Should I Become a Mathematician?

[Cod]

Well, there's no need to put it quite like that

What I meant was, if you think it's really worth it, then I guess I'll save up for Artin, and I'll just have to postpone on getting some other book.

In the mean time I'll take a look at your notes, thanks.

Keep an eye on the price of international editions. They're a lot cheaper and usually contain the same material as their US counterpart.

http://www.abebooks.com/servlet/SearchResults?isbn=9780132413770&sts=t&x=54&y=13

 

[qspeechc]

Keep an eye on the price of international editions. They're a lot cheaper and usually contain the same material as their US counterpart.

http://www.abebooks.com/servlet/SearchResults?isbn=9780132413770&sts=t&x=54&y=13

Yes, the problem is I'm not in the USA, so the price may be good, but the shipping is dreadful, the first few are all over $40 or $30 for shipping! And then there's import tax, duties, etc., which adds another 40% or thereabouts, so a $60 2nd-hand book (including shipping, I think) comes out at $84 etc.

Also, I've tried a few times to buy from bookseller in India, but they won't ship to where I am.

But thanks for the tip, I'll definitely keep my eye out for a good deal.

I'm in no rush anyway, there are many, many books I'd like to read, and maybe one day I'll get round to Artin (hopefully not too long from now).

 

[homeomorphic]

As for underlying theory, I really think it is just algebraic manipulations, like partial fractions or something.

Even I, Mr. Conceptual himself, would agree with that. However, it makes much more sense with complex numbers than real numbers. Maybe people avoid complex numbers because calc students aren't 100% comfortable with it.

Partial fractions just aren't that great of a thing.

I didn't really understand the algebraic tricks when I first saw it, so that was annoying. But after I figured out how to derive it myself, it was somewhat less annoying. It's kind of analogous to multiplying both sides of an equation by something. It just isn't anything to write home about. But it's also not something to get upset about, either.

I think the motivation in calculus is also to write things in a form where they can be integrated.

 

[synkk]

Does university reputation matter? I have an offer from a top 20 world university, and top 10 UK universities to study mathematics, but also one which is closer to home, but has less reputation? I'd prefer to go to the one with the lower reputation, as i'd like to stay at home, but I'm not sure if I should just suck it up and go to the one who should give me more career prospects. After graduating I plan on going onto actuarial, or investment banking jobs, or perhaps graduate work, if I'm good enough.

 

[Locrian]

With a very few exceptions at the very top or bottom, I would say university reputation does NOT matter going into actuarial work. Actuarial work is not like law where only going to the top few schools makes it worth the price.

 

[synkk]

With a very few exceptions at the very top or bottom, I would say university reputation does NOT matter going into actuarial work. Actuarial work is not like law where only going to the top few schools makes it worth the price.

Any ideas about investment banking? I've looked at the alumni of the less reputable school and it appears some people have gone onto investment banking, however it was from an economics degree. Though I do read around a lot on economics and I've interned at an investment bank, I don't think think I'd be able to do the degree (I'm not an essay person). Searching around, it does seem that investment banks do seem to go for target schools, however I'm not quite sure.

 

[DeadOriginal]

Any ideas about investment banking? I've looked at the alumni of the less reputable school and it appears some people have gone onto investment banking, however it was from an economics degree. Though I do read around a lot on economics and I've interned at an investment bank, I don't think think I'd be able to do the degree (I'm not an essay person). Searching around, it does seem that investment banks do seem to go for target schools, however I'm not quite sure.

If you don't graduate from a target school and you didn't have any outstanding internships that allowed you to network extensively your chances of going into investment banking are against you no matter what you studied.

On the other hand, I wouldn't do actuarial science to try to get into investment banking. Actuaries are focused in insurance. If you want to get into investment banking, the CFA exams will serve you better.

 

[synkk]

If you don't graduate from a target school and you didn't have any outstanding internships that allowed you to network extensively your chances of going into investment banking are against you no matter what you studied.

On the other hand, I wouldn't do actuarial science to try to get into investment banking. Actuaries are focused in insurance. If you want to get into investment banking, the CFA exams will serve you better.

I wasn't planning to do actuarial science to go into investment banking. Thank you.

 

[IGU]

qspeechc -

Jacobson's Basic Algebra I is available in a Dover edition. It's probably the level you're looking for and around $12 new at Amazon. Less dense than Lang, more extensive and a step up in depth from Herstein's Topics in Algebra.

It is somewhat dry, meaning you have to supply the enthusiasm.

