Fast-tracking mathematics.

For the discussion of math. Duh.

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Fast-tracking mathematics.

Postby Anpheus » Fri Nov 16, 2007 10:55 pm UTC

I'm perpetually bored by my classes and would like to proceed to more difficult and complex subjects. I've always been a fast learner--I haven't attended Calculus I for more than a month and a half and in one hour I was solving problems on the projector faster than the students around me.

I am familiar with Galois Fields, because I wrote implementations of Rijndael/AES in a few programming languages to get a feel for them. I'm familiar with modular arithmetic for the same reason, but would like to more fully appreciate it. I have an intense fascination for a particular use of Euclid's propositions to find the square root with only a compass and a ruler.

I have always been fascinated with mathematics (and computer science, physics,) and I've reached the point where I am intensely bored waiting to go to a university, upon completion of this year of community college.

So I want someone to help tutor me, I suppose, on more complicated subjects. I have not done any integration beyond integration by substitution, but I am familiar with the idea, if not the process, of integration with shells/washers, and with some but not all of the rules of finding areas and volumes.

Is anyone up for it? If necessary, we can use some wiki pages with latex formatting to assist in notation. I'll try and find out if there's any tools that will allow more direct collaboration between two people with math formatting.
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Re: Fast-tracking mathematics.

Postby aguacate » Fri Nov 16, 2007 11:43 pm UTC

All I have to offer is advice:

There is no substitute for treadging through the boring classes. You might be able to pick up the ideas and the basics, and even pass any test that you would have to pass in the boring classes, but it's not the same. If you want to go far in a subject (and I guess this depends on your goals and whatnot) you need a strong foundation, even in the boring drudgery. Plus some unexpected idea from a boring subject subject now might inspire you later on when you're working on more advanced ideas.

By all means study everything you can outside of your classwork to get ahead and get interested, I guess what I'm saying is don't dwell too much on the idea that you can bypass the stuff that seems easy to you now. I've done it myself too many times.

The only fast-track is getting a solid foundation in the basics.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 12:01 am UTC

If I wanted boring drudgery I wouldn't have submitted the request, the short answer to following your advice is no.

The advanced material is intractably out of my reach because the teaching element is missing from virtually every large textbook. I -can- learn from those books, but even then, somebody would have to recommend which ones are best.

I've found learning from textbooks to be the worst possible way to do it, but have done it nonetheless. I covered everything I learned so far in Calc I in under a week of reading a single comprehensive study guide, I believe Barron's Forgotten Calculus, which is suggested for students who have taken Calculus but haven't done math for a while and need a review.

So, please, if you're going to tell me "stick with the boring, it'll help you in the long run" you haven't helped anyone, and I find it annoying to believe that you think every class I take will truly make me a better mathematician. There are bad teachers, mine right now isn't one of them, and there are pointless classes, and I want to avoid as much of that as possible.
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Re: Fast-tracking mathematics.

Postby mike-l » Sat Nov 17, 2007 12:48 am UTC

Don't be so fast to dismiss Aguacate's advice. I definitely fast-tracked my education, and while I feel I ended up with a good result, there are a number of holes in my foundation that I constantly have to go back and plug, and cause much more advanced topics to be more difficult than they should be.

That aside, there is way more to math than what you are learning now, and people aren't good at telling students that. You should definitely look into other topics. Since you mentioned modular arithmetic and finite fields, group theory would be a good place to start. Someone else would have to recommend a good text (this is one of the subjects I skipped and picked up along the way as I needed it). Lang's Algebra is the ultimate reference, but I definitely DO NOT recommend learning from it for the first time, and besides, it's almost 900 pages long and only 50 are on group theory :).

An introductory analysis book (point set topology mostly) could be good as well (e.g. Rudin), though it's probable that some multivariable calculus (Stewart is comprehensive, example and exercise filled, Spivak is awesome mathematically, perhaps I wouldn't want to learn from it though) should precede this.

Set theory is a good place to get a glimpse at the real guts of mathematics (Naive Set Theory perhaps?), and I personally found it very interesting at first, but I wouldn't recommend going to far with it; intending no offense to any set theorists or logicians on the board, there are much more interesting avenues of mathematics to study.

