The Should I Become a Mathematician? Thread

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.

dustbin

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.

RJinkies

dustbin - I'm checking out a handful of other proof and logic books from my library (Eccles, Houston, D'Angelo, and Chartrand).

those other two books are good too, but D'Angelo and Eccles are a bit more advanced...but if you got the other books, they are good to supplement once you're a few chapters into the other ones....



[on a side note, most people think Apostol's book on analysis is a great second text on the subject if you start with one easier...]

but if anyone's tackled both texts, do you run through his calculus book and then tackle his analysis book in the next semester, or have some done both books at the same time... I'd think that both books would be a second tackling of calculus and a second tackling of analysis in the ideal world.... you need a bit of intuition starting off...

[i'd like to hear what people tried in calc or analysis before tackling those tomes]

[I heard of people doing fine with Syl P Thompson's calculus book and then going into Apostol's calculus pretty okay... which says a lot for thompson being great preparation...]

[I know some people that really wanna prepare well for Apostol or Rudin and they tried this text
- Advanced Calculus: A Friendly Approach - Witold A.J. Kosmala - Prentice-Hall - 700 pages - 1998 - going for the intimidation-free approach... anyhoo, just my three cents]




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Mathematical Thinking: Problem-Solving and Proofs - Second Edition - John P. D'Angelo and Douglas B. West - Prentice-Hall 1999 - 412 pages

[For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.]

[Offering a survey of both discrete and continuous mathematics, Mathematical Thinking begins with the fundamentals of mathematical language and proof techniques such as induction. These are applied to easily-understood questions in elementary number theory and counting. Further techniques of proofs are then developed via fundamental topics in discrete and continuous mathematics. The text can be used for courses emphasizing discrete mathematics, continuous mathematics, or a balance between the two. It contains many engaging examples and stimulating exercises.]

[Extremely Useful - Great Read]

[I ran into the first edition of this book ten years ago when taking courses at George Mason University, and really loved it. I still love it.]

[It covers proofs from all basic 'pieces' of mathematics and gives the reader a good feel for the 'proofology', both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and abstract algebra courses. It's not perfect though.]

[The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.]

[Difficult but well worth it]

[I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores. I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar 'proofs and problem solving' style books.]

[The best reference for Proofs]

[This is an advanced book, with a lot of information on every page. I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about every major branch of mathematics.]

[It's highly recommended for anyone who is *serious* about mathematical proofs. Although the book is packed with material, it's a small book, so it's one of the first I choose to take with me when I travel.]

[pretty hardcover]

[Used at University of Pennsylvania Math 202]

[they use it with - Howard Eves and Carroll Newsom - An Introduction to the Foundations and fundamental Concepts of Mathematics - Revised edition - Holt, Rinehart and Winston 1965
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and

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An Introduction to Mathematical Reasoning (Paperback) - Peter J. Eccles

[User-Friendly! Almost makes learning analysis fun!]

[If you are struggling with a first analysis course or any course that uses proofs, this is the book for you! It introduces basic analysis topics like logic, sets, and the real numbers. And it is written in almost plain english! Moreover, the author focuses on teaching proof writing.]

[Fabulous So Far]

[I'm at the end of my first discrete mathematics course and have struggled to find clear explainations of how to write a proof, meaning how to choose what method and how to choose what the next statement should be to lead to the desired conclusion. I'm only on chapter five and it is a breath of fresh air to read this. Rather than just showing the completed proof Eccles shows the "scratch" work that goes into making the proof, discusses the reasoning and alternative paths, and then has the final proof that is easily understood.]

[For a student who is just learning mathematical proofs, this book is just horrible. The examples are awful and the author shortcuts many proofs. For example only part of a proof is proven. Not only that, when giving the answer to a problem, instead of writing out the reason to why, it's just a one worded sentence. I'm in a class with about 20 students and we all agree this is probably one of the worst mathematical reasoning book out there. We got more help from using online resources then the book. For someone out there who knows the material then this book is a good review but for people learning the material do not get this book.]

[Chris Gray approved]

[Logic/set theory based introduction to problem solving and proofs, with chapters on various techniques: induction, finite and infinite sets, counting, and number theory. My current fav.]
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hope you enjoy the notes, on the other two...

dkotschessaa

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.

dustbin

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.

homeomorphic

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.

miglo

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.

homeomorphic

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.


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.


Yeah, pretty much.

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.

@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.

RJinkies

- 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.

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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.

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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....

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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.

homeomorphic

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.

miglo

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.

dustbin

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...

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.)

micromass

That was fun!!! Everybody should definitely try this one.

Here's my solution:

Let

[tex]\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)}[/tex]

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

[tex]\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)}[/tex]

Rearranging gives us

[tex]\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)}[/tex]

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

[tex]\frac{a^3 - a(c^2+bc+b^2) +bc(c+b)}{a^2 - a(c+b)+ bc}[/tex]

Rearranging again gives us

[tex]\frac{a(a^2 - c^2)-bc(a-c)-b^2(a-c)}{a(a-c)- b(a-c)}[/tex]

Eliminating a-c and we get

[tex]\frac{a(a+c)-bc-b^2}{a- b}[/tex]

Rearranging again and we get

[tex]\frac{(a^2-b^2)+c(a-b)}{a- b}[/tex]

Eliminating a-b yields

[tex]a+b+c[/tex]

AnTiFreeze3

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.

RJinkies

looked at a little bit of the problem, the pattern is interesting:


ax+by+cz
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a+b+c


not making it messy, now that's a challenge...