-IGU-

 

[mathwonk]

qspeechc: To put it another way, recall the famous quote: 'when asked how he had managed to make such progress in mathematics despite his youth, Abel responded, “By studying the masters, not their pupils.” '

 

[dkotschessaa]

Well unmotivated because they seem to just come out of nowhere. The book I'm using says do this and this and you will get this. But I don't blame it, deriving it seems tricky-- you need a bunch of clever manipulations that aren't so straightforward.

This might make you hate them more, but whenever I get stuck on something like this I always like to know something of the history of it. The first known use of partial fraction decomposition was by Isaac Barrows, in his proof of the integral of the secant function: http://en.wikipedia.org/wiki/Partial_fractions_in_integration

Following on the advice of Mathwonk to make your own exercises, (way early in this thread) this is another place where this is useful. Take something like [5/(x+5)] * [8/(x^2+2)] or something like that. Multiply it all together, then try to decompose it again. Maybe integrate it before and again afterwards to show yourself how everything fits together. Then make more complicated problems.

If you're a real math geek this will actually start to become enjoyable...

-DaveK

 

[homeomorphic]

If you're a real math geek this will actually start to become enjoyable...

Perhaps, but I wouldn't want anyone to get the impression that you have to like that sort of thing to do math. It's much more interesting than that, thankfully. I'm sure there's a place in math for those who are thrilled by things like partial fractions. But there's a place for those who are not thrilled by them.

Partial fractions? Just learn them so you can get a good grade and be better at integration and then move on to better things. It would be much more interesting to design some Turing machines or figure out how to do some ruler and compass constructions. Something that has some intellectual content to it.

 

[dkotschessaa]

Homeomorphic, you are correct. That pretty much just came out wrong.

 

[mathwonk]

I agree with dkotschessaa that partial fractions is just a way of reversing adding fractions. it may seem more natural when you study complex analysis and poles and laurent expansions.

as a general rule, there is nothing at all that has no value and no interest, it is just being taught that way. I have a friend who is really really smart, and every time i say to him that something is rather boring or uninteresting, he ALWAYS says back: well what about this?... and it becomes fascinating...

 

[dkotschessaa]

I think one issue is that all these topics get thrown into textbooks and kind of whiz by kind of quickly (this is just the nature of the study I suppose) when really we don't get the story behind them. The truth is for every section of your calculus book there was likely a mathematician or two or more who spent serious time coming up with that particular technique or mathematical idea. There are people behind those ideas. This emphasis I find lacking. Maybe it's just me.

-DaveK

 

[Locrian]

Any ideas about investment banking?

Not really my thing. I can tell you that there isn't a single answer to the question you asked. What education you'll require will instead depend on what you want to do at the investment bank. Trader? Quant? Systems? Janitor? Different requirements.

 

[Nano-Passion]

Thanks everyone. :)

I think one issue is that all these topics get thrown into textbooks and kind of whiz by kind of quickly (this is just the nature of the study I suppose) when really we don't get the story behind them. The truth is for every section of your calculus book there was likely a mathematician or two or more who spent serious time coming up with that particular technique or mathematical idea. There are people behind those ideas. This emphasis I find lacking. Maybe it's just me.

-DaveK

That is one of the things that really irk me in our education system. The history gives so much motivation and context.

 

[dkotschessaa]

It's a state of affairs that isn't acceptable in the humanities but for some reason it is in the sciences. You just have to take it up on your own.

 

[qspeechc]

qspeechc -

Jacobson's Basic Algebra I is available in a Dover edition. It's probably the level you're looking for and around $12 new at Amazon. Less dense than Lang, more extensive and a step up in depth from Herstein's Topics in Algebra.

It is somewhat dry, meaning you have to supply the enthusiasm.

-IGU-

Great suggestion, seems perfect, thanks

qspeechc: To put it another way, recall the famous quote: 'when asked how he had managed to make such progress in mathematics despite his youth, Abel responded, “By studying the masters, not their pupils.” '

Yes, a good education lasts all your life, so I suppose I will get Artin's book, and Jacobson's too. And then Lang. Whew, mathematics is a slog! (But a good slog!)

 

[Nano-Passion]

It's a state of affairs that isn't acceptable in the humanities but for some reason it is in the sciences. You just have to take it up on your own.

You mean it is a state of affair (including a bit of history and context into the curriculum) is acceptable in the humanities but for some reason it isn't in the sciences?

