# Studying mathematics efficiently

I am particularly angry at myself for the last few days. I noticed how inefficiently I work. Here is the general scenario:

I decided to study abstract algebra and analysis some days back. I tried reading Herstein's Topics in Algebra and then decided to go for analysis. Once again, I tried reading Rudin and breezed through chapter 1 and got horrified while reading chapter 2. I lost interest in analysis and came back to Herstein. Later, I saw some comments on mathoverflow by someone that he read Rudin's Analysis when in high school. I felt insulted as I thought it was cowardly to leave Rudin. So, I tried reading Rudin again with the same result.

After a month, here I am, having forgotten whatever I read in Rudin and Herstein.

My questions are :

1) How do people study efficiently?(I am seeking answers from someone doing at least graduate studies and above) I have wasted nearly a month juggling from one book to another and do not want to end up wasting more.

2) Do you find it efficient to study one book at a time or is studying two topics at a time advisable?

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possible duplicate of How to effectively and efficiently learn mathematics – Clive Newstead Sep 20 '12 at 10:41

I have quite a few years experience in what you talking about. In my opinion (and my profs too) if you want study mathematics only way is to exercise.

Find some book (I like something similar to Shaum's series) with examples and solutions and do them every day a few.

I don't think there really is another way to study mathematics.

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This is very much an example of what Matt said. I personally do not benefit from exercises at all - I either get the material or I don't. If I don't get it, I am helped by more theory and examples, but exercises hardly ever help me. – akkkk Sep 20 '12 at 13:53
@Auke In general, humans learn about 70% of what they learn by doing. – Graphth Sep 20 '12 at 18:36
@Graphth: SOME humans learn about 70% of what they learn by doing, yes, but it is a rather short-sighted statement to say that this is true in general. – akkkk Sep 20 '12 at 21:20

I don't think anyone is qualified to give you advice, including myself, since everyone will have their own method that works best for them. I'm a grad student in pure maths.

To me it sounds as if you are picking the wrong books. You need to think about two things:

1) Is the book you picked at the right level for you? By this I mean for example, do you know enough basic calculus to be doing analysis? If you do, how much analysis do you know and how much would be required to read the book you picked? Should you be revising more elementary topics before moving on to abstract algebra and analysis? Who cares about what Joe Bloggs read at high school, this is about you and not about what other people did or did not.

2) Does the book match your taste? This might be a bit subtler to figure out and fix, if not. There is an infinitdude of books about analysis and similarly about algebra. I have not learned analysis from a book and the algebra I know I learned from Gallian's Contemporary Abstract Algebra which I have read more or less cover to cover. I suggest that you get yourself the most "famous" introductory books about the topic from the library, test each by reading a chapter and then decide which to use. The things I like best about Gallian is that it's well-written, easy to read, comes with lots of exercises and on top of that solutions for the odd numbered ones.

Hope this helps.

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I find the exercises the most important part of the book. Do the books you picked also come with exercises? This point addresses your (implicit) question about how to remember stuff: by doing exercises. Try it! – Rudy the Reindeer Sep 20 '12 at 10:51
I like Herstein.And I feel that Apostol's calculus should be enough background.I come from Olympiad background, so I am not sure mathematical maturity is that big a problem. – user41685 Sep 20 '12 at 11:03