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When trying to learn a new subject in mathematics from a book I usually find myself mostly learning the theory directly presented there(reading through all the theorems, proofs, definitions etc.) not solving many(if any) of the exercises suggested after every chapter/section.

The same question always bugs me: how do I find a balance between following the book and solving the problems? On one side, if I solve the problems I get some practice coming up with proofs by myself and deepen my understanding of the studied section, but on the other hands, I could spend that time expanding my knowledge, getting to know more interesting theorems compared to the rather simple, boring ones contained in the problem sets and could get to know more different fields of mathematics sooner. What are your thoughts on this, how do you decide what you should focus on?

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  • $\begingroup$ This is definitely a question of personality. I think the root lies in whether you are a theorist or a practitioner, and your preferred method of analysis. $\endgroup$
    – cygorx
    Commented Jan 6, 2014 at 22:00
  • $\begingroup$ Yes, but there is the question of how far I can go in focusing on the theory. I'm afraid of ending up knowing "something" in a million different fields, but having a stupidly shallow understanding and being unable to solve even simple problems in any of them. Or that in my pursuit of theory I'll find myself unable to progress into more advanced topics, because I haven't developed a sufficiently deep understanding of the "lower" fields. $\endgroup$
    – Ormi
    Commented Jan 6, 2014 at 22:08
  • $\begingroup$ If you understand the theory properly, then applying it is mostly simple. Part of learning is analyzing and asking questions. $\endgroup$
    – cygorx
    Commented Jan 6, 2014 at 22:10
  • $\begingroup$ If the problem sets are "boring", maybe you should use another book. ;) I'm not sure at which level you are learning, but most of the books I have ever used asked to prove some very non-trivial results in the exercises. $\endgroup$
    – fkraiem
    Commented Jan 6, 2014 at 22:17
  • $\begingroup$ Well, some of the problems might be somewhat interesting, but still, you won't get the awesome stuff like the general Stokes' theorem or Galois theorem about nonsolvability of polynomials of degree > 4 in the problems set. So while exercises might sometimes be interesting to an extent, they still slow you down on the path to the really good stuff. $\endgroup$
    – Ormi
    Commented Jan 6, 2014 at 22:33

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It all a bit depends why you pick up the book in the first place what do you want to learn from it and why.

Are you just reading the book to escape boredom , what in itself is better than other ways to escape boredom please do so.

If you are reading the book to learn from it then apply the knowledge in the book, do the excersises or what I do more times, make your own (more interesting?) excersises and try to get an answer to them, see the book just as a trove of tools that should help you to solve your own excersize. (try to prove one of those interesting theorems yourself but without using the underlying boring theorems)

There is no need to keep close to the sequence in the book, if you are only interested in the last chapter just read that, any excersize you make yourself after reading that chapter will guide you back to earlier chapters.

but do make it fun for yourself no use just reading it (except for preventing you from doing other mischief)

Good luck

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