As far as the teaching element you are referring to, the best way to learn is probably to get a text and do all the exercises and examples you can. It may be easier to listen to someone talk and watch them write the words down than it is to read them from the text, but there is no substitute for actually doing the exercises. If you have particular questions you can post them, but I imagine it will be difficult to find someone who has the time and the willingness to teach an individual course to you over the internet.

I hope this post was helpful,

Cheers,
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Re: Fast-tracking mathematics.

Postby Mathmagic » Sat Nov 17, 2007 1:20 am UTC

Anpheus wrote:So, please, if you're going to tell me "stick with the boring, it'll help you in the long run" you haven't helped anyone, and I find it annoying to believe that you think every class I take will truly make me a better mathematician.

I have an idea for you: how about instead of strolling in here with 1 post under your belt and asking us for advice, then being completely arrogant and smug and dismissive of someone's advice (who so kindly offered it in the first place), you thank them for their insight and kindly state you're looking for other options?

You haven't even started university yet, and you're acting like you know what this whole deal is about. Aguacate obviously has experienced this before, and knows what it takes to succeed in mathematics. Your last statement:

I find it annoying to believe that you think every class I take will truly make me a better mathematician.

Is possibly one of the most ignorant and arrogant statements I've heard from someone who has posted here looking for advice. I'd be interested to find out how many other people want to offer their advice when they're going to be greeted with this attitude in return.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 1:27 am UTC

Acting like Galois doesn't make one like Galois :wink:

They are giving you good advice.

Read Lang's Algebra, or anything by Lang in general.
Read Dummit and Foote's Abstract Algebra.

Also the Math community my friend, is fond of mathematical genius in the form of papers, not in the form of "Look I am so smart".
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 1:31 am UTC

Is it really ignorant or arrogant to believe that every single class, every single hour spent in class, will make me a better mathematician? To believe that maybe I could improve myself without having to pay through the nose to get there?

You -can- get there from here, without spending as much money or as much time, and it takes a willingness to do so and some sieve for the vast quantity of information available on the web and in the books.



The book references -are- helpful, telling me I should waste my time and money in classes I don't need to take are not.


Edit: I can't prove to you I'm intelligent, I made every admission I could that I'm aware of different fields, but not well-equipped to understand them. I know about their existence, I just don't know how to best approach understanding them. I say I learn quickly, because I do, and because if anyone would give me a chance, they'd find that too. I'm not going to bring silly things like GPAs or test scores or IQs into this, because they aren't a proper measure of utility. And I really, desperately hope you aren't suggesting I should have published papers at the age of 19 that prove my intelligence when, by my own admission, my education has been to go from lackluster school to lackluster school. I was told I could not take the AP Calculus test in HS, no matter what. I was denied, twice, to take AP Calculus or test out of any of my classes, ever.

I'll be bluntly honest, I am fucking tired of this education system putting me through a long and boring track. I know there are more opportunities at the university level, but dammit, I've never even been given the chance to prove myself by any of my educators. My Principal in high school told me outright, in front of one of my parents, that, "I know for a fact you're just going to be flipping burgers in ten years down the road."


I fully expect the university environment to be faster paced, to be more educational, to be less boring, etc. But there's no reason I have to wait, is there?
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Re: Fast-tracking mathematics.

Postby Mathmagic » Sat Nov 17, 2007 1:32 am UTC

Anpheus wrote:Is it really ignorant or arrogant to believe that every single class, every single hour spent in class, will make me a better mathematician? To believe that maybe I could improve myself without having to pay through the nose to get there?

When that advice is coming from guys like aquacate and mike-l (the latter being a graduate student), then yes, it is.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 2:08 am UTC

Ok, then give me an example of how it'd hurt me in the long run.

The only way I see it hurting me is if I don't give every new area of learning a full treatment, and I don't plan on, or want to, skip over any details.

Edit: Keep in mind, society convinces people that the more you pay for something, the more valuable it really is. I hope that such a pathology isn't being demonstrated here, but when you respond to a remark that specifically mentions that the cost of higher education can be excessive and suggest that it really is valuable regardless, it makes me somewhat inclined to believe you're equating the cost of my education with the merit of it. If you're not, then please, address my comment fully. Why should I have to pay full price for an education I could receive for otherwise far less expense?
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Re: Fast-tracking mathematics.

Postby Alpha Omicron » Sat Nov 17, 2007 2:32 am UTC

Anpheus wrote:Ok, then give me an example of how it'd hurt me in the long run.

Examples were given.

