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Suggestions for learning Mathematics?

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ThunderShade
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10 Oct 2010, 11:10 am

I'm in secondary school/high school and I'm dissapointed with Maths. I feel like all of it boils down to "monkey see, monkey do" and that doesn't interest me at all. I want to learn Maths in all of its elegance, not just as "Here are 15 indefinite integrals. Evaluate them with integrating by substitution where the substitution is either really obvious or given, and even then it wouldn't of been hard". Any suggestions for reading?



ruveyn
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10 Oct 2010, 12:31 pm

ThunderShade wrote:
I'm in secondary school/high school and I'm dissapointed with Maths. I feel like all of it boils down to "monkey see, monkey do" and that doesn't interest me at all. I want to learn Maths in all of its elegance, not just as "Here are 15 indefinite integrals. Evaluate them with integrating by substitution where the substitution is either really obvious or given, and even then it wouldn't of been hard". Any suggestions for reading?


Learning to do integrals is essentially a mechanical task. The creative part of mathematics is discovering new things that you try to prove or solve open outstanding problems. Here is an open problem called the Collatz Conjecture:

Define a rule T: as follows

T(n) = n/2 if n is even
T(n) = (n + 3 )/2 if n is odd.

T(n) = 1 and STOP.

Start with an integer and keep apply to the rule successively. It is conjecture that no matter one integer you start with you will end up with 1 after some number of steps depending on the integer you start with.

Simple? Yes. But if you can resolved it: either prove the conjecture is true or provide a counter example where the applications of rule T never terminate. If you can do that you will become an instant mathematical superstar.

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10 Oct 2010, 12:49 pm

A lot of math can be characterized as transforming one expression of something into another. So, in this sense, one is simply learning rules of transformation and a vocabulary. College level math can get very abstract, and that is where it gets a lot more interesting to me.



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10 Oct 2010, 1:07 pm

I was mentioning integrals as an example of how everything that is taught in school is just a mechanical process and no more. I am not implying that I want another way to learn how to do integration. What I want to do is "free my mind" (for lack of a better term) and truly understand mathematics at a "foundational" (if that's even a word) level (set theory for example), and I was wondering what books people would suggest.



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10 Oct 2010, 1:58 pm

You want to study abstract algebra and maybe number theory. Perhaps topology in a couple years once you have more of a foundation.

I can recommend "An Introduction to Mathematical Thinking: Algebra and Number Systems" by Gilbert and Vanstone. Another more rigorous option (but much more demanding and higher-level) would be Gallian, "Contemporary Abstract Algebra" which is an excellent book. You can get a previous edition of it for probably $5-$10 on Amazon.


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ruveyn
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10 Oct 2010, 2:04 pm

Orwell wrote:
You want to study abstract algebra and maybe number theory. Perhaps topology in a couple years once you have more of a foundation.

"Contemporary Abstract Algebra" which is an excellent book. You can get a previous edition of it for probably $5-$10 on Amazon.


That is a wonderful book. I recorded it for RFBDNJ (Recording for the Blind and Dyslexic, New Jersey). I had to fight the temptation of doing the exercises while I was record the book. I did the exercises later.

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10 Oct 2010, 2:40 pm

ThunderShade, that sounds like a good approach. I wish I had thought of it when I was in school.

Recently, I have been looking into books on the history of mathematics. A couple that have received generally positive reviews are "A Concise History of Mathematics", by Dirk J. Struik, and "Foundations and Fundamental Concepts of Mathematics", by Howard Eves. (I have not read either of them yet.) You can find more information about them on amazon.com, along with related books.



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10 Oct 2010, 3:08 pm

ruveyn wrote:
Orwell wrote:
You want to study abstract algebra and maybe number theory. Perhaps topology in a couple years once you have more of a foundation.

"Contemporary Abstract Algebra" which is an excellent book. You can get a previous edition of it for probably $5-$10 on Amazon.


That is a wonderful book. I recorded it for RFBDNJ (Recording for the Blind and Dyslexic, New Jersey). I had to fight the temptation of doing the exercises while I was record the book. I did the exercises later.

ruveyn

Probably the best math book I've ever had. But I still had to attach some caveats to it. It is intended primarily for beginning graduate students or advanced undergraduates, and the author does assume some background in linear algebra, which the OP (being a high-school student) likely doesn't have yet.


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Malin
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10 Oct 2010, 3:13 pm

I'm posting my math question here rather than starting another thread as I think I have here the attention of the people who's attention I want.

As to Maths, I've studied Logic but now I'm in over my head. Within a week I want to understand Modal Logic, including a good foundation in Temporal Logic. I have PDF's on the subject, but they are all written for people with a mathematical foundation which I don't have.

I understand perfectly the explanation of (C= Certain/ the square, P= Possible/ diamond) C = ¬P¬x, where x is any statement and vice versa. The problem is all these other terms. Where do I find a simple guide to the mathematics I need to first learn before I learn Modal Logic (except for Predicate Logic, which I know)?

All aid would be greatly appreciated.

I liked maths in high school, but have left it there basically for 8 years, and besides a basic training in logic and some pop-maths books (Emperor's New Mind) I am completely ignorant on the subject, so:

As to the above puzzel, I understand how it works with most numbers, but what of 3? Wouldn't it go to 3 to 6 back to 3 and then have to repeat? 3's a legal integer, no?



ThunderShade
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10 Oct 2010, 4:11 pm

Orwell wrote:
ruveyn wrote:
Orwell wrote:
You want to study abstract algebra and maybe number theory. Perhaps topology in a couple years once you have more of a foundation.

"Contemporary Abstract Algebra" which is an excellent book. You can get a previous edition of it for probably $5-$10 on Amazon.


