundredth by changing the second 6 to a 7. The fraction is about 91.67%.
To change a percentage to a fraction, you go backward. Change the percentage to a decimal, and then put the digits of the decimal over a power of ten that has the same number of zeros as decimal places.
The percentage 13.25% becomes 0.1325. Putting 1,325 over 10,000 and reducing the fraction, you have .
Which version do you use? It’s whichever version is most helpful and informative in the circumstances. For example, the three circles in Figure 1-2 show you the same circle labelled with fractions, decimals, and percentages. Each is valuable in some format or application. Your choice.
FIGURE 1-2: Comparing fractions, decimals, and percentages.
Figuring in Financial Factors
A big application area in finite mathematics is that involving financial topics. There’s interest, dividends, amortized loans, continuous compounding, and more. And each of these topics comes with its own, special formula for performing the computations needed.
In real life, if you end up working with all this financial figuring, you’ll have all sorts of apps and programs to do all the hard work. But you still need to understand what you’re figuring and whether the result you get makes any sense. You need to know what number or form of the number needs to be input into what value. The financial overview in Chapter 11 will give you much more confidence.
But what if you’re not going into the field of finance? You still want to know what’s going on in that area. For example, when determining how much money you’ll have in your savings account after a certain number of years, you need to know that the initial deposit is entered as a decimal number, the rate of interest is entered as a decimal, the compounding value is in terms of how often each year, and the time is a number of years. So how much will you have after ten years if you deposit $50,000 at an interest rate of 4.75% compounded monthly? Here’s the computation:
You’ll have more than $80,000 — or your investment will have earned more than $30,000. You want to do better than that? Then try out some other institutions or investigate into what other processes or investment forms are available.
Finding Statistical Satisfaction
Statistical figures are part of everyone’s life. What is the average daily temperature? What does she need on the next test to get an A in the course? Does your IQ score put you in the genius category? What is the median price of a house in that lovely neighborhood?
Statistics provide a way of explaining situations, but you have to understand what is being presented and understand the possible misunderstandings or misuses when statistics are used.
One of the basic measures studied in statistics is the average. The average can be the mean, the median, or the mode. And the mean can be arithmetic or geometric. In Figure 1-3, you see a graph representing the salaries, in thousands of dollars, of the employees at a certain firm. Just looking at the figure, you can determine one of the measures for average: the mode.
FIGURE 1-3: The salaries at XYZ Manufacturing.
The mode is the most frequently occurring score. In this case, the mode is $50,000. So the owner of the company can say that the average salary is $50,000. Is this a good representation?
You can also quickly find the median from this graph. The median is the middle score, when you line up all the numbers in order. Looking at the graph, how many people or salaries are represented here? You see that one person is earning $10,000, two people are earning $20,000, and so on. Add them all up, and you’ll find 20 salaries listed. The middle is really between the 10th and the 11th numbers. So adding up the numbers associated with the salaries, you have , and you can stop there. The three people represented in the $40,000 column are the 9th, 10th, and 11th in an ordered list. The middle is between the 10th and 11th, which are both $40,000, so the salary $40,000 is the median. Is this a better representation than the mode of $50,000?
There’s one more average to check — the one you’re probably most familiar with when talking average scores — and that’s the arithmetic mean. The arithmetic mean is what you get when you add up all the scores or salaries and divide by how many there are. Adding up the 20 salaries and dividing by 20, you get
The mean average is $40,000. This is the same as the median, so it looks like this salary is the better representation of what the employees earn, on average. But someone reporting that the average is $50,000 wouldn’t be lying — they just may be misrepresenting for one reason or another. If you know what is going on, you can make a better judgment based on the statistics given.
There’s a lot more to investigate in terms of the statistics of a situation, and you get much more information in Chapter 12, to help satisfy your statistical cravings.
Considering the Logical Side of Mathematics
You hear someone make the following argument:
All cats have four legs.
All cats are mammals.
Therefore, all mammals have four legs.
You can probably do some convincing reasoning, with examples, to show why this argument is false, but what is basically wrong here? Are the assumptions wrong? Is the structure of the argument wrong? What structures work?
Aristotle is usually credited with being the first person to use — or at least record his use — of a formal logic system. Many others followed him, tweaking the subject and format and applying it to the sciences and other areas of endeavor.
Mathematics has long been a part of logic, coming from both directions. Principles of logic have been applied and incorporated into mathematical systems, and, going the other way, some mathematical findings have been utilized in further developments in logic.
In Chapter 13, you find the basics of logic, truth tables, and some applications of logic. And then perhaps, you can weigh in on Mr. Spock’s quote: “You may find that having is not so pleasing a thing as wanting. This is not logical, but it is often true.”
Unlocking the Chains
The study of Markov chains has helped in many applications in the real world. When making a prediction about a coming event, using a Markov chain, you consider only the present state, not the history of events or any other outside influences. Not all situations are appropriate for the use of these chains, but they still have been important enough to continue to study.
Consider a situation where a diet enthusiast has decided to limit her lunches to either broccoli, carrots, or kale. Each lunch consists of that vegetable, only, and nothing else. Figure 1-4 shows her choices after eating one of those vegetables and the percentage of the time she makes that choice.
FIGURE 1-4: What is she having for lunch tomorrow?
If the dieter eats broccoli on one day, then 40% of the time she’ll have broccoli the next day, 40% of the time she’ll have carrots, and 20% of the time she’ll have kale. If she has carrots one day, then the next day her two choices are only carrots again (70% of the time) or broccoli the other 30% of the time.
The diagram gives you lots of information about her eating habits, and a picture is often very helpful when trying to figure out patterns and make predictions, but there’s another format that’s even more useful for the predictions part.
