Learning Maths on Your Own: Advice and Resources

[IamNameless]

Currently I am seeking advice on how to go about learning mathematics on my own. I have some problems in mathematics that I believe is stopping my success I also hope someone here can give advice on how I can find a way to overcome them.

I'm going to break up my post into three parts, the first being what my difficulties are and the second being my specific questions and lastly the books I have in my collection. I would appreciate any response on either of this, any information would be greatly valued and appreciated.
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Part 1
My main problem is that I am unable to learn anything without substantial proof, that is I can not keep interest on anything that I'm being told to learn on faith.

I stopped paying attention and learning in school settings around the high school algebra level. I was greatly interested in solving problems, but I was never given the explanations of how we're able to solve them and they still be logically correct (except intuitively).

Once during class I remember being told of some problem that can't be solved with normal algebra, so I decided instead of continuing with the classes I would amuse myself by attempting to solve it by normal means, this continued till I left high school (I dropped out and received my ged).

I than proceeded to attempt teach myself (this being the case because I do in fact enjoy mathematics and would like to make a career out of it). I began buying random books for this purpose (I went without a clear goal in mind and bought many that have no use to me at my current level). The books that I currently own, I shall post at the bottom of this post.

I have found that there are branches of mathematics and theories that I would enjoy going into and that fit my standard of having to have everything proven (including even trivial things). There came something very clear when I began to go through the material... I can not understand any of it without going back and learning everything that I can't learn.

It appears that I can get through basic set theory and logic without anything beyond what I have. But anything I call advanced (you might still call elementary I suppose) requires as a prerequisite of some form of calculus. Calculus I'm sure I can get through, but from what I see, it starts on a very shaky foundation (not everything at this level is proven or attempted to be proven).

So basically it seems to get to a level of understanding, one must go through a long and painful time of memorizing things that they don't believe in and can't possibly understand at that time.
_________________________________________________________________________

Part two. Questions:
Most of these question I ask, are going to be a generic. I am seeking information of why we do things this way and how I can find these answers on my own (if possible).

1) As my first and most important question, I would like input on how I should go about learning on my own. I have a deep interest in gaining the knowledge that takes to understand number theory (at the level where I could read and understand research papers). I expect to be able to devote at least 5 hours on average of dedicated time a day. I can afford to buy some textbooks (maybe one college textbook every 2 weeks).

The rest of my questions are going to be unrelated, I only ask them because I'm curious and hoping someone has time to answer any of them.

2) The properties of negative numbers: I've never grasped how we've decided that we get a Negative*positive = a negative. I'm wondering is there some proof or is this how we have decided to define a negative number, have we just given this property?

3) Definitions of terminology: I've noticed that some things seem to be given definitions that don't always make sense (for example they exclude something, when it doesn't appear logically that they should (it doesn't appear that they shouldn't either though)).

"a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself."​

Is there some consequence of stating it as:

"A prime is a natural number which has exactly two divisors 1 and itself".​

My definition would allow for one to be prime, I don't see any reason why it shouldn't be, should I? I've thought that maybe it would interfere with some other proofs that rely on that specific definition, but if it was tweaked as I've said, wouldn't it allow some things to be proven as well?

4) Deriving formulas: Everyone knows the Pythagorean theorem from school, I remember having using it in so many different problems, but how did he proove it? And by how, I mean how did someone know to proof it, and how did that person go about it.

5) In regards to the previous question, how would you know to prove something. Is there something intuitive that you go by, is there some pattern? If possible could you describe a problem that would suggest there is some underlying truth and a proof could come out?

6) The basic operators of arithmetic and what we use for all computational mathematics. How do we know they work and how do we know that they work as we have defined them? In school (and in my books) it just gives properties of the numbers and no attempt to explain the properties.

7) If heard of some proofs being so very hard to solve, I'm just wondering what makes them hard? The one specific one I'm thinking of is 'Fermat’s Last Theorem'. So basically what specifically makes something hard to solve?
_______________________________________________________________________

Current books I have in my home library:

Math Proofs Demystified,
Stan Gibilisco, 2005
This book only shows proofs the way high school geometry does. It does cover basic logic though.

Algebra for college students, 4th edition
Mark Dugopolski, 2006

Proofs and Fundamentals
Ethan D. Bloch, 2006
Covers set theory, logic, some proof techniques and some more advanced topics (this ones my favorite book, I'm halfway through it).

Geometry a comprehensive course,
Dan Pedoe, 1970
This ones to advance (not the geometry I remember).

Fundamentals of Number Theory,
William J. LeVeque, 1996

Mathematics of Classical Quantum Physics
Frederick W. Bryon, Jr. and Robert W. Fuller, 1970

Basic Abstract Algebra
Robert B. Ash, 2007
I can't get through the first page... Maybe shouldn't have boughten this one.

