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## The Should I Become a Mathematician? Thread

 Quote by dustbin I figure I will ask here, rather than cluttering the main page with another of these topics... I'm currently studying calculus (Stewart) at my college. I have found it pretty unchallenging (there are exceptions of course to some concepts and problems- but I can pick up on these as well without issue) so I decided to start into Apostol. I've been reading all of the corresponding topics for my Stewart-based class in Apostol, which has proven to shed a clear light on all of the subjects. I really like to read Apostol, it makes a lot of sense (is explained very well), and the read feels just challenging enough. However, I can do very few of the problems in Apostol. Any of the problems involving calculation, I can do (that I've encountered). I can then do maybe a select few (or couple) of the more theoretical problems. Sometimes I know the "why" but I have little to no idea of how to put it on paper in a way that would give me a good score were it graded. I have the solutions to some of the answers worked out (from MIT, Caltech, etc.) which have helped, but they are still a little overwhelming for me. This is a little frustrating because, when I read Apostol, I feel like I really understand it and feel confident that I will be able to do the problems. The first few problems make me feel good and then I get smacked in the face. I still continue to read it and attempt problems since my understanding of the material has clearly shown in my Stewart-based classes (I have a 100%)... but I really want to work the more theoretical/rigorous problems and material. Is the missing ingredient logic and proof? I was reading Principles of Math (mathwonk's suggestion) and got through the first few chapters and then quit because I do not much care for the style. I'm checking out a handful of other proof and logic books from my library (Eccles, Houston, D'Angelo, and Chartrand). Is there anything else I should be studying to really be able to conquer the Apostol problems??
Apostol (and Spivak) are both known for their challenging exercises. It is perfectly normal that you are not able to do most exercises. I actually think you're already doing a good job if you can "see" intuitively why something is true.

Would it help to post your solutions (or thoughts) in the homework forum. We will certainly help you to rigorize your arguments. I think the best way of learning proofs is by doing them, making mistakes and being corrected.

 Quote by micromass Apostol (and Spivak) are both known for their challenging exercises. It is perfectly normal that you are not able to do most exercises. I actually think you're already doing a good job if you can "see" intuitively why something is true. Would it help to post your solutions (or thoughts) in the homework forum. We will certainly help you to rigorize your arguments. I think the best way of learning proofs is by doing them, making mistakes and being corrected.
Thanks for the suggestion, MM. I will certainly start posting up to get help with the problems in the help forum. I am fairly comfortable with induction and epsilon delta proofs, but beyond that I certainly need a lot of work. I'm starting through Chartrand's book (which I like so far) and really deconstructing/going through all the proofs given in Apostol and Stewart.
 Just a few things I saw in the notes there... "How to Solve it" (in it's various editions) is the classic G. Polya book on problem solving. While it contains examples, it is more philosophical and is based on getting you into a particular mindset of problem solving. It's excellent. I took my time reading it - almost a year off and on while I let the concepts sink in. There is a section on proofs, but it won't teach you anything like set theory. "How to Prove it" is Daniel Vellemen's book, which I'm using now. It's excellent. Lots of examples, and a very logical structure. I'm going through it before I take my first abstract math class to avoid the "culture shock" of such a class. The first couple of chapters introduce logic and set theory, and then different proof techniques are explained, and then some more advanced concepts in set theory. It's nothing like Polya's book, but it's a great companion to it. The title is possibly an homage to Polya(though there is no mention of this), but sometimes people seem to get them mixed up. Oh, and if you get this book, get the latest edition, because the first one had no answers or hints to any of the problems. I found that very frustrating. Fortunately I was able to swap it out for the newer edition at my library.
 I agree with you about D'Angelo's text, RJinkies. Hopefully I will have the time to come back to it at a later point, though. It looks like a very interesting text. I read a bit of Eccles and did not have an issue... but perhaps this is because I have read material on logic, proof, etc. before from Apostol, Allendoerfer, and some brief touchings on it in my college algebra class. I've found so far that Chartrand is great for me. I have started working through it since I was able to purchase it for <\$10 with shipping. Hopefully I will be cross-enrolling in an intro to proofs/higher maths course at the local university this fall. Thanks for the tip on Kosmala. It looks like an interesting read... I just requested it from the library. It is interesting how difficult of a jump it is to make from the standard mathematics education to the more rigorous material. I have been the top of my class in all math courses up to this point and, while I feel I'm quickly picking up on this new material, it is still a difficult transition. Any advice is always appreciated.

 It is interesting how difficult of a jump it is to make from the standard mathematics education to the more rigorous material. I have been the top of my class in all math courses up to this point and, while I feel I'm quickly picking up on this new material, it is still a difficult transition. Any advice is always appreciated.
I didn't even do that well up to that point, but it was actually an easy transition for me. It was much more natural to try to figure out how everything worked than to do it by rote, which drove me insane. So, when I changed majors to math and started doing upper division stuff, I felt like I was being freed from my chains. Only lasted a couple years, though, and then it got hard again.
 do grades in lower division math classes count as much as upper division when your trying to get into grad school? or is it all based on GPA? i know having research experience helps a lot but im at a community college at the moment waiting to transfer very soon, and i dont think community colleges have any research opportunities, unless i haven't looked in the right direction. i ask because i definitely plan on shooting for at least a masters in the subject for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles both linear algebra and diff equations in one class) im guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.

 do grades in lower division math classes count as much as upper division when your trying to get into grad school?
They count a little, but not as much. I wonder if getting a C in linear algebra and diff eq is a factor in why I only got into one grad school, despite strong recommendation letters and very good upper division grades. Probably not, I think. I'm guessing it's probably just that other applicants had taken more math classes or had research experience and that sort of thing.