I call troll on Anpheus.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 2:40 am UTC

I'm not a troll, and examples weren't given.

If anyone said, "Oh, I was working on <this> paper, and it took me so much longer because I skipped a class on <that>" then I missed it.

If it's going to hurt me by causing me to overlook material, then I need to be cautious about giving each subject of study a full treatment, but that doesn't warrant such an unabashedly negative response from everyone here. Everyone's advice has consisted of telling me to shut up and slog through the system.

Well, yes, eventually I have to get to the end of the system to get a degree. I plan on pursuing graduate level mathematics, so yes, I do need to understand all of this. I am not a troll trying to get someone to waste their time teaching me, I was sincerely hoping there would be someone here, of all the communities I could possibly have asked, who has a genuine wish to educate others.

I don't think I'm looking in the wrong place for that person, I just think the reactions insofar have been poor and hastily so. I don't think I did anything to warrant the insult of being called a troll, because I haven't done anything to insult anyone here. I state my feelings, annoyance, and you act as if I'm calling you idiots. You're clearly not.


So why has the conversation descended to that level? If we can't sort out my request, my plea for help, then let's sort out the communicative difficulty first. Why the hostile response? I'm honest about what I want to do, and what I want to learn about, and I'm asking for someone to help me to do that.

Edit: If anyone wants to work it out over an IM client, to keep it out of the forums, if I've offended anyone, I'm sorry. IM clients/"screen name": AIM/"Anpheus", Yahoo/"Anpheus", MSN/"aaron@frieltek.com", Gtalk/"aaron.friel@gmail.com" IRC works too, but you'll have to specify a server. I don't know where XKCD's IRC channel would be, I haven't looked.
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Re: Fast-tracking mathematics.

Postby skeptical scientist » Sat Nov 17, 2007 2:42 am UTC

Anpheus wrote:Is it really ignorant or arrogant to believe that every single class, every single hour spent in class, will make me a better mathematician?

Did you mean the opposite of what you said there?

In any case, yes, every class you take, and every hour you spend on the material will make you a better mathematician. That said, there are a finite number of hours and dollars to spend, and there's nothing wrong with wanting to be more efficient about things. I do think that classes are the best way to learn the material, and a solid grounding in calculus and other basics is essential later on.

Your best move, if it is available to you, would be to try to transfer to another section or sequence which covers the same material that you have been finding so tedious, but which moves at a faster pace. This class will give you a good grounding in the basics, but will hopefully do it in a more engaging way. Additionally, and I know you don't want to hear it, but to gain respect from professors later on when you want to apply to their school, take advanced classes, or apply for summer programs, independent study classes, or other nice things, you need to be able to show that not only have you mastered material, but you've also been able to succeed in a classroom setting.

Right now that means doing well in the class you are in, even if you find it boring, and hopefully impressing your instructor. One way of doing that might be to tell him (or her) that you find the material a bit slow, and does he have any suggestions for additional material you could possibly cover and discuss during office hours? Your instructor might not be interested in putting in the extra time on his part that that would entail, but if he is, then not only would you be able to cover some extra material, you would have someone who knows you to help you understand the material more easily as well, and you would be able to impress your instructor at the same time.

If he's not, reading a bit on your own as well would be a good thing. I would recommend Rudin's Principles of Mathematical Analysis (not the book entitled simply Real Analysis), or one of the algebra books mentioned, or else a book on naive set theory. I would start with only one book, and I would read it slowly, doing many of the exercises. I find that when learning mathematics outside a classroom, there is a temptation to move too quickly, reading the material and thinking you understand it, but not really understanding it in the same way as you would if you took a class; this temptation must be actively resisted.

I fully expect the university environment to be faster paced, to be more educational, to be less boring, etc. But there's no reason I have to wait, is there?

Yes, it will be, and no, there isn't. However,
Keep in mind, society convinces people that the more you pay for something, the more valuable it really is. I hope that such a pathology isn't being demonstrated here, but when you respond to a remark that specifically mentions that the cost of higher education can be excessive and suggest that it really is valuable regardless, it makes me somewhat inclined to believe you're equating the cost of my education with the merit of it. If you're not, then please, address my comment fully. Why should I have to pay full price for an education I could receive for otherwise far less expense?

