Is it necessary to prove everything and solve every problem in the books?

Question

I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice.

Whenever I study mathematical books (Rudin, Hoffman/Kunze, etc.), I always try to prove every theorem, lemma, corollary, and their relationships in the book. Unfortunately, that determination has been demanding huge time consumption; sometimes, it takes me days to fully understand and able to prove materials in the few pages of book. I am willing to devote my time to understand the topics, but I also wanted to devote time to my undergraduate research projects and other courses. Recently, I started to depend a lot more to the proofs presented in books and websites (like MSE), which has been causing a huge guilt and fear that I am not making the knowledge into my own.

Despite my effort to prove/solve every problem per chapter, I found myself to skip some of the problems and move on to the next chapter, which resulted huge fear as that means I did not fully understand the materials..

• How do you read the mathematics books and make knowledge on your own?
• Is it absolutely recommended to prove everything and solve every problems in the book?
• Also is it recommended to devote more time to the problems than exposition preceding the problems? I found myself devoting a lot time to the actual expositions in the book as I like to play around with definitions and theorems, try to come up with my own ideas, and formulate my own problems (I actually found that making my own problems is much more fun than problems presented in the book).

Answer 1

I would suggest you read every problem, and in your head if you can see the direction pretty clearly then no need doing that, generally big texts do have repetition, but concise books meant for only problem solving without any theory do try to make sure each problem is unique.
As far as second part of your research goes, I believe your approach gives you more fundamental clarity but definitely not an approach to get 'grades', if you're planning on doing research you should carry on this attitude but if not I would suggest some problem solving.

Answer 2

Mathematics is kind of a subject that is fun as well as scary. I always know the concept and do one or two problems per concept. As solving each & every problem is time consuming and is of no use. My advice is practice all the concepts and theorems, and do two or 3 different types of problems which based on same concept. It helps us to know which concept to apply for certain problem. If you got the concept perfectly, it's application is not difficult, so have no fear and move forward. This subject is very easy when the concepts are understand well.

Answer 3

Overall answer: no.

I struggled with the same problem as you for quite a long time and, in hindsight, I think I could have spent my time more wisely. Here are my current general guidelines at the time of this post. They may or may not work for you. The overall philosophy I employ is that exercises are usually there to get you comfortable with the material: you are probably usually expected to remember the theorems and (mostly) forget the exercises once you have done them. I expound on this below.

How do you read the mathematics books and make knowledge on your own?

Nowadays I spend more time thinking about the material in the text than doing exercises. I treat the theorems as problems and try to see how far I can get in proving them before reading the proof. Importantly, I don't spend all day doing this! A good strategy might be to set yourself a goal, such as "I want to get to such-and-such page by the end of the week", and pace yourself accordingly; if you spend thirty to forty-five minutes thinking hard about a theorem and having no ideas, perhaps take a peek at the proof and continue from there (alternatively, skip that proof and come back to it later).

If the book has examples in the text, read and understand as many as you can. Otherwise, be sure to allocate generous amounts of time towards coming up with your own examples. Note that this is not necessarily easy, and is arguably the most important stage.

Finally, if the book has exercises, be sure to do at least some of them and try to understand the general idea of the rest. Don't be afraid to ask and try to answer your own questions which might be inspired by some of the exercises.

One downside to this method is that it doesn't work for all books and all subject matters. Some authors place key theorems in the exercises and proceed to use them later, expecting you to have proven them yourself. The hope is that for each topic there exists a book for which this method is not useless.

Another downside is that, in rare cases, too many of the exercises are interesting (this appears to be the case with Rudin, among other books)! In this case it's definitely up to you how much time you spend on the exercises. Allocate time according to how interesting you find the exercises and how comfortable you are with the material in the text.

Is it absolutely recommended to prove everything and solve every problems in the book?

If you can do this and still lead a comfortable life, then by all means do so (if it doesn't hurt, then it can only help, right?). Unfortunately, attempting to do this will probably make life less than comfortable for you, so I would advice against being this extreme.

However, the point of this "advice" is that you should get as comfortable with the theorems and the examples as much as you possibly can, and this is good advice. I would advise just thinking about stuff as much as possible. Thinking about maths in the shower, on the way to the shops, while cooking dinner, etc. will get you used to thinking about the topics that interest you.

Also is it recommended to devote more time to the problems than exposition preceding the problems?

This heavily depends on both the book and the reader. However, if you prefer to come up with your own examples and fiddle around with the theorems, and if the book you are working from is not expecting you to prove key theorems in the exercises, then I would say this approach is a good substitute for doing exercises.