Reading, Learning and/or Teaching from This Book
If you want to learn a language you need to practice, get involved. It is the same for mathematics, and one of the main ways to get involved is to do the exercises. Especially for the student who is afraid of, allergic to, or otherwise distressed by mathematics, I have tried to provide at least one exercise in each group that can be done after a careful reading, perhaps more than once, of the corresponding section. The exercises are numbered so that, for example, Exercise 2.1 is the first exercise in Chapter 2. Each exercise usually has “subparts” numbered with lowercase roman numerals, such as:
(i) = 1, (ii) = 2, (iii) = 3, (iv) = 4, (v) = 5 , (vi) = 6, and so on. Some of the exercises are more di cult, some require exploring outside this book. Some exercises are “open ended” in the sense that our answers to them can always be expanded. Occasionally I include an exercise for which I do not know the (complete) answer. The majority of exercises are interdisciplinary, a mixture of math with some other part of real life. (Yes, mathematical principles are a part of each of our lives—whether anyone realizes it or not!) The goal is to deepen understanding of how things really work. Even the occasional “pure math” exercises are designed to show how mathematics works. I might add that if you see a word or abbreviation that you do not understand—consult a dictionary! As an exercise you can look up the Latin words that have the following abbreviations; thus, e.g., means “for example,” i.e., means “that is,” and viz., means “namely” or “that is to say,” and cf., means “compare.” (I’ll do the first one for you; hence e.g. is the abbreviation for the Latin exempli gratia, which means as I just said “for example.”)
Anytime you have problems reading the text or doing an exercise it is always a useful strategy to ask a friend or classmate. If two or more of you cannot figure out the answer, then your group can approach the teacher, if you have one. It is my intention that this book be useful for self study, but it is advantageous to have an instructor. The chapters were written to be read in the order presented, for the most part; however, it is quite possible to dive into any chapter and start reading without much di culty. Whenever you see a number in a form such as [28], I am citing a reference in the list of references towards the end of the book just before the index. In this case [28] refers to a book by Albert A. Bartlett, The Essential Exponential (for the future of our planet).
I need to emphasize that this is not a pure mathematics text, it is truly interdisciplinary. No one I know is simultaneously an expert in all of the
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areas visited, including me. This might cause a bit of apprehension in some instructors, it might excite a sense of adventure in others. Fortunately in most of the areas discussed there are simple “necessary observations” that mathematics can help illuminate. (For example, a stable system of finance requires an honest assessment of risk.) It is true, however, that none of us are experts in everything. Such is life, for we as citizens are often confronted with new things, forcing us to make decisions with incomplete information— requiring us to research the best we can in the time we have. In any event, with regard to any of the many topics discussed I do not claim to be o ering the “last word” on any of them. Rather consider each chapter an introduction that highlights some (hopefully) interesting information worthy of further investigation. Occasionally it turns out that an “ordinary” person discovers answers that were missed by experts.
I have heard variants of the following comments in the same day: “This book does not have enough mathematics.” “The book is very interesting, too bad there is so much mathematics.” I find it somewhat pointless to dispute the first comment, since for this book my working definition of mathematics is: the search for and study of patterns. Patterns are everywhere, thus the entire book is either mathematics or proto-mathematics. But if you accept as a worthy goal showing how traditional mathematics arises naturally, and is useful, in other subjects, then you have to spend some time talking about the other subjects while introducing the mathematics. On the other hand, the entire book is written from the perspective of mathematics. Mathematically averse students have told me that the mathematics is easier to take in the concrete contexts developed herein.
I believe that at least some of the people with mathematical knowledge or talent should devote some of their time seeking solutions to the major problems of our time, or the not too distant future: climate change; overshoot of carrying capacity and/or ecological collapse (local or global); problems associated with food, energy, pollution, environmentally induced disease; economic collapse; disappearing justice—to name a few. It follows that any insights gained should be communicated to others, via our educational institutions and other avenues. If this book is found to be helpful in this regard, feel free to use it, expand and improve it. Adapt it to any circumstances special to you.