That is a wonderful book. I recorded it for RFBDNJ (Recording for the Blind and Dyslexic, New Jersey). I had to fight the temptation of doing the exercises while I was record the book. I did the exercises later.

ruveyn

Probably the best math book I've ever had. But I still had to attach some caveats to it. It is intended primarily for beginning graduate students or advanced undergraduates, and the author does assume some background in linear algebra, which the OP (being a high-school student) likely doesn't have yet.

I do have a little knowledge on matrices, their inverses, how they represent linear transformations and Gaussian and Gaussian-Jordan elimination. Anything else I'd have to learn before reading it?



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10 Oct 2010, 4:21 pm

Malin wrote:
I'm posting my math question here rather than starting another thread as I think I have here the attention of the people who's attention I want.

As to Maths, I've studied Logic but now I'm in over my head. Within a week I want to understand Modal Logic, including a good foundation in Temporal Logic. I have PDF's on the subject, but they are all written for people with a mathematical foundation which I don't have.

I understand perfectly the explanation of (C= Certain/ the square, P= Possible/ diamond) C = ¬P¬x, where x is any statement and vice versa. The problem is all these other terms. Where do I find a simple guide to the mathematics I need to first learn before I learn Modal Logic (except for Predicate Logic, which I know)?

All aid would be greatly appreciated.

I liked maths in high school, but have left it there basically for 8 years, and besides a basic training in logic and some pop-maths books (Emperor's New Mind) I am completely ignorant on the subject, so:

As to the above puzzel, I understand how it works with most numbers, but what of 3? Wouldn't it go to 3 to 6 back to 3 and then have to repeat? 3's a legal integer, no?


Have a look at this:

http://en.wikipedia.org/wiki/Modal_logic

There are some useful references and pointers given at the bottom

ruveyn



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10 Oct 2010, 5:21 pm

ThunderShade wrote:
I do have a little knowledge on matrices, their inverses, how they represent linear transformations and Gaussian and Gaussian-Jordan elimination. Anything else I'd have to learn before reading it?

It would be helpful to know a bit about determinants. But you can get through the important stuff in that book without needing linear algebra; it just comes up in some of the examples and exercises. (Almost) everything else in the book builds up directly from the axioms it defines, so in principle you can just dive right in without any prior math background.


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Malin
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10 Oct 2010, 5:34 pm

Cheers Ruveyn, I'd already looked but did miss the most important external link which I'm now studying, but what of this:

ruveyn wrote:
Define a rule T: as follows

T(n) = n/2 if n is even
T(n) = (n + 3 )/2 if n is odd.

T(n) = 1 and STOP.

Start with an integer and keep apply to the rule successively. It is conjecture that no matter one integer you start with you will end up with 1 after some number of steps depending on the integer you start with.


I still want to know, what about 3? What am I missing?



ThunderShade
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10 Oct 2010, 5:40 pm

Malin wrote:
Cheers Ruveyn, I'd already looked but did miss the most important external link which I'm now studying, but what of this:
ruveyn wrote:
Define a rule T: as follows

T(n) = n/2 if n is even
T(n) = (n + 3 )/2 if n is odd.

T(n) = 1 and STOP.

Start with an integer and keep apply to the rule successively. It is conjecture that no matter one integer you start with you will end up with 1 after some number of steps depending on the integer you start with.


I still want to know, what about 3? What am I missing?


Reading up on the problem earlier considering n = 0, it seems to be that it is restriced to n being a positive integer and that it is actually T(n) = 3n + 1 for n odd.



ruveyn
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10 Oct 2010, 7:59 pm

ThunderShade wrote:
Malin wrote:
Cheers Ruveyn, I'd already looked but did miss the most important external link which I'm now studying, but what of this:
ruveyn wrote:
Define a rule T: as follows

T(n) = n/2 if n is even
T(n) = (n + 3 )/2 if n is odd.

T(n) = 1 and STOP.

Start with an integer and keep apply to the rule successively. It is conjecture that no matter one integer you start with you will end up with 1 after some number of steps depending on the integer you start with.


I still want to know, what about 3? What am I missing?


Reading up on the problem earlier considering n = 0, it seems to be that it is restriced to n being a positive integer and that it is actually T(n) = 3n + 1 for n odd.


Right. My error.

In any case no one has come close to resolving the Collatz. Unlike Fermats Last Theorem work on the Collats has not produced any spin off results in math or physics of great value. But it is strange that something so simple a ten year old kid can comprehend it has evaded resolution for over 50 years.

ruveyn



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11 Oct 2010, 10:18 pm

Presuming that, since this is int division any number continuously divided by 2 will equal 1. With this in mind, the only way for a number to increase in size is if there is a sequence such as

odd num
even num
odd num
even num
even num...

or odd num
even num
...
odd num
even num
even num

If there is a pattern such as

odd num
even num
even num

or

odd num
even num
even num
...
even num
the number will have decreased in size (3x/4 < x). Since an
odd num
even num
odd num
even num
can only occur if the number + 1 divided by 4 has a remainder of 0, there is an approximately 25% chance that the number will rise, and a 75% chance that it will lower. Given an infinite amount of possible turns, the 75% will always be triumphant. Also from this we can predict the approximate number of turns will be:

n2(x, y)= if (x > 1) y=y+1 n2(x / 2, y) ::: This doesn't return a number, this sets y and reads x, sorry for odd format.

y=0
x=arbitrary number > 1
z= n2(x, y)
approximate amount of turns = z + (z/4)

Haven't quite figured out the correlation to
odd num
even num
even num
...
even num
cases, so I'm unable to predict it in an exact form so far. Correct?