Should I Become a Mathematician?

[courtrigrad]

If one had the option to take either abstract algebra or real analysis, which one should he take? Also, mathwonk what would you recommend a freshman math major do over the summer?thanks

 

[JasonRox]

If one had the option to take either abstract algebra or real analysis, which one should he take? Also, mathwonk what would you recommend a freshman math major do over the summer?


thanks

That's like asking should I take Physics or Psychology.

I would simply go with which ever is of interest to you. For myself, I found both to be interesting. If you really liked Calculus, Analysis would be something to seriously consider.

 

[mathwonk]

as jason implied, both topics are important so you might want to pick based on which has the better teacher, or which you enjoy more.over the summer also you need to do what you want to, but if you want more math, there are vigre, REU, and other summer research experiences. also there is summer school, but you might want to go to the mountains and store up natural energy for the school year.

 

[balletomane]

I've found this thread extremely helpful. Thanks. :)

My question is a little strange: how can I know if I've learned something well enough and how can I improve my retention?

For example, I took calculus in high school and did well. Less than two years later I wouldn't be able to set up a double integral without looking in a book first. How can I keep this from happening, given that I have limited time to review old material.

Also, I am taking abstract linear algebra (we're using Axler) and finding it very difficult (in particular because I haven't had linear algebra before). Is it just me or is the learning curve really steep?

 

[mathwonk]

the learning curve in math is always steep, as the way of thinking and the objects of thought are not found in real life.

forgetting time is very short for this reason, and so even specialists like me forget their own research in 3-6 months.

so always review and think about the topic. try to get beyond reading and on to doing. the best way to review is without the book, trying to reproduce previously learned material. you will find you can reconstruct a good bit of it, the rest you need to review. anything you learn well the first time never needs to be relearned, but this reveals that much material is not well learned the first time.

hang in there, what other fools have done, you can do, as the great sylvanus p. thompson said.

 

[domhal]

Interesting reading. I too had been wandering recently about retention of details from courses past - in particular how it seemed like I had forgotten too much!

As an example I was looking over some analysis recently (I took analysis most recently in the Spring of this year) and had to show that the set of all limit points of E is closed. Feels like it took far too long to do!

Nice suggestion Mathwonk, I'll see how that goes.

 

[balletomane]

Thanks so much, Mathwonk.

 

[mathwonk]

one thing i do myself, is i always try to reproduce material without, looking at any books or notes. that's how i see how much i have understood. this year i am teaching grad algebra and i try every day to write down the lecture without consulting any book or even my own notes from the previous course. i do not always succeed and when i do not, i make a point of noting what it was i overlooked.

 

[courtrigrad]

so basically if one can write down what he remembers perfectly, he is an expert in that area?

 

[CPL.Luke]

^ pretty much, if you are able to reproduce all material on command perfectly without any reference, then you are obviously very knowledgeable on the subject. And if you are very knowledgeable about a particular subject, then doesn't that make you an expert?

 

[mathwonk]

i am not writing out the material entirely from memory, but reworking it, as I only remember the details vaguely.

As I rework it and rethink it I begin to understand it better, and add more and more new insights every year, than I had before. it should keep getting easier as time goes by. It should boil down to a few basic precepts, and not remain a long list of facts to remember.

One should also try to solve problems, using the ideas, and to think of ones own proofs for the theorems, simplifying and improving the ones one may have seen before.

 

[JasonRox]

One should also try to ,solve problems, using the ideas, and to think of ones own proofs for the theorems, simplifying and improving the ones one may have seen before.

Yeah, always a good idea if possible. Some proofs are just like the way it is.

I tend to attack proofs from time to time, which results with my own proof that I enjoy better.

 

[mathwonk]

here is a tiny example: i always liked group actions in terms of orbits and stabilizers, as very visual.

But i disliked cycle notation for permutations, as overly compoutational.

now i realize a cycle is just an ordered orbit, and i like them much better, and can also see why certain little conjugation formulas hold.

e.g. if s is a permutation then a standard useful algebraic fact is that

if (123...k) is a k cycle, then s(123...k)s^-1 = (s(1)s(2)...s(k)).

in orbit terms this just says if a certain element takes a to b, then if i conjugate it with an element s, the result takes s(a) to a to b to s(b), i.e. takes s(a) to s(b).

this can be visualized. i.e. if i rotate one vertex to another then fix that one, then rotate back, it is the same as fixing the original vertex. so the conjugate by s, of a rotation fixing vertex a, fixes vertex s(a).well anyway, i guess you need to find your own way here.

 

[mathwonk]

heres a deeper one: the hard part of the proof of the inverse function theorem is that a smooth map taking 0 to 0, and having derivative equal to the identity at 0, maps some open nbhd of 0 onto an open nbhd of, 0.

i finally realized that the linear approximation definition of derivative, guarantees that the original map is homotopic to its derivative on some nbhd of 0, hence wraps the unit sphere the same number of times around 0, qed.

the proof still takes work, but this is it in a nutshell, conceptually.

