How to study multiple textbooks per math subject?

[bacte2013]

Dear Physics Forum personnel,

I am a college sophomore in US with double majors in mathematics and microbiology. I apologize for this sudden interruption but I wrote this email to seek your advice regarding to utilizing multiple textbooks per mathematical field. I always thought that different books offer different perspective and approach to the math subject, and I bought several books for self-studying (and to prepare for a future courses I will take). Currently, I am studying Rudin's PMA, Apostol's MA, and Pugh's RMA for introductory mathematical analysis, and I am also studying Niven, Burton, and Apostol (analytic number theory) for introductory number theory. Those books are truly great textbooks and I have been thoroughly enjoying them but I have few problems...What I am doing is reading a chapter on one book and study similar chapters on other two books. I noticed that those books per subject cover quite similar materials but with different tone and approach. Sometimes, I get confused by their different explanations and proof approach to a given definition and theorem, ultimately causing me a reduced brain concentration. How should I study multiple books per subject? Do I actually need multiple (3~4) textbooks per math subject? Should I just select one best book and put all of my concentration to it? There are countless books on both introductory mathematical analysis and number theory, which cause me a huge temptation to read them all and doubt that my current books are not enough to cover all basic materials.

Thank you very much for your time, and I look forward to hear back from you!

PK

 

[micromass]

I think that more than 1 book is overkill. There should be one main book that you learn from. It is of course a good idea to use additional textbooks, but you shouldn't work through it like you work through your main book. Use the additional books mainly if the explanations of the main book don't really resonate to you, and for exercises.

So your main book should be written in a formal (but not too formal style). Then it's good to have an additional book that is filled mainly with intuition. And then you might want to use a book that has good problems. If you're lucky, the book you use will combine those three traits well (for example, Carother's real analysis book does all three things really really well). But most books will only be really good in one of these.

 

[bacte2013]

^
Dear Professor micromass,

Thank you very much for the advice! I noticed your explanation of three traits in my mathematical analysis books too: Apostol is quite formal and with good intuition but with easy problem sets, Rudin is very formal with very good problem sets, and Pugh is intuitive and with excellent problem sets too. I think I will go with Apostol as it really clicks with my heart. I read some portions of Carother but I am not sure if I should read it too (or later on) since it seems that Carothers cover very similar contents to Apostol/Rudin/Pugh. I am planning to advance to both Rudin's RCA & Folland or Stein's real analysis after completing those three introductory analysis books.

As for the number theory, I think Niven et al's book has formality, intuition, and really good problem sets. I think I will chose that as main one.