[eliteraspberrie]

I don't like the typical definition-theorem-proof approach of most textbook in mathematics, including these. It's great for a classroom, no good for self-study. As an alternative, I highly recommend A Book of Abstract Algebra by Pinter. If you work through that first, you may actually enjoy these two later.

[g9yuayon]

Can't agree more. Definition-theorem-proof type of textbooks is way too clean. They don't tell you how ideas came to be or why they mattered. In other words, it's hard for students to learn the intuitions behind the ideas. I wish there are list of "XXX from Ground-Up" type of books that show readers a list of problems, struggles of people trying to solve them, and how ideas emerge from the numerous attempts. Leslie's paper Paxos Made Simple was written in that way. A few chapters of Kleinberg's Algorithm Design were written in that way too.

[tnecniv]

I think math classes should be paired with history more. My probability professor often offered historical context (for example, the Poisson distribution first being used to model deaths due to horse kicks in the Prussian army) to the ideas we discussed, and the stories were often both interesting and insightful.

[craigching]

I've been refreshing on Calculus and I found that Kline's book was good at application as well as a bit of history, at least I never got the history part at University and I found it very interesting.

http://store.doverpublications.com/0486404536.html

[jostylr]

Nature and Growth of Mathematics by Edna Kramer is a fantastic book interweaving history and math.

[agumonkey]

I'd love to know all people seriously learning it to tell what they loved. Who like to be stuck on an abstract definition and figure it out on its own (ideal to real), who likes to have gradual build up (real to ideal).

My first AA book was the "European kind", all symbols and definitions, a few proofs every ten pages. It was too dry for me. I never thought other people would think that way.

I love brain teasing but I also need a minuscule amount of inspiration to power my neurons.

[mathgenius]

When you see Lemma/Proposition/Theorem think of it as an API that you can interface with.

Skip the proofs on the first read (this is the implementation, and may or may not be enlightening.)

But, number one rule with learning maths is: you got to do it yourself. Play with it somehow. It's similar to learning a new (or first) programming language (or API): have a project in mind and try to do it using that language.

Seriously, you absolutely cannot learn this stuff just by reading. Or, at best you may learn a very small fraction of it.

IMO, this text is far from "typical definition-theorem-proof". There is plenty of other prose and examples there aswell.

[codeofthedamned]

I agree. Pinter's book is very easy to read and great for self-study.

[tzs]

And it's a Dover book, so is quite reasonably priced (around $12).

[vitriol83]

this is always a tension in writing mathematics textbooks. at one end of the extreme you have e.g. Bourbaki which are very dry, but prove a great deal very efficiently and in the utmost generality. on the other hand you have textbooks which may be not as comprehensive and will intersperse the text with illuminating examples which historically would have been the original motivation for the subject. which is best really depends on your point of view and level of sophistication in the subject. usually I try to have both types of book at hand.

what would be great is if typesetting tools improved sufficiently so that one could choose 'beginner' or 'advanced' mode when reading a maths textbook. perhaps that is too fanciful!

[sound_of_basker]

You can say that again. The other day I was looking at proving the Pythagorean theorem in R_n. Merely starting the problem formally is non-trivial. :-(

[andrepd]

You can prove it through mathematical induction. Show that if it's valid for n dimensions then it's valid for n+1 dimensions. So then if it's proven for R_2 it's proven in general.

[trentmb]

If anyone else got curious about induction on the reals, my first interesting result was this Math.StackExchange post: http://math.stackexchange.com/questions/4202/induction-on-re...

[mturmon]

The induction in question here is of course on the dimension of the vector space, which is a natural number -- not the members of the vector space itself.

[trentmb]

Yes- it just kick-started my curiosity is all!

[JadeNB]

No need to start with R^2; start with R^1, where it's easy. Starting the proof with R^2 essentially means that you must do the induction step twice.

For an amusing instance of induction on dimension, you might enjoy the proof of the AMGM inequality, which proceeds by upwards induction that doubles the dimension, followed by downward induction: https://proofwiki.org/wiki/Cauchy's_Mean_Theorem#Theorem .