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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.

__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.**

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__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 text books (maybe one college text book 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?

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__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