Welcome to the world of positive integers.
I believe that the two fundamental ideas that we must focus on are the set of positive integers, and the operators between them, such as addition, multiplication, powers, factorials, and so on. I consider the positive integers as unsigned integers >= 1, therefore the descriptor (positive) is really unnecessary. However for clarity, to ensure that you know I am not including the negative integers, I will retain the words positive integer. When I use negative operators such as subtraction and division, I choose to restrict them such that the results are integers greater than zero.
My focus on only the positive integers may seem a bit simplistic, however, as you are undoubtedly aware, the positive integers alone can occupy a lifetime of research. Number theory is filled with unfinished ideas that involve only the positive integers. Primes, prime twins and Goldbach's Conjecture are good examples, and there are many more. The mystery of the prime numbers is the primary focus of many number theorists. The fact that there is no known function that will predict all primes successively, provides much motivation for number theorists, both amateur and professional alike.
The many unknowns and conjectures of number theory lead me to believe that to begin to fully understand mathematics, we must first fully understand the positive integers. I believe that by focusing on the positive integers, we may be able to unravel the secrets that lie within the many astounding and fascinating theorems that mathematicians of the past have developed.
Most of the ideas that I will present (now and in the future) are probably not new, and much of what I have developed on my own, I later learned had been discovered long ago - sometimes hundreds of years ago! Although this is disappointing at times, the experience gained through "reinventing the wheel" is valuable. Mathematics has become extremely complicated and often confusing; and as a result, the average person is intimidated by it. My only hope is that by focusing on the positive integers we will be lead to a simpler mathematics that will be both logical and intuitive.
Enjoy.