 

[mathwonk]

the point is that i am always turning every concept over and over in my mind, trying to make it my own. I want to live there, to see the objects, and not have to depend on some memorized argument to understand them.

i hope this comes across in all my explanation here. i am never parroting some learned formulas, unless that all i have to offer. then i say so. for this reason for a long time some people failed to understand that my explanations of strange objects, were even about the same objects they had memorized versions of.

to a mathematician the objects are real, not dependent on some book learned representation of them.

 

[AlbertEinstein]

I haven't gone through all those 18 pages, yet I have a question: does speed of problem solving a necessary quality of a mathematician?

 

[mathwonk]

no. depth matters, and creativity matters. not speed. unless you compete with faster people for the same results. if so, do not tell them how far you have gotten unil you are finished.

of course you have to be fast enough to finish before you die. that's about it.

 

[JasonRox]

of course you have to be fast enough to finish before you die. that's about it.

I'm certain that if I'm given enough time I can prove Riemann's Hypothesis. I'm just not fast enough!

 

[courtrigrad]

Can Spivak's book (Calculus) be used as an intro to analysis text (i.e. before a real analysis course?) How would you change or edit the following curriculum:

Calculus 2
Calculus 3
Linear Algebra And Differential Equations
Computer Science
Intro to Analysis (Spivak Calculus)
Real Analysis
etc..Also what are your opinions about Real Mathematical Analysis by Charles Chapman Pugh?

Thanks

 

[Ki Man]

Any good math books for High School students to read?

I'm sure those calculus boks you recomended are great but I'm pretty sure they will be a tad too advanced

 

[mathwonk]

spivak wrote his book for college students who were very bright but had no calculus, so it could precede calc 1 and 2, but maybe a course in calc from say thomas would be wise.

a good book that anyone can read in high school, is "calculus made easy", by sylvanus p. thompson, about 100 years old.

it was a book studied in high school by one of my friends when we took the spivak course as freshmen in college.

this thread is so long no one can be expected read it all but i think i have already given recommendations for high school, junior high, etc. let me try to find them.

 

[mathwonk]

yes, they are in post #8, page one of this thread.

 

[mathwonk]

a high school student can and should try to read anything he/she likes. actually high school students may be brighter than college students.

and often more motivated. so plunge right in.

i read principles of mathematics in high school, and it had lots of great book references at the ends of the chapters. then i went to the college library and looked up those books

i still remember sitting in the stacks puzzling over the proof that there exist an infinite number of prime integers.

let p1,...,pn be any finite set of primes. and then consider the integer

N = 1+p1p2...pn, 1 plus the product of all the primes pi.

we claim no pi divides N, because if say p1 did divide N, then since

1 = N - p1p2...pn, p1 would also divide 1, which is false.so none of the primes pi divide N. But N is larger than 1, so we claim some prime must divide N. I.e. among all divisors of N greater than one, there is a smallest one say q.

then q cannot have any factors larger than 1, or they would be smaller factors of N.

so q is a prime factor of N, but q cannot equal any of the pi. so the finite list p1,...pn, is not the full list of all primes.

hence there is an infinite number of primes.

 

[Ki Man]

actually high school studnets ARE BRIGHTER THAN COLLEGE STUDENTS. (i think i was brightest when i waS ABoUT 15, at least based on my IQ scores.)

thats what i was thinking too =P

I need stuff that would be available at a local library... would those be on their shelves

 

[mathwonk]

public libraries do not have good math books in my experience, but maybe that's because i live in the south. you can look. but most people can gain access to a university library somehow. and many books are available free, like my graduate algebra text, which is accesible to anyoe with enough patience and who knows something about matrices. so maybe the place to start is learning matrices. there should be books on that available most places. again my book on my website is free, but very concise. there are many free books on linear algebra on the web. ill send you some adresses if you cannot find them.

 

[Ki Man]

I live in the west, near disneyland. there's lots of universities and bookstores around here. I probably live within a half hour drive of at least 10 state colleges and universities and caltech is around 40 minutes away or less

i guess for now i could ask my math teachers to let me borrow books but maybe next year i'll start looking into more sources of books

E: what's 'your site'

 

[mathwonk]

heres my website:

http://www.math.uga.edu/~roy/

 

[mathwonk]

have you noticed? even though there are almost 14,000 hits to this thread it still is not a sticky?

 

[JasonRox]

It might even surpass the hits for Physicists!

 

[Ki Man]

maybe they kind of assume mathmaticians are a level of sub-physicists

physics is nothing without math, but math is still just math =]

My math teacher let me borrow a book called freakonomics bye Steven D Levitt and Stephen J Duber