Should I Become a Mathematician?

[mathwonk]

I agree with most of what the previous poster said except the somewhat cynical tone. Also i would suggest trying the Putnam just for fun and education. And I am a professional mathematician.

 

[Alpharup]

I recently bought Apostle calculus for self study...It is much cheaper than Spivak in India...I love Apostle's calculus and it is very thought provoking...I have read a few pages and the way the subject is presented is great...The use of inequalities and method of contradiction, induction for proofs is much logical...I have never known how simple axioms can be used to prove many results...But it is a little bit time consuming...For undertstanding a single result, it takes many strategies like linking many axioms, using comparisons, etc...All these are little difficult for beginner like me...So please suggest some simple strategies for undertstanding the mathematics of Apostol in a much easier way by and in much lesser time...
(Note; Iam in vacation and after my college is opened, I won't have time...So, I want to cover as much material as possible within short duration)

 

[IGU]

So please suggest some simple strategies for undertstanding the mathematics of Apostol in a much easier way by and in much lesser time...

This stuff is hard. What you're doing is learning a new way of thinking. Apostol will give you the best kind of start, but I don't think you'll find a way to make it easy and quick. Even just doing a couple of chapters, working the hard problems (not just the ones that are for practice), will give you a big advantage going into an engineering calculus class. You'll notice when they're not being rigorous (this proof is beyond the scope of this book, or we'll assume this lemma), and you'll feel more in control.

It's somewhat of a cliche, but the more you put into it (the harder you work), the more you'll get out of it. And once you work through some Apostol, the class you take will probably seem easy in comparison.

-IGU-

 

[mathwonk]

the time cannot be decreased. the point is to try to realize how much you are learning in a few pages of apostol. i.e. time spent on apostol expands. a few pages will last you a long time and take you a long way.

 

[Alpharup]

I don't know why Apostol like books can't be used for engineering mathematics..
I compared the topics covered in Engineering Mathematics Textbook(Erwin Kreyszig) and Apostol and found that they almost match in topics. Moreover, The engineering mathematics is not so rigorous in the approach. What I feel is that lack of rigour discourages mathematical learning. There should be continuity in ideas. I feel that Apostol gives the continuity of ideas. After reading a few pages, I got immersed in it and I didn't refer any other textbook. I think it is more self contained in concepts. On the contrary, when I read engineering mathematics, there is a need to refer some other book for results, proofs, etc..Many tough proofs are omitted and it irritates a lot. Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

 

[IGU]

Please do comment on the idea whether Apostol like textbooks can be used for Engineering mathematics.

Obviously you can use Apostol, but for most engineering students the proofs are uninteresting and irrelevant, taking time away from practicing usage of the new mathematical tool. Apostol teaches math, not engineering. And he created and refined his books while teaching the material to Caltech freshmen and sophomores with no calculus background, who were much more about science than engineering (still applied math, but a little less so).

As I wrote earlier, you will benefit from learning calculus from Apostol. It will give you an advantage over your peers, engineering students who don't understand the math as well. So go for it! Just make sure you don't neglect practicing the application of what you learn to real-world problems.

-IGU-

 

[QuantumP7]

I became fully deaf about a year and a half ago. I've always had problems with my hearing and severe depression, so no degree yet. I've been studying finance so that I can try to make some money and get some cochlear implants (Medicaid in my state doesn't pay for it), and get off of SSI. I REALLY miss studying pure math, though. *sighs*

 

[mathwonk]

I have learned what I know of calculus by teaching it from several different books, learning something different from each one.

They include Spivak, Courant, Kitchen, Apostol, Thomas (an older edition), Cruse and Granberg, Edwards and Penney (several editions), Fleming, Loomis-Sternberg, Bers, Sylvanus P. Thompson, Stewart, Lang, ...

 

[nitro_gif]

Ok, got a few books on the go right now, in particular Lang's Basic Mathematics.

I like the content, but how can I retain and absorb more information? I feel like I read stuff but don't retain what I should, so I reread it again and still don't retain enough.

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

 

[epenguin]

Is it worth writing notes from the text as you are reading?

Maybe it is worth making notes after reading and then find out if you know what you have read.

(Do what I say not what I do. )

 

[dkotschessaa]

When reading a math text, how does one approach it from an active standpoint rather than a passive standpoint?

Is it worth writing notes from the text as you are reading?

Yes, you pretty much have to. Except perhaps for some exceptional people, if you're not at least doing some pencil and paper work while reading, you're not really going to learn much.

Math textbooks are dense and leave a lot of stuff out, intentionally. Proofs in particular, with good reason, do not show all the "background" steps involved in getting from point A to point B. So you need to fill in those blanks, and you need to "convince yourself" that the things the books is saying are true.

If something is abstract, you may need to scratch out some concrete examples. For example, if you were reading an algebra text that tells you that a^xa^y=a^(x+y) then you'd want to plug some numbers in there to see that it "works."

