Other Should I Become a Mathematician?

[dfff333]

Ehm, hi. I was going to post a new thread because there may be other questions that this doesn't directly jump at, but I decided to post in here.

Anyways, I should probably get this beginning part out of the way first, albeit this probably won't be that long, anyways : I'm only 14 years old. I have had a love for mathematics since my tutor first came. I am actually not home schooled, but I do get him weekly. Ever since then when I was 10 I have had a considerable passion for it.

I have read a lot of the posts here, but not all, so I'm sorry if I missed one where it addressed the topic of getting into college early.

First of all I should mention that I'm not asking about what my school laws are here for early admission into college, I am simply asking what books I should read, what references I should go to, and perhaps any techniques I could use to convince either the administrators, or the professor/professors themselves that I do actually have the capability and maturity (which includes social maturity and up-to-par etiquette) of entering college early.

I was looking for matrice theory and linear algebra references/books?*(1), and some books that might go a bit more in depths in the topics listed? : Differential equations, calculus, modular forms, elliptic curves. I can look them up online but I was just looking for some opinions on the actual works. Also, I tend to formulate ideas in my head when I work, and I noticed one thing, the half-derivative. Would there be any chapters of books that you could recommend on that?

My final question is the same as the last part of the former of the former paragraph (P. #4). Replies that are saying my option of doing this is not smart is fine too, I just really need some help or discussion about this.

Also, my grades are not that good. But I wouldn't necessarily blame that on my intellectual or academic abilities, merely because I am bored, and thus can't focus as hard as I normally would. Sounds like a poor excuse, but boredom really kills me. And it's not the math that is boring, it's the class, the teacher, and the disruptive environment.

Suggestions.
Comments.
Questions.
Critiques.
All of the above are welcome. Please.

*(1) : I saw mathwonk's link up there.

 

[mathwonk]

Unfortunately poor grades, regardless of the cause, will keep you out of the schools your talent may make suitable for you. So if you want to get into a college that is interesting to you, you will almost surely have to demonstrate an ability to excel in the school that currently bores you, or change schools and demonstrate it in a better one. I have actual experience with this. I mean a school like Harvard is going to have to decide which superbly qualified students to admit so they are not going to admit any questionable ones.

The early pages of this thread have many book recommendations. One way to test yourself is to read and work the problems in Spivak's calculus. In differential equations I recommend Martin Braun's, and on a higher level: Arno'ld's ODE.

It is hard to advise you without knowing more about you. What books have you mastered, found easy, or hard?

'There are also some good prep schools out there like Andover and Exeter. and good summer programs like Duke's TIP program.

 

[fora]

liberal arts is always evergreen and interesting...

 

[fora]

i know a site called www.liberalartscolleges.org... which gives u a idea for liberal arts colleges... i am sure it has maths related infomation...

 

[Robert1986]

here i
s the linear algebra book link:


http://www.math.uga.edu/%7Eroy/4050sum08.pdf

*GASP* just when I was starting to really enjoy your advice I learn that you are a UG[sic]A professor! I don't think I can take advice from you being a GaTech student.


Just kidding, I really enjoyed reading through this thread and getting advice from a pro.

 

[mathwonk]

i understand your horror.

but maybe you can go for help to some of my students who are ga tech profs, like ernie croot. http://www.math.gatech.edu/users/ecroot

there is actually quite a bit of interaction between the two schools.

matt baker is a former uga prof that tech hired away.
http://www.math.gatech.edu/users/mbaker

 

[Robert1986]

i understand your horror.

but maybe you can go for help to some of my students who are ga tech profs, like ernie croot. http://www.math.gatech.edu/users/ecroot

there is actually quite a bit of interaction between the two schools.

matt baker is a former uga prof that tech hired away.
http://www.math.gatech.edu/users/mbaker

I haven't had Prof. Croot, but he has filled in a few times for some of my profs and I like him. And his website is fun.


I haven't attended a lecture of Prof. Baker's per se, but I have a class that is next to his number theory class and starts an hour later and I sit outside to hear his lectures as they are very entertaining (he does magic sometimes!)