 

[Mépris]

I still find it somewhat distressing that there are so many gaps in my pre-calculus background, which is something I expressed here a few months back. I have other commitments, so have not been able to fully immerse myself in that. The truth though, is that I don't particularly find learning (high school) algebra interesting. Sure, there are some parts I am intrigued about but the very thought of many of the things, such as being able to show that "if p and p + 2 with p ≥ 5, are both primes then the number p + 1 is always divisible by 6", leaves me unmotivated and a to a greater extent, frustrated. Looking forward, all I see is a series of hoops called "Pre-calculus", "Single variable calculus", "multi-variable calculus", "Linear algebra", etc...

I understand there are few things with my algebra that do need taken care of, but I figure I can take care of those loose ends as I move forward. It feels more tedious than actually fun. Perhaps it's the "collecting and not reading books syndrome", where people feel guilty about not reading books they were supposed and end up just collecting them. At any rate, I find probability and differential equations (very, very basic stuff, such as simple y = kx models for increase/decrease in number of bacteria or fish, but I know there's more cool things to be done with those) quite interesting - not to mention fun - and would rather learn those more thoroughly, so that I can start learning from a proper intro physics text and perhaps get into some more applied math.

I also recalled a post you (mathwonk) made, where you said that studying from a book should be done with the aim of learning something from it, not necessarily reading it line by line. (I'm paraphrasing here...) Can that be applied here? I really would rather just get ahead but, if that wasn't obvious enough, I'm at a loss here.

Another question. I noticed that MATH 25 and 55 at Harvard, Rudin's text is used. http://www.math.harvard.edu/pamphlets/freshmenguide.html seems to suggest that for MATH 25, the students could do just fine even if they've had a rather limited exposure to both subjects. Is this not somewhat premature? I thought learning from Rudin's book was usually after one had studied proof-based calculus courses, say both volumes of Apostol.

 

[Nano-Passion]

At any rate, I find probability and differential equations (very, very basic stuff, such as simple y = kx models for increase/decrease in number of bacteria or fish, but I know there's more cool things to be done with those) quite interesting - not to mention fun - and would rather learn those more thoroughly, so that I can start learning from a proper intro physics text and perhaps get into some more applied math.

I find that studying differential equations is much more fun than diving straight into learning how to find every derivative and integral of elementary functions. I feel very unmotivated when I finish one section of integration and the next one is just the same, akin to "you've learned how to integrate this type of function, now learn how to integrate that type of function." It is too computation based--something that I can leave for computers to take care of.

I find it much more interesting to study differential equations, and referring back and forth when required to differentiate or integrate x, y, or z function. And in fact, you feel much more motivated once you are in that context. Is everyone under the belief that learning everything in a linear manner is the best way? Because I certainly don't. And I find that it takes all the fun out of everything. I hope more people will realize it and things change.

 

[mathwonk]

In past years, math 55 at harvard has used a variety of books, all aimed at someone who has already had preparation comparable to apostol. they have used apostol's mathematical analysis, dieudonne's foundations of modern analysis, flemings functions of several variables, loomis and sternberg's advanced calculus, and notes by Wilfried Schmid. All of those books are more high powered and I think better than Rudin.

Most upper level courses at harvard can be very very advanced, and math 55 is one of the most ridiculously hard courses in the country.Sure read whatever interests you and use that as motivation to go back and learn more elementary stuff when you need it.

 

[qspeechc]

This might be interesting for undegrads looking to go to grad school. It's the topics Columbia grad school expects all entering students to know:
http://www.math.columbia.edu/programs/main/graduate/gradknowledge.html

 

[mathwonk]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

 

[SophusLies]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

This is excellent advice. When I was an undergrad I bit off way more than I could chew multiple times. There were a couple of quarters that I took 8-9 math/physics courses. I survived but at the time I thought I was learning more because I was covering all the bases. This couldn't have been farther from the truth because I was just learning everything in a trivial way.

Now in grad school I'm almost done with all my classes. Grad classes are much harder but you don't take as many each term and the ideas are fully developed. I feel I'm learning more than I ever thought possible recently because I can focus entirely on a smaller amount of material than a surface scratch of a whole bunch of subjects.

 

[dkotschessaa]

I know this is absurdly hard to do, but still I assure you it is worth it, to try consistently just to understand a small amount of mathematics, i.e. one idea at a time, really well. Do not make it a goal to read a whole book. That is ok, but the point is to learn one idea at a time. I speak from experience.

That's what I try to do in my own time, but unfortunately undergraduate classes are structured to give you as many different topics as possible in the shortest possible time. I feel like I'm always saying "Wow, that's a really cool idea...wish I had time to understand it."

