Help me decide between these two books?

[dkotschessaa]

For some independent study this summer, I have a choice of two books, available electronically, to study.

John Stillwell - The Four Pillars of Geometry

Abel's theorem in problems and solutions



The decision is fiendishly difficult. The first book plays on some of my strengths - I have a good background in basic euclidian geometry though the study of math history, which is where the book starts. Then it gets into a lot of things I've been wanting to learn and strengthen. (Like my ability to work visually).

The second is just fascinating - and covers a lot of topics I will encounter formally later in my studies. If I get through this book now I think I'll have a major heads up on a lot of those classes down the line. However, it also contains some things I feel I might lack the background for, like set theory and proofs, which I am self studying with "How to Prove it," by Velleman,

My background is three semesters of calculus and early history of mathematics. I haven't had any abstract yet, or linear algebra. I think it's a choice between really solidifying and enhancing existing knowledge vs. learning something completely new and a bit daunting.

Just can't seem to make up my mind!

-Dave K

 

[micromass]

Wow, that book about Abels theorem is really good. It touches on several different parts of mathematics and it has nice explanations. You should have a good time learning it.

As for the Stilwell book, I think there might be better books out there. An excellent book is "Introduction to geometry" by Coxeter. It touches on all kind of different parts of mathematics and offers nice insights.
If you want a book on euclidean geometry, then perhaps "Geometry: Euclid and beyond" is something you'd like.

 

[dkotschessaa]

Yeah, I like how it starts with the theorem and the overall aim of the book, then dives into all these different mathematical concepts (seemingly unrelated, to me) to build back up to the theorem. I just hope I can tackle it. The language around 1.2 is where I start feeling un-prepared, but I'm still wondering if the Velleman book can help me with that. I have about 6 weeks to prepare for this and study ahead...

-Dave K

 

[Sankaku]

I haven't seen the Abel book before. Since it looks pretty concise, you may want a supplement for the basic Abstract Algebra. I often recommend Pinter as it is very readable as well as inexpensive.

Stillwell is a nice author - all of his books that I have seen are well written. I remember reading through some of that book back when I did a geometry class. I liked the approach, it starts pretty gently.

 

[sponsoredwalk]

I seen this thread the other day & just today got them both out of the library as I had literally been meaning to finish them both off this summer anyway once exams were over so definitely feel free to post specific questions here about the books, I know I'd definitely like to talk about these two.

I've read most of the Abel book & some of the Stillwell book already. Another one extremely similar in style & philosophy hopefully read along with the second half of the Abel book is https://www.amazon.com/dp/0883856182/?tag=pfamazon01-20. I don't see any reason why you shouldn't read both books along with Steenrod, I'd say you'd be far better off doing it that way.

You'll learn a fantastic exposition of group theory based on thinking of groups in terms of permutations (read this to find out why it's done that way) & these (or these or these) lectures will be really helpful in clarifying the stuff sped through in the Abel book. Also, most of chapters 1 & 2 of the Abel book independently (recommended). Topology comes up when discussing the fundamental theorem of algebra & winding numbers & I'd recommend reading Steenrod along with these sections for a more complete exposition of that material (a real analysis course such as this one motivates a lot of this material).

However you said you had problems with the language in section 1.2, is that in Abel? Is the language of transformations & surjections completely new to you? In this context a transformation is another word for a bijection & in the case of finite sets the word permutation is also used equivalently. If this material is giving you problems then I'd recommend these, they deal with proofs & develop the necessary material (the discrete structures videos have some algebra stuff in there that covers some of the Abel stuff like groups, subgroups, cosets & Lagrange's theorem):

http://www.youtube.com/playlist?list=PLF2DF6C3C8015DF5F&feature=plcp
http://nptel.iitm.ac.in/video.php?subjectId=106106094
https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20
https://www.amazon.com/dp/007154948X/?tag=pfamazon01-20

 

[mathwonk]

you might get some interest if you start up a thread here where you read the abel book as a group, helping each other. then you could ask questions as you go along.