The sad truth is that the university education serves a twofold purpose: not only does it educate you, but it also proves to others that you have been well educated. The first purpose can be accomplished much more cheaply at a public library by a studious and dedicated individual, but the second cannot.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 2:51 am UTC

Skeptical Scientist, I'm glad you pointed out that quote, and yes, I should have a negative in there! I can't believe I overlooked it.

I have no option to excel (no advanced classes available) at this point and won't for nearly a year. Nor do I have the money to do so. Again, in a year I should be able to go to university classes with no problem.

Thank you for your response, as well. And of course, in order to pursue degrees in mathematics and be published, I would have to prove my education.


Edit: To elaborate on the advanced classes, this is small-town Iowa. There really are no advanced classes available to me without them costing thousands of dollars more than I want, or can spend. I do not want to apply at UNI just to spend a few thousand dollars when I don't have any financial aid set up at the moment. Again, in a year it'll be an option, as will likely going to the university of my choice.
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Re: Fast-tracking mathematics.

Postby btilly » Sat Nov 17, 2007 3:00 am UTC

I'm going to disagree with everyone else.

First some background. The education system really does a bad job until you get to advanced material, and I fully understand the OP's frustration. Furthermore I'll note that surveys have found that people whose IQ is in the top 2% drop out of school at higher rates than the average population - almost certainly because of how poorly our education system handles gifted people.

Now I'll be first to admit that foundations matter. http://www.perlmonks.org/?node_id=26380 and http://www.perlmonks.org/?node_id=70113 are two essays I wrote on another site about learning mathematics, and in both you'll find that I strongly emphasize the importance of fully mastering foundations and building up from there. However I'll also say that not paying attention as you sleep through boring classes is not a particularly good way to get those foundations.

Next I'll submit that since the educational process will require that he sleep through those classes there is no point in his learning all of that material on his own. What will he accomplish? Oh, right. He'll guarantee that when he reaches those classes he'll be even more bored!

Math is a big subject. Rather than try to master math on your own, I'd suggest that you start trying to learn some corners of math that sound interesting. Then go deeper or look elsewhere as your interests take you. For instance number theory. It is very dated now, but I liked Hardy and Wright's The Theory of Numbers. Or some linear algebra. I'm going to suggest either Finite-dimensional Vector Spaces or Down with Determinants! Or perhaps some combinatorics - there are lots of books on combinatorics and graph theory to start with. They don't require a lot of background to get into. (Plenty of teenagers have published good research papers in combinatorics.) And there is lots of tie ins to computer programs you might want to write.

If you have anyone available who is good at math, you might want to pick a topic they know something about. That way they can recommend good books, and answer questions. Believe it or not, many professors will react well to a smart kid who genuinely wants to learn about their stuff, and who is able to do it without too much hand holding. Not all professors, of course, but you only need to find one or two.

There are lots of options. Most will work out pretty well. Just remember that you're trying to enjoy yourself. And also keep yourself honest - if you've read something and are fuzzy on the details, read it again. Go back to fundamentals. Perhaps review something that you'd thought you'd learned a long time earlier. Maybe a subtlety slipped by you before.

Oh, and don't forget: Good luck!

skeptical scientist wrote:
Anpheus wrote:Is it really ignorant or arrogant to believe that every single class, every single hour spent in class, will make me a better mathematician?

Did you mean the opposite of what you said there?

In any case, yes, every class you take, and every hour you spend on the material will make you a better mathematician. That said, there are a finite number of hours and dollars to spend, and there's nothing wrong with wanting to be more efficient about things. I do think that classes are the best way to learn the material, and a solid grounding in calculus and other basics is essential later on.


I know this is commonly given advice, but I've personally found it to be horribly wrong for me. Spending unnecessary time in a frustrating way is demotivating, and the demotivation can easily hurt more than the putative benefit.

Economic constraints prevented me from finishing my PhD in math (I needed to support my wife through medical school), but I think that the answers I've given in other threads demonstrate that I've learned a wee bit more math than just "2+2=4".

Seriously: how hard is it to not double-post?
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Re: Fast-tracking mathematics.

Postby tetromino » Sat Nov 17, 2007 5:09 am UTC

ok, I'll answer the OP's questions without resorting to condescension.

1. I believe that in many parts of the US, high school math classes below AP level really are pretty useless. so I can sympathize with your position. however, university math classes are most definitely not useless. so if you are going to go around telling grad students and postdocs, people who may not have taken a low-level math class in 10 years, that classes don't help, you are guaranteed to get a negative reaction.

