My Math Forum How to begin advancing in mathematics?

 February 22nd, 2010, 05:31 AM #2 Senior Member   Joined: Nov 2008 Posts: 199 Thanks: 0 Re: How to begin advancing in mathematics? This is an interesting and not entirely straightforward question. At university, at least in my experience, maths essentially starts again from scratch with the reintroduction of basic notions such as functions etc. so in one sense there are next to no prerequisites in knowledge. However, in practice a solid grounding in basic mathematical technique and a certain amount of developed intuition about what is being discussed are necessary. The essential difference between school level maths and higher maths is 'rigour', so while seemingly simple ideas are dealt with at first they are done so with a level of attention to detail that most students find very difficult, even those with a high degree of competency in school maths. Another key element of higher maths is abstraction; structural properties of 'concrete' systems, e.g. numbers, are isolated and abstracted so they can be studied in themselves, giving rise to more abstract structures. These abstractions can then be abstracted themselves and so on. While in principle the abstraction does not rely on the thing it abstracts to make sense, in practice if you don't see where the thing comes from it's not going to mean anything. The cornerstones of modernish (late 19th, early 20th) century maths are analysis and abstract algebra. To approach an understanding of these you should get a decent knowledge of number manipulation, leading to gcse level algebra (2x +4 = 9 etc.). From here A level calculus should be approachable, which, after much practice, makes analysis start to be understandable. Abstract algebra in turn builds on the symbolic manipulation of school algebra. I'd recommend starting by going through the gcse and A level material on these subjects. School level geometry is also a good bet. Also, statistics is a very useful thing to understand in our society. Seriously, people are in prison because the courts did not understand stats properly.
 February 22nd, 2010, 11:00 AM #3 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: How to begin advancing in mathematics? Pseudonym's answer contains most of the information I would give (and then some more), but a few more specific suggestions: A very good idea is to find a book (or better yet, but harder, a course) on "discrete mathematics". These courses tend to introduce mathematical rigor, and provide most of what you would need to prepare for studying algebra, combinatorics and number theory. Typically, in such a class, you would learn (very) basic set theory and logic, the foundations of number theory, permutations/combinations, sums, and basic graph theory. Set theory and logic will be necessary to go anywhere in the world of mathematics, number theory is fun and beautiful, and the last three (permutations, etc) will provide the foundation for combinatorics and probability.
 February 22nd, 2010, 02:41 PM #4 Newbie   Joined: Feb 2010 Posts: 7 Thanks: 1 Re: How to begin advancing in mathematics? What would you recommend as a good book on discrete mathematics? I too am beginning to self-study and have started reading Spivak's "Calculus" as a starting point.
 February 22nd, 2010, 04:48 PM #5 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: How to begin advancing in mathematics? I'm not sure, but I'll look around. My class used the book by Rosen, which is apparently fairly standard, but I found it to be pretty bad.
 February 22nd, 2010, 05:56 PM #6 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: How to begin advancing in mathematics? I've looked a bit at Amazon reviews, and this is how it seems: Susan Epp's book is the one that seems to have the most positive reviews (from people who didn't just sound like disgruntled students.) This book by Lovasz and Pelikan may be another good choice. I know that Lovasz and Pelikan are both top mathematicians in the field, and everything I've heard suggests that they (like all of the famous Hungarians...) are excellent instructors. There is a Dover Book which might be a good supplement, as Dover books tend to be, but the also tend to be a bit obtuse. How to Prove It, How to Read and Do Proofs and How to Solve It are all excellent texts which will be very useful for self study. (To be fair, I haven't read any of them completely, but I've heard nothing but glowing praise about them... Polya is as renowned as an educator as he is as a mathematician, and what I've read from "How to Solve it" is fantastic.) I hope those help. (Fine print: I have not read any of the books I'm recommending, so do not kill me if not good books. The first 3 seemed to have broadly positive reviews from mathematicians, and from students who seemed at all competent. The last 3 I have heard constantly suggested for people just getting into higher math.)
February 22nd, 2010, 07:12 PM   #7
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Re: How to begin advancing in mathematics?

