Hi,I'm a physics major student outside US.

As a non math major student, I have always been curious what is the right way to study or do mathematics.

I usually hang around forums on physics but I thought If i came here, a lot of smart math guy will read it and can give me advice in different perspective for me but typical in their circle.

Can someone describe their general practice/process/routine for learning mathematics?

My personal difficulties are...

How do you come up with a solution or proof easily? I can do simple few line proof. How does one do proof that is couple pages long?

How do you memorize theorems ,definitions and formula easily? They are too logical statement that there is nothing to grasp with, like visualization and etc. I always try to remember it in verbatim, hold one or two for minutes and have to refer a page again later.

How do you study for long time? I easily exhausted after 2-3 hour, feeling faint.

If you know books about this topic, please recommend.

## How to study mathematics in general. Please, give me some advice

### Re: How to study mathematics in general. Please, give me some advice

Short answer: mathematics isn't easy.rr00 wrote: How do you come up with a solution or proof easily? I can do simple few line proof. How does one do proof that is couple pages long?

Longer answer: Sometimes, long proofs are actually just a lot of easy steps. Those ones are not such a big deal - you know exactly what you have to prove, but maybe its like 10 different things you have to check. These are kind of easy "verification" proofs. The harder proofs are when there's just one statement to prove, and not an obvious way to do it - usually, when I produce a long proof of this kind, it's the result of a lot of trial and error. I usually attempted one proof, and found a hole. Then, in attempting to resolve that, I need a lemma or two, so I have to prove those [and coming up with these requires you to have an intuition for what's happening, generally], and then maybe separate out a couple of claims for clarity. It would definitely be rare for me to immediately see how to do it. It usually takes me several days of thinking about it, and many false starts, before producing such a proof.

First, you do you get used to logical statements. But you don't really learn a theorem by memorizing the statement word for word - you have to ask yourself "What is this really saying?" and figure out the real meaning/intuition to really understand it. It can take a long time to absorb a theorem - you definitely don't read math books at the speed of a novel. This afternoon, I spent several hours reading one page of a book.rr00 wrote: How do you memorize theorems ,definitions and formula easily? They are too logical statement that there is nothing to grasp with, like visualization and etc. I always try to remember it in verbatim, hold one or two for minutes and have to refer a page again later.

It's very hard to be productive for a long time without a break. Generally, I work on a problem (or reading, whatever) until I get tired, and then I walk around, talk to somebody, or whatever, before coming back to it.rr00 wrote: How do you study for long time? I easily exhausted after 2-3 hour, feeling faint.

### Re: How to study mathematics in general. Please, give me some advice

It depends what you mean by mathematics. If you're talking about proofs for problems in a math class (or even contest math), it's in large part about whether or not you've seen it before. In math, you learn by doing. Period. For example, proofs using group actions or the Sylow theorems typically follow similar steps. Some proofs are more sophisticated of course, but having seen the basics before gets you a long way. That said, upper level math is certainly more difficult than the computational calculus courses that freshmen (or physics majors?) study.

Now research mathematics is a different story. In my REU, there was a basic statement about a computational fact I knew was true, but couldn't think of a proof for. I tried all sorts of things, like showing a function defined on Z_p was one-to-one, but got nowhere. Later I came up with a nice computational proof using elementary number theory. This took me all day to figure out. I ended up figuring it out by looking at a specific example (let n = ...). A couple months later I came up with a one-line proof using some theorems in algebra when I wasn't even thinking about the problem anymore.

Sometimes after learning so much math in so many different areas, you just have ideas come to you. I had an officemate tell me about one of his problems once, for example, and I immediately told him it reminded me of continued fractions. He asked me if I'd studied this problem before, because there were some papers on the subject linking certain aspects of the problem to certain continued fractions. Which leads me to my next point: it can be useful to bounce ideas around other smart minds, whether that means people in your office or people who attend your talks at, for instance, a number theory seminar in the department. Plus your advisor may give you some good direction, too, of course.

As for remembering theorems... in some classes, it just makes sense. In some classes, there are a few very important theorems you need to basically solve any problems you'll be tested on (maybe that means Sylow in algebra, or DCT in measure theory, etc.). Sometimes the instructor makes it memorable by giving certain examples or just being a good teacher. It can be instructive to think of good examples. Now as much as many people claim to hate memorization, it can be useful to some extent in math.

Finally, you start your work early and let it sit in the back of your mind. Then a proof will come to you when you're at the gym or in the shower or in bed.

Now research mathematics is a different story. In my REU, there was a basic statement about a computational fact I knew was true, but couldn't think of a proof for. I tried all sorts of things, like showing a function defined on Z_p was one-to-one, but got nowhere. Later I came up with a nice computational proof using elementary number theory. This took me all day to figure out. I ended up figuring it out by looking at a specific example (let n = ...). A couple months later I came up with a one-line proof using some theorems in algebra when I wasn't even thinking about the problem anymore.

Sometimes after learning so much math in so many different areas, you just have ideas come to you. I had an officemate tell me about one of his problems once, for example, and I immediately told him it reminded me of continued fractions. He asked me if I'd studied this problem before, because there were some papers on the subject linking certain aspects of the problem to certain continued fractions. Which leads me to my next point: it can be useful to bounce ideas around other smart minds, whether that means people in your office or people who attend your talks at, for instance, a number theory seminar in the department. Plus your advisor may give you some good direction, too, of course.

As for remembering theorems... in some classes, it just makes sense. In some classes, there are a few very important theorems you need to basically solve any problems you'll be tested on (maybe that means Sylow in algebra, or DCT in measure theory, etc.). Sometimes the instructor makes it memorable by giving certain examples or just being a good teacher. It can be instructive to think of good examples. Now as much as many people claim to hate memorization, it can be useful to some extent in math.

Finally, you start your work early and let it sit in the back of your mind. Then a proof will come to you when you're at the gym or in the shower or in bed.

### Re: How to study mathematics in general. Please, give me some advice

Thanks for advices!

After seeing those advices, I thought I have to be more specific.

When you are thinking, Do you write down you thoughts?

When I am thinking, It's hard to do more than 3 steps of syllogism in my head.

1.What is the best way to sustain long chain of thought?

2.What I hate most when I'm doing mathematics is the moment when after I struggled for couple hours, then look up proof or solution. then I 'm disappointed with myself by the fact that the proof or solution uses knowledge I have already known. In other words, everything in every steps of proof is things that I know. but I couldn't make those steps. or take a lot of time than reasonable. What is cure for this?

After seeing those advices, I thought I have to be more specific.

When you are thinking, Do you write down you thoughts?

When I am thinking, It's hard to do more than 3 steps of syllogism in my head.

1.What is the best way to sustain long chain of thought?

2.What I hate most when I'm doing mathematics is the moment when after I struggled for couple hours, then look up proof or solution. then I 'm disappointed with myself by the fact that the proof or solution uses knowledge I have already known. In other words, everything in every steps of proof is things that I know. but I couldn't make those steps. or take a lot of time than reasonable. What is cure for this?

### Re: How to study mathematics in general. Please, give me some advice

I definitely write down a lot of my thoughts. Often times I don't see the next step until I've written down what I'm currently thinking about. Also, sometimes something seems to make sense in your head, but when you write it down you realize there is a gap (e.g. additional assumption needed, or something). If I'm really stuck, sometimes I just start writing down various facts I know that might be applicable, even if I don't see how yet. Sometimes it just takes a lot of time thinking about it in the back of your mind, when you step away a new idea may come to you in the shower!