 

[mathwonk]

Those are both extremely bright guys and well thought of. here are some links about their actiities:

One recognizing Ernie for solving a famous problem while still a student.

http://www.uga.edu/columns/991011/campnews4.htmland one advertising a conference this spring with Matt as principal speaker.

http://www.math.uga.edu/~xander/bellairs2011.html (edit: By the way, I am also a magic amateur and Matt bought some of my best card magic books when I pruned my bookshelf a few years ago, but I have not seen him perform.)

 

[Robert1986]

Those are both extremely bright guys and well thought of. here are some links about their actiities:

One recognizing Ernie for solving a famous problem while still a student.

http://www.uga.edu/columns/991011/campnews4.html


and one advertising a conference this spring with Matt as principal speaker.

http://www.math.uga.edu/~xander/bellairs2011.html

I like Prof. Croot's hair in that picture; I didn't realize that Carl Pomerance was at UGA, either.

 

[mathwonk]

Carl was there for 20 or 30 years and he seems to be at dartmouth now:

http://www.math.uga.edu/dept_members/carl.html

 

[yik-boh]

Is it that hard to be a Fellow Actuary in US and also in my country (Philippines) (if you just know)?

 

[dkotschessaa]

Ok, this is going to seem like a dumb question. Do you go through a LOT of paper?

 

[QuantumP7]

Ok, this is going to seem like a dumb question. Do you go through a LOT of paper?

I do!

 

[mathwonk]

This is a totally unrelated remark.. But some people ask me to be "friends" and I almost never say yes. It is just because I have no clue what that means. I am a fairly private person and do not want to be notified when someone else posts a post or whatever. I apologize for what may come across as disrespect which is not at all my intention. I love you all, but i am a little reluctant to get on any social media level. I have no facebook presence and do not really understand current modes of communication. Bless you, and thank you for offering me this compliment. Please forgive me for my shyness.

 

[mathwonk]

wow. i just found the thread "math and science learning materials" on here:

https://www.physicsforums.com/showthread.php?t=174685
i downloaded elementary math lectures by lagrange and they look great, for high school math say. it is amazing how much one can learn even about elementary math from a great mathemtician.

e.g. in discussing elementary quafdratic equations, he points out that if one is given say the sum b of two numbers and also their product c, then one can rediscover the numbers if one knows their difference x.

but just knowing the usual trick about relating the square of a difference to the square of the sum, one has then x^2 + 4c = b^2, hence x = sqrt(b^2-4c).

this is equivalent to, but much simpler than the usual high school derivation of the quadratic formula by completing the square, and would be much easier to teach in some cases I wager. Of course here the letter b stands for minus the second coefficient in the quadratic equation, and x stands for the difference of the two roots.

so one gets (almost) the usual formula for the roots themselves by adding b to the formula above and dividing by 2: i.e. the roots are of form (1/2)(b ± sqrt(b^2-4c)).

that is so much simpler than the usual derivation. i know i had a lot of trouble in high school following the usual completion of the square argument in our book, (and i won the state algebra contest).

there are also free copies of vector analysis by josiah willard gibbs, higher mathematics for scientists and engineers by ivan sokolnikov, and euclid's geometry.

 

[QuantumP7]

So, I've become completely addicted to mathematics. I'm too broke to go to school now, so I've been studying math on my own. First, I was brushing up on math so that I could become a theoretical physicist. Then, I got sucked into the world of pure mathematics, as if pure mathematics was some kind of black hole. So there's no getting out of this.

I think that I'm so driven, maybe, because I didn't even know about pure math until last year, and I'm 28 now. I just feel like I have SO much catching up to do.

So, yeah. /mini blog... cause this thread was kinda dead.

 

[mathwonk]

yes indeed no posts for 16 days, so do you have a question? or else we die.

 

[QuantumP7]

I have been wondering if there is an Eastern counterpart to the Western Algebraic Geometry Seminar. Anyone know if there is?

 

[Chris11]

Honestly, the math that you do in physics is tedious; go into pure math.

 

[Nicholasng925]

Honestly, the math that you do in physics is tedious; go into pure math.