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 Originally Posted by cknapp I've looked a bit at Amazon reviews, and this is how it seems: Susan Epp's book is the one that seems to have the most positive reviews (from people who didn't just sound like disgruntled students.) This book by Lovasz and Pelikan may be another good choice. I know that Lovasz and Pelikan are both top mathematicians in the field, and everything I've heard suggests that they (like all of the famous Hungarians...) are excellent instructors. There is a Dover Book which might be a good supplement, as Dover books tend to be, but the also tend to be a bit obtuse. How to Prove It, How to Read and Do Proofs and How to Solve It are all excellent texts which will be very useful for self study. (To be fair, I haven't read any of them completely, but I've heard nothing but glowing praise about them... Polya is as renowned as an educator as he is as a mathematician, and what I've read from "How to Solve it" is fantastic.) I hope those help. (Fine print: I have not read any of the books I'm recommending, so do not kill me if not good books. The first 3 seemed to have broadly positive reviews from mathematicians, and from students who seemed at all competent. The last 3 I have heard constantly suggested for people just getting into higher math.)
Hey thanks a lot! I saw the reviews on Amazon for Rosen's book and it looks like people either hated it or loved it. Those last three books I think would help me a lot. I went to school for electronic engineering, but never got deep into math the way I would like to, such as doing proofs and truly understanding a formula. I'll check out those books, thanks a lot again!

February 23rd, 2010, 05:37 AM   #8
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Re: How to begin advancing in mathematics?

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 Originally Posted by youngmoney I saw the reviews on Amazon for Rosen's book and it looks like people either hated it or loved it.
You'll see this a lot with math books... You can't just sit back with a math book and a cup of tea on a sofa, and get anything out of it-- you need to have a pen and paper ready, and work through stuff. Especially if you are studying by yourself, and don't have anyone to guide you and say what to emphasize, and how, really to do it.
Most people don't realize this, and think that a book is "confusing" and "obscure" if they need to do this.
On the other hand, you'll get books which really don't teach anything-- the examples are bad, the exercises are worse, and the prose rambles-- which people like, because it's "approachable".
C'est la vie, I guess.

Anyway, cheers, and feel free to ask any questions you have here.

February 23rd, 2010, 07:51 AM   #9
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Re: How to begin advancing in mathematics?

Quote:
 Originally Posted by cknapp You'll see this a lot with math books... You can't just sit back with a math book and a cup of tea on a sofa, and get anything out of it-- you need to have a pen and paper ready, and work through stuff. Especially if you are studying by yourself, and don't have anyone to guide you and say what to emphasize, and how, really to do it.
"I read this juggling book, but I still can't juggle!"

 February 25th, 2010, 09:07 AM #10 Newbie   Joined: Feb 2010 From: Canstable Posts: 2 Thanks: 0 Re: How to begin advancing in mathematics? Thanks to however moved this thread, and thanks cknapp and pseudonym for your considered responses. It is evident that the study of maths is a subject to be studied in its own right! I would be lying if I said I wasn't still confused, in fact I may be more confused than ever. Is discrete maths what I want to be looking at? I've been getting definitions off wikipedia, which of course is known to be less than reliable. The problem is, as I think Pseudonym may have described (I was confused) that to understand one idea I must read on several other ideas, which in turn require me to read on several more, in an endless branching out of ideas. I think the best approach may be to get some GCSE books and 'go back to school' with it all, but I'm not too keen, I don't trust the education sector (at least at high school level) to be adequately teaching ideas, more getting students through exams to create seemingly better and better exam statistics. So if there is an 'independent' way I can do it, I'll be far more inclined to choose that way. Would I be a fool to consider getting a 'Maths for Dummies' type of book?

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