Introduction to Mathematical Thinking: Algebra and Number Systems

Question

I'm currently in my final year of high school, and want to pursue some recreational maths before I go to university. I've been scouring the internet and various book stores for a nice textbook to learn something new from, and have come across this book a few times, called Introduction to Mathematical Thinking: Algebra and Number Systems, by Will J. Gilbert.

In the abstract of the book, it says:

Topics covered in this comprehensive introduction range from logic and proofs, integers and diophantine equations, congruences, induction and binomial theorem, rational and real numbers, and functions and bijections to cryptography, complex numbers, and polynomial equations.

Basically, my question is; is this a good book to learn from?

I found that it's used in some university first year courses (i.e. MATH 135 at Waterloo University), and so I figure it has some credibility. It also seems to cover useful, relevant, and interesting topics, such as binomial theorem and complex numbers.

Has anyone else every used this book? If so, was it good? Or are there other ways of learning that you would recommend, either through other books, or through other means?

Answer 1

I'm not familiar with the text you reference in your post, (haven't read it or used it), so I'm not in a position to compare it with other great books I've encountered.

A great option to consider is Thinking Mathematically by Mason, Burton, and Stacey. It explores the types of thinking required to do math beyond the level of applying rules and calculating things, and it does this by exploring various topics to illustrate this.

Another consideration you might want to take a peek at is Velleman's How to Prove it: A Structured Approach. Again, the text includes problems from various domains of mathematics as a means to illustrate different proof methods, problem solving strategies, etc.

Both texts are frequently given as references; perhaps consider conquering the first, and then the second, or a mix of both! You can explore the table of contents for each book via the links provided.

Answer 2

There is one major point that I don't see adressed in your question: are you going to study mathematics or not. The answer will probably depend for a great part on this question.

I must say that I'm unfamiliar with the book you mention, but if you're not going to study mathematics and really are going for the recreational part of it, some mathematicians have written books that you might find interesting. Not necessarily on different topics, but still: Marcus du Sautoy, 'Finding Moonshine' or Barry Mazur, 'Imagining Numbers' might be the kind of books you're looking for. (and there must be more like these lieing around)

If you're going for something more rigorous, yet still recreational in a way, the book "Proofs from the Book" may be an interesting option. There's many different fields of mathematics in there and beautiful proofs. It might well be a bit too hard, though. Or if you're going for a different direction, Pólya's "How to solve it" could be useful.

All in all, there's lots of different books you may consult, most of which I probably don't even know. And it all depends on what you really want to do or read.

Answer 3

Unless you are already well-versed in it, I would recommend looking at an introduction to set theory. I like Paul Halmos' Naive Set Theory. It introduces the reader to some nonbasic concepts from a "naive" standpoint (a naive axiom is described by natural language). Since set theory will appear in essentially any course you take, it's a good idea to look into it.