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?
Best Answer
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