-Dave K

 

[mathwonk]

one of my best math teachers, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

 

[dkotschessaa]

my best math teacher, the great maurice auslander, said if you are not writing 5 pages for every page you read you are not learning anything.

Fantastic!

I've discovered the joy of the whiteboard now. I have a standard one, plus sticky-whiteboard sheets plastered all over my office wall. I am enjoying the hours of lively activity, working out examples, proving theorems, writing definitions until I know them from memory, ironing out all the details and just generally mathematically playing around. I've found it is better for someone as hyper as me, rather than trying to sit still, hunched over a desk. I'm learning quite a bit this way.

-Dave K

 

[nitro_gif]

Fantastic!

I've discovered the joy of the whiteboard now. I have a standard one, plus sticky-whiteboard sheets plastered all over my office wall. I am enjoying the hours of lively activity, working out examples, proving theorems, writing definitions until I know them from memory, ironing out all the details and just generally mathematically playing around. I've found it is better for someone as hyper as me, rather than trying to sit still, hunched over a desk. I'm learning quite a bit this way.

-Dave K

I have considered getting a white board. Sitting is no fun to me.

 

[Mandelbroth]

I suppose this is the place to ask this.

I'm just entering my junior year of high school. I like to consider myself a "mathematician," though I don't really do it professionally.

I like to consider myself as talented, though this is really a biased opinion. I'm really a pure math person, but I've been interested in application to medicine for a long time now. However, I've recently been reading some papers on applied math, and I'm having trouble dealing with the estimations and approximations. My pure math background is much stronger than my applied math background. I've tried treating $\approx$ like an equivalence relation, but I have issues with its transitivity. Even then, I see a lot of what I call "abuse of equality."

Do any mathematicians have advice for how to jump that hurdle? I want to go into applied math, but I have no idea how to get passed this. Or, if this continues to bother me as much as it does, should I even go into applied math?

I have considered getting a white board. Sitting is no fun to me.

I got a whiteboard.

Best. Christmas. Ever.

 

[Dens]

Whiteboards are great. You start to write and "create" things you would never do with pencil and paper. I have experienced the amount of creativity output with it.

 

[dkotschessaa]

I suppose this is the place to ask this.

I'm just entering my junior year of high school. I like to consider myself a "mathematician," though I don't really do it professionally.

I like to consider myself as talented, though this is really a biased opinion. I'm really a pure math person, but I've been interested in application to medicine for a long time now. However, I've recently been reading some papers on applied math, and I'm having trouble dealing with the estimations and approximations. My pure math background is much stronger than my applied math background. I've tried treating $\approx$ like an equivalence relation, but I have issues with its transitivity. Even then, I see a lot of what I call "abuse of equality."

Do any mathematicians have advice for how to jump that hurdle? I want to go into applied math, but I have no idea how to get passed this. Or, if this continues to bother me as much as it does, should I even go into applied math?

I think just a better understanding of what approximation means would be useful. You don't need to be deciding about pure vs. applied math yet.

Any application of mathematics (except perhaps in some areas of computer science) will require some approximation. To make yourself more comfortable with that, you should understand that different applications of mathematics will require different degrees of approximation. A good engineer or applied mathematician will now what degree of precision is needed for the task at hand. If you're measuring the radius of a planet it is ok to be off by a few feet (or maybe a hundred or a thousand for all I know). If you're putting a pond in your backyard then "3" is probably as good as "pi". There is just no such thing as an exact answer in the "real world."

Also keep in mind that you will deal with approximations in pure math as well. 3.14159 is an approximation of pi. 1.4142 is an approximation of the square root of 2. This is true no matter how many digits There's an entire field of mathematics called approximation theory which can still be considered pure mathematics.

-Dave K

 

[Mandelbroth]

I think just a better understanding of what approximation means would be useful. You don't need to be deciding about pure vs. applied math yet.

I agree with this. I like to mentally chew on ideas for long periods of time, though, so thinking about what to do long before is helpful for me.

Any application of mathematics (except perhaps in some areas of computer science) will require some approximation. To make yourself more comfortable with that, you should understand that different applications of mathematics will require different degrees of approximation. A good engineer or applied mathematician will now what degree of precision is needed for the task at hand. If you're measuring the radius of a planet it is ok to be off by a few feet (or maybe a hundred or a thousand for all I know). If you're putting a pond in your backyard then "3" is probably as good as "pi". There is just no such thing as an exact answer in the "real world."

I understand that numerical answers are important, but if you give me something like $\displaystyle \int\limits_{(-\infty,+\infty)}e^{-x^2}~dx=\sqrt{\pi}$, the LHS and RHS are both cool. However, the fact that they are equal interests me. I think equality is the most beautiful part of that expression, and indeed in most of mathematics. I feel like by approximating things like $n!\approx \sqrt{2\pi n}(\frac{n}{e})^n$, we lose a lot of that beauty, which we could have left more precisely with $\lim_{n\rightarrow +\infty}\frac{n!}{\sqrt{2\pi n}(\frac{n}{e})^n}=1$.

