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Friday, February 18, 2011

Worst Math Books Ever

I know it is very common for math people to discuss among themselves their favorite math books, just like normal people discuss their favorite novels. And I can name my favorite math books, but I wanted to do something less discussed, the worst math books you ever read.

I am sure there a lot of terrible math books out there. And what I write will only represent the horrors I have experienced. Perhaps you have some more terrible ones.

Elementary Differential Equations by Boyce and DiPrima: Stay away from this book if you want to learn anything about differential equations. It is a piece of garbage. As the title suggests this is an elementary course in differential equations. It does stay honest to its name. This is an elementary book. You just really only need to know calculus to be able to study this textbook. This book has no rigor at all, just a discussion of different differential equations. And I am not complaining about that. It is okay for a book to have a lack of rigor if it actually manages to teach. The theory of ordinary differential equations will use analysis that most undergraduates will not know. So it is perfectly acceptable and understandable to write an elementary textbook that teaches some ideas and methods without much theory. However, this book fails at teaching. I remember I once wanted to learn about stability and non-linear equations. It was terrible, I did not learn anything. I also remember that I wanted to read the extra chapter in the book on Sturm-Liouville theory, and again, I was really unhappy, it was hard to learn anything. The authors do a terrible job at explaining. This book belongs in the trash or at a book burning festival, not in colleges.

Partial Differential Equations by Walter Strauss: I never actually studied out of this book but I did take it out of a library because I was interested in learning some of the material in it. I was really unhappy with it. No matter what chapter you turn to the author has no ability to explain himself. Even the very introduction to the book he cannot explain himself at all. But what is worst of all is the ridiculous price on this book. Books that are this bad do not deserve any kind of price to them. Strauss should be paying me for reading out of this monstrosity. If you want a great introductory book on PDE's that is like 1/3 the price get yourself "Basic Partial Differential Equations" by Bleeker and Csordas. It is an intro book on PDE but it does develop basic theory also. I was happy with it.

Advanced Calculus by Gerald Folland: Basically this is an advanced calculus book in general Euclidean spaces (multivariable). The book starts off well. Folland's first chapter on basic topology in n-space is well written. His second chapter on differentiation starts of alright but it progressively gets worse until the very end. By the time you reach the implicit function theorem of chapter three you ask yourself why you are reading this book. That chapter is a disaster. His fourth chapter on multivariable integration is lousy. And the fifth chapter on vector analysis is terrible, it gets progressively worse as one reads through the chapter. You get the impression that Folland does not want to be writing this book. The sad thing is that Folland is generally a good author. I have briefly looked over his Fourier Analysis book, though it was not appealing to me (a matter of personal preference) it did look well written. I also had an occasion to study a little bit out of Folland's real analysis book in one of my courses. I thought it was written in a reasonable manner, not perfect, but good enough. If you want to learn advanced calculus in several variables avoid this book entirely. There are better books. Any of Apostol's books would dominate Folland, so avoid it.

A First Course in Probability Theory by Sheldon Ross: The title is honest. This is a first course in probability. But the book sucks more dick than a gay porno (which I happen to watch, by the way, sometimes, to get a break from straight porn). The thing about this book is that it never explains its concepts. Its first chapter on combinatorics is reasonable. But after that the book falls apart. Math is a completely precise language. In math books concepts should be defined clearly. Nowhere in the book does he talk about what a random variable is. He mentions it a bit, but never really defines it. It is hard to follow him in the later sections of the book. This book also has non-ending examples. I know students often complain about books and say they do not have enough examples of exercises, well this book is filled with them. It does not stop. A single chapter will have a little to mention and then be bombarded with examples. The exercises at the end of the chapter also never seem to end. Too few examples is terrible, but an uncountable many, is not good either.

Discrete Mathematics by Goodaire and Parmenter: This is without any exaggeration the worst math book I have ever read. I am not even sure what makes it so bad. It is just bad. Worst of all it is boring. It cannot explain the material in any kind of exciting way. I brought this book upon first entering colleges because I was hoping to learn some basics of graph theory. I hate it.

Algebraic Topology by Allen Hatcher: What would you say that somebody writes a book on mathematics without having any definitions? You would certainly think that such a person destroyed the subject he wanted to write. Well, that is exactly what Hatcher did. I do not really have a problem that he wrote such a book. I have a problem that the book is presented as if it is some formal treatment of algebraic topology while it is largely handwaving. There are no definitions. He sets up intuition, which I have no problem with, but never really defines his terms. Then his proofs are hard to follow. First, he cannot explain himself well. Second, you have no idea what the definitions he is using since he never defined them rigorously. Third, he skips over a lot of steps. This book is homotopy equivalent to a trash can. And it shocks me that so many professors use this book as an introduction to algebraic topology. There are books on algebraic topology that are good (Massey), Hatcher's book serves no purpose. The only good thing about the book is that it has a lot of exercises and it is available for free online. Other than that, it sucks. It is disgrace to mathematics. And again, do not misunderstand me. I got no problem with informal treatments of mathematics, as long as they make it clear. I did enjoy Needham's book "Visual Complex Analysis" even though it was a non-formal geometric treatment of complex variables, but he never claimed the book to be a substitute for rigorous treatments of complex variables. This book mascarades as if it is some formal text on topology, and it is just not.

Mathematical Methods for Physicists by Arfken and Weber: Here is a general good rule about math books. Math books should be written by mathematicians, not by physicists. Physicists do not know math. They know how to use it in solving problems but they do not actually know the mathematics. And it really shows when a physicist is trying to explain mathematics, especially more advanced mathematics. When I read through Afrken and Weber (by the way, if you are wondering, no I did not read every page! the thing is like 1500 pages long) the impression I got was that the authors actually do not really know what they are talking about. If you had a professor who was confused and did not really explain a subject then you would get the impression that he does not know the subject. That is exactly how I felt reading this book. This book does not explain. It tries to but it ends up not explaining anything. There is not a single chapter that is well written. Even as a reference book I would not recommend it. The only good thing about the book are the exercises. It has enough exerices. But you are not really going to solve the exercises as the book fails at explaining the material. If you are wondering if there are any good books on mathematical physics, the answer is: yes! You can read, "Mathematics of Classical and Quantum Physics" by Byron and Fuller. This book is only like 15 dollars and of excellent quantity. I have read through some of their sections (the one on Green's functions is really good). Compare that to the trash by Arfken and Weber which is like 10 times more expensive.

Luckily, I do not have much experience with bad math books. I usually read reviews about them to know if they are good or bad. But a few of those books did come up, usually in courses I took, and so I had this unpleasant experience with them.

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