 

[homeomorphic]

That's what I try to do in my own time, but unfortunately undergraduate classes are structured to give you as many different topics as possible in the shortest possible time. I feel like I'm always saying "Wow, that's a really cool idea...wish I had time to understand it."

Unfortunately, so are many graduate level classes, so I felt the same way in grad school. Good thing I'm done with classes.

 

[mathwonk]

then try it after the class is over. take just one theorem from the class and really try to understand it. eventually you will have a few key ideas that you really understand, and everything else will seem like a simple corollary of those. e.g. after decades of teaching studying and writing about it, I can say that all of the structure theory of an advanced linear algebra class, jordan form, rational canonical form, and so on, is a simple consequence of the euclidean algorithm. So if you want to understand the structure of finitely generated modules over Euclidean domains and then pid's, first learn well the euclidean algorithm. then see if you can understand why this is all there is at work in those other theories.

for non commutative algebra, a basic idea is a group acting on a set.

for commutative ring theory, a fundamental result seems to be the noether normalization lemma.

in manifold theory, the basic theorem is the inverse function theorem, and then the implicit function theorem. In many situations, a key result is green's theorem, and then its generalizations, the general stokes theorem.

 

[mathwonk]

On the theory that you can do more math if you live longer i feel this link is relevant.

http://www.nytimes.com/2012/04/15/h...tied-good-habits-to-longevity-dies-at-97.html

Basically a public health professor proved long ago statistically that you live about 10 years longer if you:

"do not smoke; drink in moderation; sleep seven to eight hours; exercise at least moderately; eat regular meals; maintain a moderate weight; eat breakfast.”Now that's not so hard.

Moreover a person who does fewer than three of these is only as healthy at 30, as someone who does at least 6 of them is at 60.

 

[Cod]

Does anyone have any topic ideas for a numerical analysis project? Instead of a final exam in my numerical analysis course, my instructor wants us to put together a formal proposal and provide a written professional report or paper. Other than that, my instructor provided the following guidance:

You proposal should include a description of the problem and your approach to solving it. For example a project could be to construct a general text classifier and the approach to the solution could be Bayesian statistics or least squares with the text turned into vectors.

I'd like to do something as it applies to computer science (my major) or baseball (my hobby). I've found a few papers online dealing with subjects of interests, but mostly at a graduate and beyond level. Any ideas, guidance, etc. is greatly appreciated.

 

[dkotschessaa]

And get married: http://wellbeingwire.meyouhealth.com/physical-health/married-men-live-longer-than-bachelors-study-says/

Having said that, it's 9:30, and my wife and I are going to bed early, so we can get up and have breakfast, for I have a Calc III test...

Good night!

-DaveK

 

[Plaristotle]

How important is it for math majors to have experience with programming? I'm able to take only one more first-year course and I'm considering taking an introductory course on programming (using Python). Specifically I'm interested in pure mathematics and I'd eventually like to go to grad school, if that helps. Any advice?

 

[chiro]

How important is it for math majors to have experience with programming? I'm able to take only one more first-year course and I'm considering taking an introductory course on programming (using Python). Specifically I'm interested in pure mathematics and I'd eventually like to go to grad school, if that helps. Any advice?

You should have done an introductory subject by the end of your undergraduate and along the way you will probably do stuff in something like MATLAB, Maple, R, SAS, Excel, or something else.

Even with writing papers it can be handy to know simple constructs if you have to do fancy stuff for generating figures and so on.

One other thing is that if you want to test statistical theorems, then it's a really good idea to have some programming experience to test your ideas before going the whole nine yards and proving something. This happens a lot in mathematics and unfortunately lots of people don't see the entire plot, but the climax so to speak.

Also one thing to remember is that if you can't get a job in pure math but there is stuff available that is applied where you work on a computer, produce models, run simulations and write reports or give advice, then this is a good thing to have under your belt in comparison to if you had no idea what a for loop is.

 

[dkotschessaa]

Does anyone have any topic ideas for a numerical analysis project? Instead of a final exam in my numerical analysis course, my instructor wants us to put together a formal proposal and provide a written professional report or paper. Other than that, my instructor provided the following guidance:

You proposal should include a description of the problem and your approach to solving it. For example a project could be to construct a general text classifier and the approach to the solution could be Bayesian statistics or least squares with the text turned into vectors.

I'd like to do something as it applies to computer science (my major) or baseball (my hobby). I've found a few papers online dealing with subjects of interests, but mostly at a graduate and beyond level. Any ideas, guidance, etc. is greatly appreciated.

Merely expressing my jealousy. I wish we had this option!