2.
I covered everything I learned so far in Calc I in under a week of reading a single comprehensive study guide, I believe Barron's Forgotten Calculus, which is suggested for students who have taken Calculus but haven't done math for a while and need a review.
[...]
I am familiar with Galois Fields, because I wrote implementations of Rijndael/AES in a few programming languages to get a feel for them.

you will not learn any mathematics from Barron's study guides. you may learn enough to pass a test, but those kinds of books don't teach you actual math; they only cover the mechanics of differentiating and integrating elementary functions.
similarly, claiming that you've learned galois theory by implementing aes is kinda ... ludicrous. you cannot understand galois theory without a solid grounding in abstract alebra.

3. in my area, a graduate student tutor will cost you $40/hour. do not expect someone to give you a college-level education for free, the world doesn't work that way.
if you can't take university classes and can't afford a tutor, and learning alone from a book is too slow going, try to find someone who is in a similar position to you and do group study. remember, you learn the fastest by explaining the material to someone else.

4. speaking from personal experience, if you are not a genius, skipping a lower-level math class will hurt you later. in my educational career, I've done it 3 times: skipped basic calculus, ODE's, and undergrad-level abstract algebra. and all 3 (especially that last one) hurt me badly later on, when I had trouble with more advanced material because I didn't have the basics solid.

5. some actual advice.
to learn basic analysis (if you already know some calculus), read Marsden and Hoffman's Elementary Classical Analysis. it has some annoying typos, but the exposition is good, the exercises are great, and it certainly covers more than you will ever learn in a lower-level class. this was the book that made me fall in love with mathematics.
for abstract algebra (stuff you need to know to actually understand galois theory), I would recommend Herstein's Abstract Algebra.
and for introductory topology, see Munkres's Topology; most of my friends swear by it.
those 3 books will likely give you enough background to audit (or even take) grad-level classes. and that's where the fun starts. good luck!

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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 5:36 am UTC

I don't mean to say, "I know Galois Fields." I said I'm familiar with them. I am familiar with cats, but I don't know what makes them tick. I am familiar with AES, but I don't know what makes it a good cryptosystem.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 5:59 am UTC

The suggested books should take you a good few months to get through, if you work at it daily.

Get going, and make us all proud :wink:

Honestly though, you would probably love the real Galois. Not the mytical one. You would fight him, constantly though.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 6:11 am UTC

Sounds like what I need to find is an Erdos here.

Anyone willing to take amphetamines?
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 6:20 am UTC

Erdos to do what, teach you math over the internet?
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 6:36 am UTC

From what I understand, he had a vigor for mathematics and was known for teaching specialists something new about their own fields. Also, he was willing to work for very little for very long.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 6:38 am UTC

True, he also took all the fundamental courses :wink:

I want to try and be like that, once I get into Med School, and then reject them. (And so no longer have the burden of wanting to be in Med School).
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Re: Fast-tracking mathematics.

Postby ikerous » Sat Nov 17, 2007 7:20 am UTC

Can you take the derivative of ln(x), sin(x), tan(x), 2^(9x^3), sin(sqrt(x^2-9)), (e^x)/Sin(x), x^x ?
Can you integrate ln(x), 1/(x^2 + 9), (7x - 5) / (2x^2 - 3x + 1), x^2 * e^x ?

That's pretty much the basics of taking derivatives/integrating. You should at the very least be comfortable with all of those.

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Re: Fast-tracking mathematics.

Postby arbivark » Sat Nov 17, 2007 7:23 am UTC

If you go to the boring classes, you might meet people who could use your help.
I understand small town iowa, I was in Hannibal for a year before law school.
Is it feasible to drive to (iowa city, ames, des moines) to check out a good university library? Tell them you are thinking about applying, get to know faculty, suck up, use human engineering. Same as where you are at - teachers there would be thrilled to find a student who gets it and is motivated.
Part of succeeding in college is learning the stuff in classes. But another half is them testing your ability to jump thru pointless hoops. If you can do that in college, you might be able to do it on the job - it's a skill they value.
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I flunked out of college 4 times before I found the motivation to go back and graduate 4.0 so I could get into law school.