Also keep in mind that you will deal with approximations in pure math as well. 3.14159 is an approximation of pi. 1.4142 is an approximation of the square root of 2. This is true no matter how many digits There's an entire field of mathematics called approximation theory which can still be considered pure mathematics.

I always thought approximation theory was more to do with series expansions and not approximating constants, but I'm sure there's something in there.

Thank you for your response. I appreciate your input.

 

[WannabeNewton]

Taylor expand and drop all 2nd order and higher terms :)

 

[Dens]

Hey mathwonk, does it look bad if you take an art class for your final year even though you have maxed out all the art credits you need? Going into pure math, but only a stats class is available for the taking and I am not particularly interested in statistics.

Do you also think it is necessary to explain one W and a "bad" grade ("bad" = A-, I also think my transcript will show class averages.)? Particularly, if the course has some relevance to what you are doing?

Thanks

 

[mathwonk]

don't sweat it. math is art, right?

 

[Dens]

Even if it is a language class? How does the committee even view stat classes? Especially when my school is lacking pure math classes?

 

[mathwonk]

I never worry about this sort of thing. Being able to read a foreign language is very useful for a mathematician. And it is a lot easier to earn a living in stat than in math. I am not a good person to answer these sorts of questions. I care about the subject, not the perception of it by committees, and I believe committees also are best approached just by being well qualified and not worrying about how your record "looks". Can you hold an intelligent conversation about math?

 

[dkotschessaa]

Even if it is a language class?

The community of mathematicians is small (compared to other fields) and very internationally diverse. I think having language skills endears you to this community in a very positive way.

 

[Seydlitz]

I was reading this thread from page 170 when I noticed intelligence trick involving integrating $\ln x$ without integrating by parts. Anyone recall the fact? I have already closed the page before noting it.

 

[muzak]

Hello,

I will be applying to graduate school soon and have no real idea of where to apply. I was wondering if any of you know of any schools geared towards the pure end of mathematics, primarily real analysis and functional analysis and/or variations of the two, etc. I've looked into a few, but I was hoping to get a more general sort of list due to many of you guys that are probably more aware of groups involved in these fields. I'm not looking for top 20 or anything, just somewhere that is relaxed enough taking in an average student with no math research in pure or otherwise.

I've taken Introductory Real Analysis I and II and Topology, enjoyed the former more than the latter (due to the fast pace and algebraic part, was a bit too advanced for me at the time) and have been exposed to oh so rudimentary levels of functional analysis. I just really enjoyed the "building up from foundations" aspect of analysis and the elegant proofs that I understood and was able to follow. I'd appreciate any ideas you guys might have.

 

[Shivam3013]

Hello; if you don't mind, can you (mathwonk) please message me your email?

 

[mathwonk]

please post a specific question here, for best results

 

[NATURE.M]

I was kinda wondering if I completed my undergrad in physics, could I still possibly go to graduate school in mathematics. Note, I will be taking all the fundamental math courses (real analysis, topology, complex variables, etc.).

 

[mathwonk]

yes. just read the requirements for admission to a grad school in math. i suspect you will never find a requirement that your undergrad degree is in math. i believe Ed Witten majored in history at brandeis. he then apparently enrolled in grad school in first economics, then applied math, then graduated in physics. then he received a fields medal in pure math essentially.

 

[Crake]

yes. just read the requirements for admission to a grad school in math. i suspect you will never fins the requirement that your undergrad degree is in math. i believe Ed Witten majored in history at brandeis. he then apparently enrolled in grad school in first economics, then applied math, then graduated in physics. then he received a fields medal in pure math essentially.

I'll never understand how a person like Ed Witten majored in history.

 

[dkotschessaa]

I'll never understand how a person like Ed Witten majored in history.

I do. I think the sooner we realize there is no formula for greatness the sooner we can stop questioning whether we are doing the right thing and just get on with it.


-Dave K

 

[epenguin]

I'll never understand how a person like Ed Witten majored in history.

He swotted hard enough to scrape through the exams.

 

[Alpharup]

can learning topology help me to design electric and electronic circuits better? I have not finished analysis... But if topology helps me in some way to design efficient systems, then I could self study both analysis and topology in these four years of my electronics engineering...

 

[Cod]

I'm having trouble understanding how to apply specific topics to specific events. For example, I enjoy solving systems of linear equations, matrix operations, and the like; however, I have no idea how this knowledge can translate to a research topic, job, etc.. Basically, I understand the application portion when I'm looking at textbook examples, but cannot seem to come up with my own applications.

Bottom line is, I really enjoy linear algebra and numerical analysis, but have little idea how to use these outside of the popular applications (cryptography, computational fluid dynamics, etc.).

Any thoughts are greatly appreciated.