Mathematics and its applications
Cozzens, Porter
This ones about discrete mathematics (very basic though).

How to prove it
Daniel J. Velleman,2006

Trigonometry, 8th edition
Lial, Hornsby, Schneide, 2005

Algebra II
Kaplan

Modern Calculus and Analytic Geometry
Richard A. Silverman, 1969

Calculus the Easy Way, 4th edition
Downing, 2006

 

[IamNameless]

Little longer than what I had intended to wright... Sorry for the long post, I think I'll edit in the morning and shrink it down considerably.

 

[rudinreader]

Yes your post is too long.. but I would like to try to be helpful so that if you do go ahead and study math, you will feel encouraged that when you run into problems someone on PF may be willing to help (especially on the HW help - but this is the internet so you can't always expect..)

Anyways, my two cents is that you shouldn't study number theory without studying calculus, though some computer scientists may disagree in the context of "finite math", but still..

If you can't understand calculus or at least "read the book" in 2-3 months coming from some other technical background, then you probably can't understand any difficult technical subject very well..

So for that matter I recommend "Spivak Calculus" as the project book you should look towards.. kind of like a 500 page electronics project book where you work problems out in your garage over a long perod of time (6 to 24 months).

You should have the attitude of "proving calculus"..

Anyways, that's my two cents.. under the assumption you really want to study math and you have the commitment required to do it..

 

[MathematicalPhysicist]

there's a list of books in maths from some guy called:
George E. Hrabovsky in the mathematics corner of sas.org, I think you should give a look at his recommendations they are best suit for independent study.
I don't remember the date of the column which he published his recommendations, you should search google.

 

[redrzewski]

Browse the course webpages for the math course sequences at major universities. That'll give you recommended textbooks, problem sets to try, lecture notes, etc.

 

[Vid]

Have you attempted to find explanations for your problems by yourself, especially the positive times a negative and Pythagorean Theorem? These problems are about as simple as math comes and aren't hard to work out on your own.

 

[IamNameless]

rudinreader: Thanks for the suggestion, I suppose I will have to go through a calculus text (I'll check out your book you suggested, they probably have it at my local bookstore). The main reason I'm sticking away from it at this point is although it does rely on proofs it doesn't start at the beginning (it doesn't prove the proofs that the proofs rely on -- I hope I said that right).
If anyone can suggest a book that proves everything outright. Something like Euclid's Elements (except for calculus) That would be wonderful.

loop quantum gravity: Although I can not find the article with the book suggestions, I have found several articles that I have found interest in, thanks.

redrzewski: Not a bad idea, that'll give me a good opinion of what levels the books are and in what order I should study them.

Vid: I have gone over many, but I find that many proofs (not specifically those ones) require that other proofs be proven first, it appears to go on like that forever. I'm trying to find some that are basic proofs literally directly from the axioms. Finding them isn't easy (not sure what they're called) and proving them myself doesn't seem to be a realistic goal (I have no idea how to go about finding ones that would be considered useful and ones that can derive others).

 

[arildno]

2) The properties of negative numbers: I've never grasped how we've decided that we get a Negative*positive = a negative. I'm wondering is there some proof or is this how we have decided to define a negative number, have we just given this property?

As an axiom of inequalities, we say that if p,q>0, then p*q>0.(1)
Suppose p<0.
Adding (-p) to both sides of this inequality, yields (-p)>0.(2)
Let q>0, then from (1) and (2), we get q*(-p)>0. (3)
Now, we may prove that q*(-p)=q*((-1)*p)=(-1)*(q*p)=(-(q*p))

Thus, adding q*p to (3) yields the desired result, q*p<0, if q>0 and p<0

3) Definitions of terminology: I've noticed that some things seem to be given definitions that don't always make sense (for example they exclude something, when it doesn't appear logically that they should (it doesn't appear that they shouldn't either though)).


"a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself."

Is there some consequence of stating it as:

"A prime is a natural number which has exactly two divisors 1 and itself".
My definition would allow for one to be prime, I don't see any reason why it shouldn't be, should I? I've thought that maybe it would interfere with some other proofs that rely on that specific definition, but if it was tweaked as I've said, wouldn't it allow some things to be proven as well?

What would happen is that you cannot any longer say that we can uniquely factorize a number in terms of primefactors, for example 4=1*2*2, but also 1*1*2*2

Apart from such sillinesses, and that all theorems we now have about prime numbers would hold with the added caveat "except for the prime number 1", there really wouldn't be much difference.



That's some answers, I hope.

 

[Feldoh]

For learning Calculus I would suggest, as much as you and most of the people on this board hate it, a somewhat less rigorous approach the first time through. Then go back through with a rigorous one.