Human history becomes more and more a race between education and catastrophe.
H.G. Wells
In this book I often find it necessary to be critical, a position that is sometimes not appreciated by all. On the other hand, I hope that I have been su ciently critical where necessary. To remind me of this I have a picture of a landscape brutalized by the hand of man hanging above my desk emblazoned with the following quote from Shakespeare: “Forgive me thou bleeding piece of earth that I am meek and gentle with these butchers.”
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Some Details on How to Read This Book. If you have a command of fractions and are somewhat mature (this is often independent of age), you can dive into just about any chapter that captures your interest. If you want to start by studying “pure” mathematics, you can start with Part II, often abbreviated simply as, II. If you want to warm up to II, we recommend Chapters 1 (on climate change), 2 (financial collapse), and 3 (some basics). If you are uncomfortable with powers of 10, e.g., 1012 is a trillion, a 1 followed by 12 zeros, read Section 3.4 and do Exercise 3.11 at the same time you are reading Chapters 1 and 2. One more technical detail: in order to read Chapter 20, and later parts of Part VII, you need to master the Σ notation of Chapter 10.
Part I, or just I, is rather long. It is not necessary to read all of Part I before reading other parts of the book. One approach would be to read, say, the first three chapters of I, and then go on to II; returning to I as the mood strikes.
There is enough material to easily occupy a one semester course, and hopefully a su cient diversity of topics to catch ’most anyone’s attention.
Someday I hope to communicate with you via the Web sites, MartyWalterMath.com or MartyWalterMath.org. For example, when I get the time I plan to post therein additional helpful hints (maybe yours!) on how to teach and learn from and otherwise use the material in this book. By the way, the equation on the cover is the Greenhouse Law for CO2 and is discussed in Section 1.9.
I (penultimately) close these introductory comments with a bit of humor3 directed at everyone, including myself.
I finally close with comments on a couple of things to expect and not to expect from this book. What I have tried to do as often as possible is “give
3Cartoon reproduced with the permission of Gahan Wilson.
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a reason to learn a given topic.” As one of my students said recently: “Your book, for the first time, gives me real reasons to learn some mathematics.” This, I believe, is one of the main attractions of this book. Once a topic is motivated and introduced, I give only a few exercises. This is because in my many years of teaching I have found that finding a real, attention-getting example that captures, inspires, and motivates the “general student” is the hardest part. If a student truly understands one or two examples of the math involved, he or she does not need endless repetition. Much repetition is useful in gaining proficiency in implementing a given algorithmic solution, but students who “need” this are often unable to perform if the algorithm is changed slightly. Repetition bores the gifted students and most often bores, does not really help, and is unappreciated by the math averse. Some concepts are repeated in this book, for example, use of geometric sums; but each time the mathematics is motivated by a real situation.
Consider the following which is typical: I introduce “percent change” in the second exercise in Chapter 1. If either the student or teacher wishes at that time to do many more exercises on “percent change,” such exercises are available on the Web or in easily available library books. (I would be happy to explicitly list some of these resources on MartyWalterMath.org, or even post some additional exercises in the spirit of this book, if I get su cient feedback from readers desiring such.)
I have chosen to use precious printed space to introduce new topics, contexts, and pressing environmental information and issues, rather than “repeat exercises.” The Web is ideal for providing repetition of a topic once discovered; this book is useful for singling out and emphasizing topics not necessarily easily found by someone working individually.
In Part II we study some elementary mathematical structures from a mathematically mature perspective. One of the goals of Part II is to give the “general student” the ability to manipulate mathematical expressions with “letters” or symbols in them that represent numbers. In my interaction with professors in a variety of disciplines, one common request came up again and again; namely, please teach students how to deal with mathematical expressions and equations which contain letters that represent numbers, but do not contain specific numbers. In Part II we learn that numbers are mathematical objects that have certain properties. Thus if a letter represents a number, the student has “handles” or tools that can be used to manipulate it—regardless of the discipline.
Without being encyclopedic, throughout the book we focus on fundamentals: both mathematical and environmental.