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What's new besides showing modern advancements in modern mathematics as well as eloquently written notes also contains some good advice to young people like me in this room (career advice:!). in particular Does one have to be a genius to do maths? terry tao debunks the myth of genius in math and says that one need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the “big picture”.

i used bold font so that you see more clearly my problem: this summer i worked through stewart (rip) calculus, i didn't do any exercises but i understood concept very well, i could also do the examples for myself. but now i'm doing every exercise of stewart, and that takes a lot of time. at the same time i don't know whether i should go study other fields instead of doing these exercises. so my question is when to stop doing exercises and moving forward and when to stop learning and beginning to do exercises? (in other words) how to balance hard work and learning other fields and tools? another question what is hard work in maths, is it just doing exercises?

also i'm also doing imo exercises, should i stop that and do real mathematics instead?

thanks

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  • $\begingroup$ why do learning and doing exercise have to be mutually exclusive; in fact they go hand in hand. $\endgroup$ – abel Mar 15 '15 at 17:11
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    $\begingroup$ @abel there are just too many exercises, and i'm afraid that i wont be able to move further if i pass my time doing all of them $\endgroup$ – user153330 Mar 15 '15 at 17:54
  • $\begingroup$ when you do a whole lot of exercises, your brain/something tells you i know what i am doing. one you hear that, you move onto new set of exercises. you don't how many times i have heard from my students that they understand the concept but are unable to do problems. if i don't need to be diplomatic, i will tell them that they dont understand a concept until they can apply to solve problems; but i can't do that. $\endgroup$ – abel Mar 15 '15 at 18:07
  • $\begingroup$ @abel what about ma last question i'm also doing imo exercises, should i stop that and do real mathematics instead? $\endgroup$ – user153330 Mar 15 '15 at 18:29
  • $\begingroup$ @abel But solving problems doesn't always entail much understanding, you might agree? Someone with some mathematical experience could read the definitions and theorems and then abuse caffeine/ solve the exercises without needing to think about it on anything but a formal level. $\endgroup$ – Jacob Wakem Jul 27 '15 at 4:10
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Somebody clever once said you don't get to be a concert pianist by watching somebody else play the piano. I would say that the exercises are crucial because it is too easy to read over someone else's writing - particularly that of a good teacher and think that you have it. There is however a diminishing return on just doing exercises based on a single topic at the end of a chapter on that topic. Once you have it in your mind that the answer is in the chapter you have just read it acts as a big clue and you arguably stop making the connections between different areas. I would do between 5 and 10 exercises on each chapter to clear up any misunderstanding and then move on.

You really don't have to be a genius to be good (really outstanding might be different) at maths. I earned a 1st class degree from a decent university and always considered myself to be a bit thick compared to other people on the course. That was based on 50 - 60 hours per week, every week for 3 years. A little less during holidays and considerably more coming close to exams. That was enough to learn a reasonable enough range of mathematics to solve a range of problems. That probably broke down as 10-12 hours lectures another 10-12 hours discussing lectures and the rest problem solving.

The question of what is hard work in maths is complicated a little by the fact that most people who learn it to the level you are talking about love the subject. The work may be hard but the work isn't actually work. People naturally fall into spending that amount of time on it because that is their leisure time as well as their work time. If you have the passion you will find yourself putting more and more time into it.

The question of other fields and tools is really one of specialization. It is talking about within maths or maths related disciplines and that kind of specialization is really graduate level/ doctoral level study. An undergraduate course will provide you with a number of different courses and broadly speaking set out the amount of time you spend on each. Even if you don't have the resources or the time to do that kind of course, download a syllabus for a first year course and see what is required. There might be between 6 and 8 topics studied in a given year.

Finally I would say - depending on your experience you may find that mathematics at university level is not quite as numerate a subject as you might expect. The sort of calculus that I think is in Stewart certainly has its place on a maths course but you could find it in an engineering or natural science course as well. Certainly "Pure" Mathematics at higher levels tends to work in the abstract and takes calculus as the foundation for abstract spaces of objects. Applied mathematics tends to be a little more like the mathematics in Stewart's book.

This exchange here is all the opportunity you need to talk to other mathematicians.

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  • $\begingroup$ what about ma last question i'm also doing imo exercises, should i stop that and do real mathematics instead? $\endgroup$ – user153330 Mar 14 '15 at 15:01
  • $\begingroup$ If you are talking Mathematical Olympiad exercises then no you should not stop. These are generally difficult problems which are however accessible to people who have not come across undergraduate material. You should however balance it with learning more advanced topics and theory because sometimes difficult problems at on stage are seen to have solutions as part of a general theory at the next. If you are capable of doing these Olympiad problems you are certainly capable of further study. These problems are real mathematics. $\endgroup$ – Michael E Mar 16 '15 at 12:31
  • $\begingroup$ i gave you +250 for your effort =) $\endgroup$ – user153330 Mar 19 '15 at 15:07
  • $\begingroup$ i will accept your answer (thanks to you) if you help me with one last question: because i saw a user named Ittay Weiss who is a researcher say that IMO problems are nothing compared to learning real mathematics, so how do i balance them? or should i rather become more interested in the real mathemtics (like topology, real analysis....)? $\endgroup$ – user153330 Mar 19 '15 at 15:09
  • $\begingroup$ Thank you for the +250. I have to respect the opinion of Ittay Weiss - he/she has gone further than I have but I still think you should find time to do problems. You will likely find that if you study things like topology and real analysis you will find yourself naturally wanting to test your understanding by doing exercises but there really is no restriction. $\endgroup$ – Michael E Mar 19 '15 at 22:52

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