Books for a high school and undergraduate math education.

I am writing this hub because of David Fisher's reminder to me on the previous hub that mathematicians in fact read books. They do! Even I read books. In fact, I like books. I have never sold a book to a book buyer and since the beginning of my career have been collecting a library.

David would like to suggest that contrary to what I was saying in the previous hub, books are a major barrier to entry, giving professional mathematicians a large advantage over amateurs. (Perhaps I can continue to call them affectionately amateur/cranks.)

I want to restrict this particular hub to a subset of the books a research mathematician needs, namely those books which contain the ingredients of a high school and undergraduate mathematics education.

 The typical professional mathematician first shows some mathematical promise by high school. He takes courses in high school which are as advanced as possible and might participate in contests. In college, he becomes a math major, studying a program designed to be preparatory for graduate school. In this hub, I will try to provide  short list of inexpensive books with the property that anyone who masters them will have attained the level of mathematical proficiency required of a beginning graduate student. How do I define that level of proficiency? Below find a link to the graduate program at my university. There you will find sample tests for what are call the Tier 1 exams in algebra and analysis. These cover what should be undergraduate level material in mathematics. In fact, our graduate students are not required to pass them until their fourth semester. If you can pass them easily, you have reached the level of proficiency of our very best American trained graduate students.

 David Fisher's argument about barriers to entry revolves around the existence of a number of classic and beautiful texts which are priced exorbitantly because University students who are already paying outrageous tuition will spend almost anything for books. This, basically, is a scam. Proficiency simply requires mastery of a relatively small number of ideas which are in the public domain and are suitably exposed in books that have been largely forgotten but printed in sufficient quantities to be widely available.

However be warned that mastery takes a lot of work. Not only do you have to read these books but you have to understand them. You have to reach a level of understanding comparable to that which you would get from having invented the subject. This means you have to understand the basic principles of each subject, why they are true, and have an internal mental blueprint for reconstructing the subject from those principles. Being able to solve all the exercises is also a necessary condition for proficiency. Students in classes don't dream of approaching their texts this way. But being an amateur/crank is a higher calling. That also means it requires a different kind of book than the common textbooks of today. (Especially in very elementary subjects.) Often the amateur/crank does best with books written 100 years ago or more, and often these are quite cheaply available.

The recommendations below could be useful to the aspiring amateur/crank who wants to start from scratch, to the grade school student who has suddenly developed an interest in math and wants a head start, to the high school student who might want to prepare for contests or college, to the graduate school bound college student, or to anyone in between.

 A typical college bound student needs two years of algebra. Currently these are called Algebra 1 and Algebra 2. A hundred years ago, they were called elementary algebra and college algebra. A typical book in elementary algebra was a small volume you could hold in your hand that drilled you in solving simple equations and systems of equations, solving word problems, and manipulating polynomials. One such book was much like another.

College algebra took a more refined and sophisticated view of essentially the same material. A fine example of such a book is that by Paul Ryder, who was a statistician at Washington University in St. Louis in the 1940's. His book is widely available and contains challenging exercises which only a handful of our undergraduates can solve today.

 Everybody needs a course which is essentially watered-down Euclid. Why water down Euclid? Well, it depends on your taste. But Euclid is pretty verbose and not really rigorous anyway. "Plane and Solid Geometry" by Wentworth will do the trick as would other British books of the Victorian era. It helps also to have a little book of trigonometry and a little book of analytic geometry. A lovely book of Coxeter called "Geometry revisited" is available in paperback if you want to study a little deeper.

 Now to test proficiency, we'll get some contest books. If you can score over 100 on the old AHSME's, you're reasonably proficient. If you can do olympiad problems with ease, you're at the level of the very best entering freshman at exclusive places. If you are a high school student and can do it, you can get a free ride at some exclusive places.

 You're a successful enough crank to pass high school math. It's good to remember why you're doing this.

Now we're ready for college. Let me try to set you up for the absolute minimum you need for our tiers. You need some calculus.The situation with calculus books is really, really bad. Most are hideously expensive, anti-rigour, and discourage understanding. Good Calculus books, those of Spivak, Apostol, and Kitchen are ridiculously expensive or impossible to find. I compromise by suggesting Lang as the least bad alternative. If worried about rigor, read your analysis text first! I intend to try a partial remedy later by publishing hubs on Calculus for  Cranks.

You need several variable Calculus too but preferably linear algebra first. Dover's tensor themed books do the trick nicely

You need a couple of books on analysis and something on abstract algebra. (The best book,that of Herstein, is again prohibitively expensive.) Its good to have a more concrete point of view so I recommend Cox, Little and O'Shea's underappreciated book.

 Don't buy more than one or two of these at a time.Each book must be lovingly and deeply studied for any good to come of these. It's just a list to get started with on which each book won't set you back too much. You can take up studying mathematics on the cheap, but not all of this background is required to begin doing research. That we'll start in the next hub.