Symbolic logic is a class you could easily teach yourself in your spare time. Might be a dummies guide type thing around.
MIT is putting all its lectures, class notes etc online, you might find that helpful, or not.
Hang out in the math/puzzle forums here, some smart people.
I do share your frustration with math not being very accessible. By now it should be a simple matter of signing up for some automated online tutorials, but that hasn't happened the way we thought it would by now.
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Re: Fast-tracking mathematics.

Postby Aviatrix » Sat Nov 17, 2007 7:40 am UTC

(Side note: I am in a different situation, about which some of you know, and I've found your advice quite pertinent to my situation, so it's not falling on deaf ears.)

There's a very common saying about accomplishing something, which I'm sure you've heard. "Fast, good, cheap: pick any two." What you're asking is to have all three. You're clearly frustrated at the education system, and with apologies to my friends in Boone, you're stuck in Iowa. I can say with some level of confidence that you'll be happier at the university level because you'll be challenged there. If you aren't challenged by required jump-through-the-hoop undergrad courses (I was typing this as that ninja arbivark posted), you'll have more opportunities to volunteer to work with people doing actual research, or to partner with a mentor who can get you started on those first papers. That usually doesn't happen at the community college level. However, you might Google all of the math instructors at your college and see if anyone is doing active research. If so, find them and tell them you're bored. IOW, actively seek a mentor locally.

The folks here have given you some really fine suggestions. Pick one and run with it. When you get stuck, come back and ask. Many really do want to share their love of mathematics.

The educational system has a cookie-cutter mentality, but if it's any consolation almost everyone gets the short end of the stick that way. (That sounds really, really bad.) You're lucky. You're not on the short end where you can't keep up with the classes.

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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 8:34 am UTC

ikerous wrote:Can you take the derivative of ln(x), sin(x), tan(x), 2^(9x^3), sin(sqrt(x^2-9)), (e^x)/Sin(x), x^x ?
Can you integrate ln(x), 1/(x^2 + 9), (7x - 5) / (2x^2 - 3x + 1), x^2 * e^x ?

That's pretty much the basics of taking derivatives/integrating. You should at the very least be comfortable with all of those.

I made one error on the differentiation, wrong rule, and failed to get x^x until I looked up the solution, you take y = x^x, take the natural log of both sides, integrate with respect to x, and solve for dx.

I did the first and second problems using the rules I know, had to look up a rule for the third and fourth but I think if I applied integration by parts I'd have come up with a solution. Shouldn't have been lazy but I'm hazy on integration by parts, which is unfortunate, but I have a calc book I can look at for it.
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Re: Fast-tracking mathematics.

Postby Owehn » Sat Nov 17, 2007 8:36 am UTC

Here's some fun practice, then: Try differentiating an arbitrary base and exponent f(x)g(x). Now see if you can invent (and prove) a rule for differentiating an arbitrary expression where x appears n times.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 8:38 am UTC

take the natural log of both sides, integrate with respect to x, and solve for dx
You mean deferntiate with respect to x right?

Here is something fun, I managed to get it on my own and was quite proud. :D

Integrate x^x from 0 to 1.

Tips:
You will have to find the general integral first.
It doesn't have a closed form
You need to be familiar with taylor series
You need to be familiar with Integration by parts

If you are not familiar with the last two, well there is a good thing to learn. Both are fundamentally important.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 8:57 am UTC

Yes I did, sorry.

y = f(x)^g(x)
ln y = g(x) * ln(f(x))
(1/y)y' = g'(x)*ln(f(x)) + (1/f(x))f'(x)g(x)
y' = f(x)g(x) * (g'(x)*ln(f(x)) + (1/f(x))f'(x)g(x))

Reality check:

y = x^x
lny = x * ln(x)
(1/y)y' = 1*ln(x) + (1/x)*1*x = lnx + 1
y' = xx * (ln(x) + 1)


By "arbitrary expression where x appears n times" do you mean x^x^x? And subsequent power towers? Or do you mean f(g(h(i(...(x)))...)? The chain rule can be extended arbitrarily:
For a function A = f(g(x)), A' = f'(g(x))g'(x),
For a function B = f(g(h(x))), B' = f'(g(h(x))g'(h(x))h'(x)

For a power tower, you'd use d/dx f(x)g(x) = f(x)g(x) * (g'(x)*ln(f(x)) + (1/f(x))f'(x)g(x)) with the chain rule, no?



Edit: I don't know how to generate a taylor series, nor do I know how I would find the general integral of something. I read 4chan's discussion forums, and recently there was an (epic?) thread about x^x and it was generally stated that no finite number of expressions can express the indefinite integral of x^x.

