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#1 2007-09-14 05:05:30

bossk171
Member
Registered: 2007-07-16
Posts: 301

...now what

I've finished high school, and I'm taking a few years off before college, the problem is: I love math.

My question is, what should I study (by myself) while I'm waiting to go back to school? And when I start to study it, how should I go about it, websites, books, something else...?

I've completed Algebra and did quite well. Then I did geometry (hated it) then Calculus, which was pretty fun.

I've also taken Physics (great) and Chemistry (didn't really care for it) and did well in both.

Just to keep my mind working, I've been doing some low-level programming and reading a little on the history of math (the times dealing with Leibniz and  Newton are most interesting) but I miss doing pure mathematics.

So my question is, now what? What do you recommend I do to keep myself mathematically stimulated? I have a particular fondness for infinite series, but I only touched on them in calc so have virtually no formal training in them.

Also, any comments on what your favorite courses were and why would be great, because it'll give me an idea of what's really out there.

Thanks.


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#2 2007-09-14 05:42:51

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: ...now what

Well, you can do one of two things (or both really).  You can either prepare yourself for introductory math courses at the college level or start to study pure mathematics. 

Typically when you get into college, you have one to two more years of calculus before you get into pure maths. If you have covered integral calculus up to integration by parts, integration around the axis,and the like, then you would be ready for multi variable calculus and differential equations.  You should also be studying introductory linear algebra and vector geometry during these.  Once you finish multi variable calculus, you can move on to vector calculus, which is like vector geometry and multi variable calculus combined into one.  This was my first 3 semesters of math courses.

This type of calculus typically falls under applied mathematics.  It is what the engineers and the physicists use to do their work.  On the other hand, there is pure mathematics.  This starts off with an introduction to proofs.  After that you can go into a variety of introductory subjects: Modern Algebra, Advanced Calculus (a pure (theoretical) approach to the real numbers), Combinatorics, Discrete math, Linear Algebra, and probably others I can't think of at this moment.

You can go right into proofs without learning calculus at the college level.  Why my university requires differential equations and multi variable calculus for an intro to proofs class, I haven't got a clue.  So it is really up to you.  Myself, I love the pure mathematics, and I have a distaste for applied such as calculus.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-09-14 06:08:59

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

I think I too like "pure maths." I liked calc, but only the mathematical parts, not the application.

I do know integration by parts, is integration around the axis when you have a curve spun around the axis and you have to find the volume of the solid it generates? If so, the I know that as well.

So I guess that means that I'm ready for "multi variable calculus." I have the very (very) basics of linear algebra down (up to and including inverse matrices), but I guess I could keep going in that.

What books (or websites) would you recommend to get me started on multi variable calc?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#4 2007-09-14 07:17:50

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ...now what

An introductory course in linear algebra would also cover transpositions, determinants, properties of determinants, solving systems of equations, and perhaps even an introduction to eigenvalues and eigenvectors.

As for a book for multi variable calculus, I'm not certain.  The book I used was Calculus with Early Transcendentals by Stewart, but this book is big enough to take you through introductory and second semester calc, multi variable calculus, and even has a section on differential equations.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2007-09-17 11:40:40

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,564

Re: ...now what

I think you should study parametric equations, or maybe I should, come to think of it.
Here's an applet that graphs them.
The x and y values are like an Ech-A-Sketch; each has its own equation.
http://www.ies.co.jp/math/java/calc/sg_para/sg_para.html

Last edited by John E. Franklin (2007-09-17 11:50:24)


igloo myrtilles fourmis

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#6 2007-09-17 14:48:05

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

I touched on in breifly in my Pre-cal class, but it didn't really impress me. Maybe a second go would be a good idea.


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#7 2007-09-17 16:17:23

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: ...now what

It's never a bad time to study number theory big_smile, if you can't find anything else to do, investigate that

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#8 2007-09-18 03:46:10

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

Identity wrote:

It's never a bad time to study number theory big_smile, if you can't find anything else to do, investigate that

How would I go about that? Do you know of any good books?


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#9 2007-09-18 04:28:35

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ...now what

As I've said other places, I highly recommend you take an intro to proofs course before studying number theory.  It is certainly possible to go straight into it, but you'll have a much easier time if you do proofs first.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#10 2007-09-18 06:06:40

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

Do you know of any (non-textbooks) books that would be a good place to start with intro to proofs? What's in an intro to proofs class? I've read many proofs (mostly famous ones) and they didn't seem to difficult to follow. Is there something more to it?

Last edited by bossk171 (2007-09-18 06:07:07)


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#11 2007-09-18 07:09:12

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ...now what

Proofs isn't about following them.  It's about coming up with them.  A proofs class starts out in basic logic and truth statements, goes on to methods of proving.  Through it, you learn basic definitions in math (positive, even/odd, divisible, etc).  Then an introduction to set theory, functions, relations, and cardinality.  Also normally included is operators and other things such as that,


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#12 2007-09-18 08:34:13

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

Maybe I'm not fully understanding the depth of what you're saying, but I feel like all of this is stuff I already know.

Last edited by bossk171 (2007-09-18 09:53:56)


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#13 2007-09-18 11:00:37

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ...now what

Well, the easiest thing to do is try a sample problem:

Let A be a set (need not be finite) and let ~ be an equivalence relation on A.  Let f be a bijective function from A onto B. 

(a) Prove that the equivalence classes partition A.
(b) Prove that there exists a unique equivalence relation on B which depends solely on ~ and f.

Please, no one else post solutions until bossk171 has done so.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#14 2007-09-18 14:27:24

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: ...now what

Point made. I don't understand a word of that.

Positive, even/odd, divisible, functions, and relations are all very fimiliar terms, but I guess their meaning runs a little deeper than I realize.


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#15 2007-09-18 14:42:47

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: ...now what

Right now I'm reading two books

Math Proofs Demystified McGraw-Hill 2005
Elementary Number Theory - Clark

btw I got them off a torrent xP they're from the biiiig math torrent on torrentspy

Last edited by Identity (2007-09-18 14:44:30)

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#16 2007-09-18 15:39:41

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: ...now what

Just to let you know that was a "I have a firm grasp on everything that should be learned in an intro to proofs course" type question.  It is probably a bit advanced for someone who has just taken a proofs course, although not be too much.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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