Yes, Maths in Physics is tedious! A careless mistake on the symbols or signs and you're doomed. I like Pure Maths (Calculus, Number Theory, Algebra) and Applied Maths (Statistics)! :D

 

[mathwonk]

pure math is my favorite, but physics and engineering are so well based in real life phenomena that they offer insight that pure mathematicians can only hope for. go to pure math if that is your love, but even so do not neglect the advantages that physics can provide. as a pure mathematician, i have always envied the intuition and seat of the pants knowledge that physicists have.

 

[snits]

A long time ago Mathwonk discussed the School Mathematics Study Group series of books. I managed to find a list of some of the books that were put out by SMSG on a webpage for the utexas archives for the SMSG. I thought it might be useful to have the list here if anyone was trying to track them down.

Edit: This series is a different series called the New Mathematical Library put out by the SMSG as supplemental texts for interested students. The different units for the textbooks are listed at http://www.lib.utexas.edu/taro/utcah/00284/cah-00284.html .

Numbers: Rational and Irrational, Ivan Niven
What is Calculus About?, W. W. Sawyer
An Introduction to Inequalities, E. F. Beckenbach and R. Bellman
Geometric Inequalities, N. D. Kazarinoff
The Contest Problem Book I: Annual High School Mathematics Examinations 1950-1960, compiled with solutions by Charles T. Salkind
The Lore of Large Numbers, P. J. Davis
Uses of Infinity, by Leo Zippin
Geometric Transformations I, I. M. Yaglom, translated by A. Shields
Continued Fractions, by Carl D. Olds
Graphs and Their Uses, Oystein Ore
Hungarian Problem Books I and II: Based on the Eötvös Competitions, 1894-1905 and 1906-1928, translated by E. Rapaport
Episodes from the Early History of Mathematics, A. Aaboe
Groups and Their Graphs, I. Grossman and W. Magnus
The Mathematics of Choice, Ivan Niven
From Pythagoras to Einstein, K. O. Friedrichs
The Contest Problem Book II: Annual High School Mathematics Examinations 1961-1965, compiled and with solutions by Charles T. Salkind
First Concepts of Topology, W. G. Chinn and N. E. Steenrod
Geometry Revisited, H. S. M. Coxeter and S. L. Greitzer
Invitation to Number Theory, Oystein Ore
Geometric Transformations II, I. M. Yaglom, translated by A. Shields
Elementary Cryptanalysis: A Mathematical Approach, A. Sinkow
Ingenuity in Mathematics, Ross Honsberger
Geometric Transformations III, I. M. Yaglom, translated by A. Shenitzer
The Contest Problem Book III: Annual High School Mathematics Examinations 1966-1972, compiled and with solutions by R. A. Artino, A. M. Gaglione and N. Shell
Mathematical Methods in Science, George Polya
International Mathematics Olympiads: 1959-1977, compiled and with solutions by S. L. Greitzer
The Mathematics of Games and Gambling, Edward W. Packel
The Contest Problem Book IV: Annual high School Mathematics Examinations 1973-1982, compiled and with solutions by R. A. Artino, A. M. Gaglione and N. Shell
The Role of Mathematics in Science, by M. M. Schiffer and L. Bowden

 

[Dougggggg]

So I actually am in a course that is my first real test of pure math. It is basically an intro to things like sets, proofs, and logic. I was wondering, what are some interesting areas of research within pure math? I'm still young in my degree, but having time to learn some of them would be of great benifit in helping me prepare for deciding a graduate program when I get to that point.

 

[wisvuze]

Hey, I would like some clarification about going into mathematics (pure mathematics ) and getting a master's degree? I've heard that you should go straight into a PhD program after your Bachelor's, but I'm not sure if I will have the GPA for that. I'm in my first year as an undergraduate, going into my second year; but there's one CS course I'm taking I think I'll do really poorly on (50-60s, or fail depending on the exam). I have 80s and up for my math courses though.
I don't get why getting a masters first is a bad thing though? Wouldn't you get more experience and mathematical maturity? thank you
I'm not in the states though, I'm in Canada ( although I'd imagine that they have the same attitudes with grad schools )

 

[Ryker]