 or is it all based on GPA?
GPA doesn't matter very much. Most programs just require a 3.0 minimum, but that's it. Good overall GPA is sort of a sign of a consistent, hard worker, which they like. But mathematical ability is more important.

 for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles both linear algebra and diff equations in one class) im guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is.
Yeah, pretty much.
 @miglo: Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources. @homeomorphic: I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct I just need more practice and experience.
 - for lower division i guess that would be anything below calculus, the whole 3 semester calculus sequence plus intro to linear algebra and differential equations (the college i attend bundles linear algebra and diff equations ) - im guessing upper division begins with intro to analysis, or a class aimed at helping students learn what proof based mathematics is. It depends where, usually the lower is calculus i ii iii iv linear diff eqs intro to analysis but some linear and diff equations can be considered upper if they deal with some analysis and extra stuff. sometimes you can see the regular linear being second year [sometimes the 200 levels], yet the honours classes can be the upper division [at the 300 levels] same with diff eqs, regular classes could be 200 level, and honours at the 300 level. ----- with analysis, some like to bunch it with the calculus classes, others as a separate course of half a year or a whole year, and then your next class will be a 300 level/upper division one geometry classes can be upper or lower too depending how intense, and some schools that do really lite courses on abstract algebra can be lower division. often they'll use linear algebra as a prerequisite, though im not sure it should really matter that much. ------ For my money, all one needs to really focus on is a. calculus i ii iii iv b. basic analysis and more analysis and then probably more analysis all the rest is filler.... and neat if you're doing good, and a drag if it's painful... [where you're skipping too fast and not going deep enough] heck with a super duper calculus text, and two supplementary texts maybe you got (a) and (b) both as for proof and rigour, you can face that at any stage, first year extra hard calculus texts, second year linear with tons of abstract spaces and forcing you to generalize/do proofs, or you can get hit hard with analysis classes or abstract algebra with it.... A lot can really depend on your choice of textbooks... and a really limited and inflexible curriculum i think is why you get people who face these 'hard' things somewhere up the ladder, and sometimes the higher the costs of education [and textbooks] the curriculum gets worse by being more bare bones.... --- if i had my way, a uni would be a library where you get a duotang for all the math texts, year 1 2 3 4, and the reading lists [and options] and a duotang for the physics texts, year 1 2 3 4, and the reading lists... exams? what exams, doing one chapter and all the problems, is your damn exam lol spend 40 hours on a chapter, and no not pass go, till the time clocks says 40 hours.... might take 8 years to get your degree, but it'd be like 'speed learn' in the Prisoner, 100% Entry, 100% Pass.

 I do not find the material itself difficult... I am just having to work very hard at gaining comfort in proofs (both reading and writing). Prior to a few months ago I had never even seen or worked one out (save for my trig teacher proving the quadratic formula). I am particularly terrible at brevity... I just finished a proof using Rolle's Theorem which was two paragraphs long. I compared it with another answer to the problem which was only 3 sentences. At least my answer was correct I just need more practice and experience.
The interesting thing about brevity is that one of my profs said he was really impressed with my brevity, yet I never gave any thought to it. I think it's probably because my thought process is very conceptual. In trying to understand something deeply, you usually want the simplest explanation possible. When you make that rigorous, often, the proof ends up being short. That doesn't always happen, but it's my theory as to why my proofs tended to be shorter than most people's. I tend not to plow through stuff using technical brute force. It could also be that after a while, I knew which steps I could skip when I wrote the proof down, since they were clear enough.

 Quote by dustbin @miglo: Check out REU's (Research Experience for Undergraduates) to find opportunities for research. There are several sites that have lists and other resources.
so even though im at a community college i can still apply for REU's? i always thought that only applied to undergraduates at universities.
 Miglo, some are only for undergraduates at universities. I have found some that are open to all undergrad students. When you look, just look through the requirements and such info about the application. Some do not specify, which I assume means they are open to all undergrads...
 Recognitions: Homework Help Science Advisor here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith: {a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)

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 Quote by mathwonk here is a problem from an 1895 high school algebra book, Treatise on Algebra, by Charles Smith: {a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. any takers? (show work.)
That was fun!!! Everybody should definitely try this one.

Here's my solution:

Let

$$\frac{a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)}{a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}$$

be our expression. We multiply numerator and denominator by abc to get

$$\frac{a^3 (c - b) + b^3 (a - c) + c^3 (b - a)}{a^2(c - b) + b^2(a - c) + c^2(b - a)}$$

Rearranging gives us

$$\frac{a^3 (c - b) - a(c^3-b^3) +bc(c^2-b^2)}{a^2(c - b) - a(c^2-b^2) +bc(c-b)}$$

We can eliminate c-b from numerator and denominator to get

$$\frac{a^3 - a(c^2+bc+b^2) +bc(c+b)}{a^2 - a(c+b)+ bc}$$

Rearranging again gives us

$$\frac{a(a^2 - c^2)-bc(a-c)-b^2(a-c)}{a(a-c)- b(a-c)}$$

Eliminating a-c and we get

$$\frac{a(a+c)-bc-b^2}{a- b}$$

Rearranging again and we get

$$\frac{(a^2-b^2)+c(a-b)}{a- b}$$

Eliminating a-b yields

$$a+b+c$$
 Recognitions: Gold Member I got the same thing as well, Micro. I would show my work, but I did it on some graph paper that I had nearby. That was fun to do though. I solved mine a little differently than you, so I might take of picture of my work and show it that way. I haven't taken the time to get Matlab or LaTeX or anything like that yet, so I don't wanna just type in all of my math and have it be a huge, ugly mess.
 looked at a little bit of the problem, the pattern is interesting: ax+by+cz ------------ a+b+c not making it messy, now that's a challenge...