Finite Math For Dummies®
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ISBN 978-1-119-47636-8 (pbk); ISBN 978-1-119-47643-6 (ebk); ISBN 978-1-119-47644-3 (ebk)
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.
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.
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.
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.
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
You hear someone make the following argument:
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.”
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.
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, you find different strategies and net results. This still doesn’t guarantee success, but it gives you important information.
Chapter 2
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.