You can put the information from the diagram into a rectangular format — yes, a matrix. To read the matrix that corresponds to Figure 1-4, you read from the left side, representing the current state or what was eaten today, and then down from the top, representing the next day’s choice.
Reading from the matrix, if the dieter eats kale one day, there’s a 60% chance she’ll eat carrots the next day and a 0% chance she’ll repeat the kale. You see that each row adds up to 100% — covering all the possibilities for the next day’s choice. Also, what you find in the long run is that when the dieter uses this particular pattern of choices, she ends up eating broccoli 34% of the time, carrots 59% of the time, and kale 7% of the time. This is useful information when planning on future purchases. How were these percentages determined? You find all you need to create the same figures and matrices and the resulting patterns in Chapter 14.
Getting into Gaming
When you hear or read the words game theory, you may dismiss them as being something to do with gambling or with video games or with some of those fun apps on your tablet or phone. You wouldn’t be completely incorrect, but there’s so much more to game theory than just fun and games.
Game theory is applied to adversarial situations, which can be wars, competing for business, gaining votes in an election, making money, and much more.
When studying game theory, you see many of the mathematical structures and processes that are also used for other topics — matrices and sketches and solving equations are all incorporated into the study of game theory.
You can use some game theory when deciding how to invest that $100,000 you inherited from your great-aunt Lucy. You go to an investment firm and are given some figures on what may happen if you invested all the money into either a money market, some bonds, or some growth stocks for about five years. Of course, the gain (or loss) will depend on whether the economy is stable or inflationary. Here’s what you’re shown:
So what’s the game here? How would you play it? Safe or risky? In Chapter 16
Chapter 2
Lining Up Linear Functions
IN THIS CHAPTER
Writing equations of lines in several different forms
Graphing lines for visual satisfaction
Comparing one line to another
Rewriting equations for convenience
Linear function is just a fancy way of saying line. Yes, a function is something special, so not just any line can represent a function. But some lines are functions and can be very useful. And the basic rule that it takes just two points to determine one special line still holds with a linear function. Those points on the line are in the form of coordinates, (x, y). Those points are also considered to be solutions of the equation representing the line.
When performing computations or investigations in finite mathematics, you usually want one of two different forms of the equation of a line: the slope-intercept form or the standard form. One form is helpful when graphing and solving systems, and the other works better when using matrices.
The graph of a line is helpful in many ways. It gives you a visual answer to a question, such as, is it rising quickly? It also helps you determine how different values are grouped or limited. Graphing more than one line allows for comparisons and the creation of areas that you can look at for an answer. Lines can be drawn in a solid form or with dashes; these are subtle differences that have distinctive meanings.
Creating the graphs of lines is also a way of doing quick comparisons: Are they rising or falling? Are they parallel or perpendicular? Is this just the same line? For those who like a picture to look at to better understand, graphing the lines is your method.
Equations of linear functions crop up throughout the chapters of this book, so it’s good to get these important things covered right here and now.
Recognizing Equations of Lines
The equation of a line can take on many forms. The same line is represented by the equations , , , and so on. So what’s the big deal? Compare the form of a linear equation to a picture you’ve copied and are now editing. Do you need to crop the picture to focus on one part of it? Does it need to be rotated in one direction or another? Will you be changing the contrast to illustrate something better? These are all about the same picture, but you’re changing it to suit your particular purposes.
Identifying slope and its scope
One of the more recognizable and popular forms of the equation of a line is the slope-intercept form.
The slope-intercept form of the equation of a line is . The letter m is the value of the slope; it can be positive or negative or even zero. The letter b represents the y-intercept, where the line crosses the y-axis. The y-intercept can also be positive, negative, or zero.
The line has a slope of 4 and a y-intercept of (0, –3). Because the slope is a positive number, the line rises as you move from left to right. The negative y-intercept tells you that the line comes up from the lower left of the graph, crosses both axes, and continues on through the upper right of the graph.
What about the line ? What is its slope? The equation is in the slope-intercept form, but you don’t see the slope because it’s 0. Another way to write the equation of this line is . Now the slope is obvious! But what does a slope of 0 mean? Any line with an equation of the form , where k is some number, is a horizontal line. Horizontal lines are important and come in handy.
The same thing can happen with the y-intercept. If it’s 0, then the number doesn’t show in the equation. An equation with a y-intercept of 0 has the form . So the lines , , and all have a y-intercept of 0. They all go through the origin of the coordinate plane, the point (0, 0).
Creating different forms of the equation
Another very handy form of the equation of a line is the standard form.
The standard form of the equation of a line is . You can determine the slope and both intercepts from this form. The slope is , the y-intercept is , and the x-intercept is . This isn’t as handy as finding the slope and y-intercept in the slope-intercept form, but it saves you from having to change the equation into that form.
The standard form is very useful when working with matrices. You put all the equations in the same form and then have to use the numbers only when doing matrix work. You can see how all this works in Part 2 of this book. For now, you get to use the whole equation, letters and all.
You can change from one form of a linear equation to another by using basic algebra. The choice of the form of the line just depends on the particular process being performed.
Changing to slope-intercept form
To change the equation to the slope-intercept form, you first isolate the y-term on the left side. To do that, subtract 4x from each side, and you get . Then divide each term by –5; the final equation is . You can immediately tell that the slope is and the y-intercept is at –4; the coordinates of the y-intercept are (0, –4).
Changing to the standard form
In order to change the equation to the standard form, the first thing to do is to multiply each term by 8. This gives you . To put it in standard form, you add 3x to each side; the standard form is . The slope of and y-intercept of 7 were more obvious in the original form, but you can pick up the x-intercept by using ; the x-intercept is at the point .
Writing the equation of a line
You can write the equation of a line if you have the slope and a point on the line or if you have any two points on the line. Either of those choices creates the equation of the line. The equation represents those points and none other.