Can you elaborate on the 'general form' of x^x?
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 9:11 am UTC

By general form, I meant, not just the value of the integral from 0 to 1. But rather such as
The integral of 3x^2 from a to b is: x^3 from a to b.

Really about the epic thread? There is a thread back here, I made with the solution, don't look it up!

Yes it is impossible to write the solution with finite number of terms, thats what I meant when I said there is no closed form solution. That comes as an easy consequence of the answer to the question itself.

About taylor series. Well my friend, get reading! they are important.
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Re: Fast-tracking mathematics.

Postby Anpheus » Sat Nov 17, 2007 10:53 am UTC

This is interesting. Will continue later (tomorrow as per bottom of post.)

I figured out the pattern, and am using:
α = xx
αn = dn/dxn α
β = ln x + 1
βn = dn/dxn β

d/dx α = α1 = αβ
(coincidence that read in German it's 'ass' or 'assss...' later on)

d2/dx2 α = α2 = α1β + αβ1
d2/dx2 α = α2 = αββ + αβ1

d3/dx3 α = α3 = α2β + 2α1β1 + αβ2
d3/dx3 α = α3 = (αββ + αβ1)β + 2αββ1 + αβ2
d3/dx3 α = α3 = αβββ + 3αββ1 + αβ2

d4/dx4 α = αβ4 + 6αβ2β1 + 3αβ1β1 + 4αββ2 + αβ3

Will find a general rule for expansion of αn tomorrow.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 11:19 am UTC

What you are doing is very interesting. Keep going with it.

Consider taking my advice as well, learning about taylor series will not hurt.
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Re: Fast-tracking mathematics.

Postby btilly » Sat Nov 17, 2007 1:09 pm UTC

While talking about Erdos...

3.14159265... wrote:True, he also took all the fundamental courses :wink:


Glancing at an obituary, he was homeschooled by a mathematician, and he was put into university early enough that he finished his PhD by 21.

There simply isn't enough calendar time left in there for a bored Erdos to have sat through many courses like the one the OP is suffering through. Furthermore his later disregard for all forms of conventional academic life suggest to me that he wouldn't have had much patience if he had been told to take them.
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Re: Fast-tracking mathematics.

Postby Aviatrix » Sat Nov 17, 2007 3:48 pm UTC

btilly wrote:
3.14159265... wrote:True, he also took all the fundamental courses :wink:
Glancing at an obituary, he was homeschooled by a mathematician....
Pi-ishness may have been too flippant, but "homeschooled by a mathematician" says a lot. Total immersion learning is much faster and much more solid than taking courses. I think the OP would love to be in a "homeschooled by a mathematician" environment, and it's a pity he or she can't be.

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Re: Fast-tracking mathematics.

Postby Yakk » Sat Nov 17, 2007 4:51 pm UTC

Problems are important to work through.

Being able to express your mathematics in a way that someone else can check is extremly important. To test this, you need someone else to check your work.

Being asked to solve problems that aren't directly related to the last sentence you read (ie, being able to solve problems from material learnt 5 months ago) is important.

Show up to class, and bring a text, instead of skipping the class. Read the text you are interested in, and keep tabs on the class to see if they cover any material. Do the busy work: if you are good at it, it should take you a fraction of the class time to do, and will give you practice for later on.

If you pick up a dense text and read through it, you will "get" parts of it, but you won't master it to the point where you'll have the instant grasp of the term mentioned on page 3 when you hit page 50 and that term is key to a tricky proof.

When working on texts, do every single damn exercise in the text. You will be lacking a bunch of support you get when being taught a text, so doing all of the exercises (and not just waving your hands and saying "that's obvious!") will help make up for that.

Book: Calculus, by Spivak. Reasonably formal, pretty looking, and goes from A to B nicely.

...

Now, some questions. How do you think? Symbols, shapes, patterns? Visually? Tactilely? Colours or movement? Gears or Water? (just trying to get a feel for what kind of math you might like).

...

Do you know the principle of mathematical induction? (and no, I don't mean "have seen it" or "have googled it")

Do you know the epsilon delta definition of a limit? Continuity? The construction of the real numbers from the natural numbers?

Know any geometry? Proof based? How is your trig?

Know the chinese remainder theorem? Proved it?