Stewart's is pretty good for a first time through.
If you're looking for proofs and such I'd use Tom Apostol's Calc I & II (calc intro to multivariable and vector calculus)

 

[mathwonk]

i suggest you look at the lists of books on my thread "who wants to be a mathematician?"

your list is quite random, very good and ridiculously bad mixed together.

 

[symbolipoint]

IamNameless,
You need to really learn Introductory Algebra before you move beyond this. Some things are intuitive at this level but they are well illustrated in good instructional books, so that their concepts are extremely clear. Much of what you begin learning in Introductory Algebra are axioms and properties of numbers; they are mostly NOT proven, but are understood from common experience, and can be illustrated. From these, other things are proved; and such things proved are shown as instruction in the textbooks.

So, you need a good book, any old good book, on Introductory Algebra. Study it from start to finish. This should take you maybe 3 or 4 months. If you have any difficulties, you can have help from physicsforums.

Get good at Introductory Algebra. Then your next course should be either Geometry (the college preparatory kind that high schoolers study, with proofs) or Intermediate Algebra. For your own satisfaction, you might want Geometry before Algebra 2, because you express a strong interest in seeing and studying proofs.

 

[approx]

Hello, IamNameless
I have often been interested in ways that math can be proven in the shortest number of steps from basic principles. I think you are looking for something similar. My suggestion, which other people might disagree with, is that you find a college class on INTRODUCTORY Abstract Algebra, buy the book they are using, and read it slowly. Skip things you don't understand - keep in mind that the easiest things are NOT ALWAYS FIRST. When you get to a place that you are sure you won't understand anything in later chapters, go back to the beginning and start over. Take a few minutes each day to read a book on number theory also - reading them both at the same time might help you make some connections in your mind. Ask ALOT of questions on here. Oh, and also find some books on recreational mathematics - Martin Gardiner has lots of puzzles that can stretch your mind. I'm sure everyone here will try to help.
Work Hard/Have Fun!

-Approx.

 

[Gib Z]

Not to mention, the definition of a prime number you posted did say two 'distinct' factors.

Also, the simplest proofs for the Pythagorean theorem require nothing more than basic algebra. You might not like them as they are geometric and you may think they lack rigor, but I don't usually mind geometric proofs for predominantly geometric theorems.

 

[IamNameless]

Thank you all for your advice.

Currently I'm going over a college algebra textbook, I have fond that I know about 3/4 of the material. I should be able to get through the rest in a week or two. In regards to the suggestion on abstract algebra and number theory: I have a book on both, but I've found that I'm not able to get through any of it at this time. Once I progress some, I'll try again with number theory and when I have a decent background in calculus, set theory, and logic I'll retry abstract algebra.

Gib z: Yes I know this, I was asking whether we arbitrarily picked definitions for such things, I was using the definition of primes as an example. Sorry for the confusion.

SymbolicLogic: I think I'll take your advice Unfortunately though the book that I'm currently going through does not explicitly state what is an axiom, definition, property, what needs to be proven, etc. Nor does it give proofs (accept maybe a few here and there). Could you suggest either a book or a website that can server for a reference so in that I can compare them?

Also are properties the same thing as axioms or are they derived from them? -- Never mind on this one, I found proofs online for each property in the book, so it appears that properties are not axioms themselves.

Thanks again for you help,
-Iamnameless

 

[masnevets]

I would like to add a quote from Ravi Vakil's page http://math.stanford.edu/~vakil/potentialstudents.html in regards to your comments about not wanting to take things "on faith":

"Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.) "

 

[approx]

Calculus v.s. Abstract Algebra

I guess my confusion lies in the fact that I've never seen any calculus in a BEGINNING abstract algebra text, only things like groups, rings, fields, etc. By the way, I think it is easier to start with the simplest - groups, then develop definitions of rings from your understanding of groups, and then develop definitions of fields from your understanding of rings. Some texts do this in the opposite order, probably because the real numbers is a field and so we are familiar with one field already.

 

[epenguin]

Not able to answer or even read all your post but you seem well and soundly motivated IMHO. Working things out for yourself you will understand them more from the inside. Just remember that although it is good to prove and work out things for yourself, you will not be able to recreate centuries of collective effort by a world scientific community by yourself. I have often found in solving a problem it is a great help to know the answer! (Sometimes I seem to find simpler proofs.)

A useful and short book to add to your list is “How to solve it” by G. Polya. It is basically half-a-dozen tips, illustrated by examples. One thing I think students probably often fail to use, and it has often shown me the way out to remember is “Have you used all the information you are given or possess?” And another one I am sure will suit you and you will use a lot is “When you have solved the problem, the work is not finished!”.

As to how to have ideas, how Pythagoras thought of his theorem, that is a hard one. I am hoping to see some answers here! Certainly a lot of playing around and questioning come into it, so does inspiration, so does experience. At least your active approach gives you an advantage over most students and is essential so