Hey, I would like some clarification about going into mathematics (pure mathematics ) and getting a master's degree? I've heard that you should go straight into a PhD program after your Bachelor's, but I'm not sure if I will have the GPA for that. I'm in my first year as an undergraduate, going into my second year; but there's one CS course I'm taking I think I'll do really poorly on (50-60s, or fail depending on the exam). I have 80s and up for my math courses though.
I don't get why getting a masters first is a bad thing though? Wouldn't you get more experience and mathematical maturity? thank you
I'm not in the states though, I'm in Canada ( although I'd imagine that they have the same attitudes with grad schools )

If you're in Canada, then I think it's pretty much standard that you first get a Masters and then a PhD (if you apply to Canadian schools, that is). I've noticed some universities now started offering a straight path to a PhD or the option to transition into it after a year or so into your Masters. But with the latter, you still apply for the Masters first, and then later switch.

 

[mathwonk]

if you enter grad school, some schools will try to interest you in a PhD program if you seem to qualify. The thinking is that it saves time for you to go straight towards the ultimate goal. from your point of view, if you have the time and can afford being poor longer, you may feel more confident if you enter a PhD program after learning the extra background a masters would provide.

Talent is a valuable commodity. PhD programs are always looking for students who seem to have the ability to do research. If you have this and can demonstrate it, they will usually take you.

I myself lost focus (during the vietnam war) the first time I enrolled in grad school in a PhD program and left with only a masters. I was then recruited later into another PhD program, but because I had a masters was allowed only three more years to finish. this was very hard for me.

How long you are allowed to stay in a PhD program depends on available money for support and other factors that vary from time to time, like desire to upgrade the program by making it more difficult or more efficient, or general level of difficulty of the school. E.g. an average state school probably let's you stay longer than an elite private school.

 

[nobelium102]

I think phd in us is around 6 years
wheras in Canada you do 2 years of masters and 4 years of phd (it varies of course)
so in the end it's basically same thing

 

[wisvuze]

Thanks all for the clarifications/insights. And this thread is awesome, thanks for keeping it up

 

[mathwonk]

bless you wisvuze. it is the questions that keep it alive. fire away.

 

[sponsoredwalk]

Mathwonk, here is one definition of a differential equation:

"An equation containing the derivatives of one or more
dependent variables, with respect to one of more independent
variables, is said to be a differential equation (DE)",
Zill - A First Course in Differential Equations.

Here is another:

"A differential equation is a relationship between a function
of time & it's derivatives",
Braun - Differential equations and their applications.

Here is another:

"Equations in which the unknown function or the vector function
appears under the sign of the derivative or the differential
are called differential equations",
L. Elsgolts - Differential Equations & the Calculus of Variations.

Here is another:

"Let f(x) define a function of x on an interval I: a < x < b.
By an ordinary differential equation we mean an equation
involving x, the function f(x) and one of more of it's
derivatives",
Tenenbaum/Pollard - Ordinary Differential Equations.

Here is another:

"A differential equation is an equation that relates in a
nontrivial way an unknown function & one or more of the
derivatives or differentials of an unknown function with
respect to one or more independent variables.",
Ross - Differential Equations.

Here is another:

"A differential equation is an equation relating some function
ƒ to one or more of it's derivatives.",
Krantz - Differential equations demystified.

Now, you can see that while there is just some tiny variation between them,
calling ƒ(x) the function instead of ƒ or calling it a function instead of an
equation but generally they all hint at the same thing.

However:

"Let U be an open domian of n-dimensional euclidean space, &
let v be a vector field in U. Then by the differential equation
determined by the vector field v is meant the equation
x' = v(x), x e U.

Differential equations are sometimes said to be equations
containing unknown functions and their derivatives. This is
false. For example, the equations dx/dt = x(x(t)) is not a
differential equations.",
Arnold - Ordinary Differential Equations.

This is quite different & the last comment basically says that all of the
above definitions, in all of the standard textbooks, are in fact incorrect.

Would you care to expand upon this point if it interests you as you know
a lot about Arnold's book & perhaps give some clearer examples than
dx/dt = x(x(t)), I honestly can't even see how to make sense of dx/dt = x(x(t)).

A second question I really would appreciate an answer to would be - is there
any other book that takes the view of differential equations that Arnold does?
I can't find any elementary book that starts by defining differential equations in
the way Arnol'd does & then goes on to work in phase spaces etc...