Seen the proof of the infinitude of primes?

Unique factorization into primes? Modulus of rings? Know what a group is? A ring?

Are any of the following claims clear to you:
There are the same number of even numbers as there are numbers.
There are the same number of integer fractions as there are integers.
There are a different number of real numbers than integer fractions.

...

Some classical algebra would be interesting to throw your way. Sadly, I can't think of a text: I learned it from private course notes of the instructor.
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Re: Fast-tracking mathematics.

Postby SimonM » Sat Nov 17, 2007 5:06 pm UTC

Most of that list can be found from the notes in the resources thread
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Re: Fast-tracking mathematics.

Postby gmalivuk » Sat Nov 17, 2007 7:00 pm UTC

SimonM wrote:Most of that list can be found from the notes in the resources thread

Not really. You can read all those websites through completely, and you still won't have ever proved the Chinese Remainder Theorem yourself, or used it.
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Re: Fast-tracking mathematics.

Postby 3.14159265... » Sat Nov 17, 2007 8:48 pm UTC

btilly wrote:
Pi wrote:True, he also took all the fundamental courses

Glancing at an obituary, he was homeschooled by a mathematician....
:oops:

I am sure he did cover all the material, in a systematic manner for the fundamentals.
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Re: Fast-tracking mathematics.

Postby bray » Sun Nov 18, 2007 1:10 am UTC

A lot of people seem to be acting as though Anpheus asked if he could learn math without doing any work. Yes, the answer to this is "no". We get it. But it's pretty clear that that's not what he's asking at all. What was asked is if he could learn math by working hard on his own instead of wasting time sitting in classes that move too slowly for him. The answer to that is "yes". I taught myself calculus and it doesn't seem to have hurt me at all. There's no secret to it: take your calculus book and start reading. If there's something you don't understand, go back and read about that until you do. And don't just read: do lots of problems (they don't have to be boring problems from the book, they can be interesting problems from the book or even ones you make up yourself) to make sure you're actually understanding stuff rather than tricking yourself into believing that you do.

Learning calculus (and multivariable calculus) well is probably the first step but if you really want to do actual math (which is quite different than what you typically see in calculus classes), you should work on getting comfortable with logical arguments and proofs (it might sound insulting to suggest that you're not comfortable with logical arguments but, trust me, there's plenty of math majors who aren't). I actually don't know a good resource for this but the table of contents and reviews for "How to Prove It" look pretty promising. If you find it interesting, graph theory might be a good place to start working on your proof skills: it's not demanding abstraction-wise and is easy to play with but has an infinite number of little problems to hone your skills on (e.g. show that any connected graph has two vertices which can be removed without disconnecting the graph).

I don't think that set theory or point-set topology would be a great idea at this point (it usually takes people a while to become comfortable with the kind of abstraction they involve) but you could probably handle real analysis, number theory, introductory abstract algebra, combinatorics, or linear algebra. Take a look, pick the one that looks most interesting to you, and then ask for book suggestions (and then ignore anyone who suggests Lang's Algebra). If you read and work at it, asking questions here will probably get you help when that time comes.

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Re: Fast-tracking mathematics.

Postby Alpha Omicron » Sun Nov 18, 2007 1:40 am UTC

arbivark wrote:Read a lot. Godel Escher Bach, A Beautiful Mind, Cryptonomicon...

Reading, read, read. Huzzah!
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Re: Fast-tracking mathematics.

Postby notzeb » Sun Nov 18, 2007 6:45 am UTC

I tried to fast track math a while ago, and I can say from experience: don't do it! There's quite a bit of algebra I had a hard time with, because I never bothered to practice multiplying polynomials together enough to see why x^3 + y^3 + z^3 - 3xyz is 1/2(x + y + z)((x - y)^2 + (y - z)^2 + (z - x)^2). In fact, it's much better to go in the exact opposite direction:

Know your basics to the Olympiad level. Get involved in highschool math competitions - you will be amazed by the beauty and difficulty of real geometry (do you know Ceva's theorem? How about the properties of the Lemoine point? Pascal's theorem?). You might want to check out artofproblemsolving.com, which has online classes (which cost $$ but are way worth it).

Edit: if you really want to jump right in, I guess you can always check out mathreference.com...

Edit2: I'm sorry, I mistook you for a high school student. I am very embarrased: :oops:

I hope there is some way to make this up to you.
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