ince the beginnings of humanity, the task of counting was always very important. The development of human society had always been based on counts.
In the beginning, the simple Arithmetic was enough: counts of people, food, game, stones, days...
There were many symbols to represent the counts. The Roman Numerical System was one of them, but it wasn't practical. The set of mathematical symbols that we use today was originated with the Hindus, was improved by Arabs, and it's a Decimal System just because we have 10 fingers!
While we survived only with the counting of whole numbers, several imperfections went unnoticed. One of them, and of which solution brought significant progress, was the lack of the zero number! The zero number, for many centuries, was an inconceivable truth: after all, even if there were people without fingers, it was difficult to imagine a practical use for zero. In few words: the zero always existed, but we weren't prepared for it, at the beginning of civilization.
The sufficient maturity came with the abacus, a Chinese invention proper to do large counts. For the abacus' processes to work correctly, the knowledge of zero (no account), had to be well established.
The acceptation of zero ( ) brought a lot of novelties. Starting with expansion of our numerical system, because such as the zero, the negative numbers became evident too. Thus, it became easier to comment about previous days, lost points, regressive sums and other kinds of counting... Our universe was no more contained into the Natural Numbers [N]. But, since those days, it became possible to describe the universe with the Whole Numbers [Z], positive and negative.
But that was not enough. New measures were not represented at that scale yet. There were the fractional numbers that, for a long time, was the cause of fight between the men. While it was possible to round the fractions, the agreements happened. But numbers like square roots, pi number and other rational numbers, made us understand that the application of our scale of Whole Numbers was extremely limited: in a straight line, which could be all positive and negative numbers, it was also possible to align intermediate numbers, like very small fractions, as many as they were possible to imagine. Concomitantly, the existence of infinitesimal quantities, had led us to the infinitely large quantities and vice versa, through its divisions by the unit ( ).
In this way, the set of Real Numbers [R] arose, those that could be understood and therefore used, because our universe was no more contained into the Whole Numbers.
And we live in peace in our Mathematical Cosmo for many and many generations... No one could imagined that we would have more surprises. Our Line of Numbers looked like perfect...
It was when the Algebra arose, and with it the possibilities of working with numerical representations instead of numbers themselves! The equations could be handled with some abstraction, independent of the quantities of which they meant. And, once a day, the quantities related to the square root of negative numbers ( ) arrived! They were the new "black holes" of our Mathematical Reason.
Luckily, by then, always when we faced a mathematical problem without solution, our understanding of the world increased. Thus, our Line of Rational Numbers became the plan of Complex Numbers [C]. Our simple numbers, that once upon a time was used to count sheeps, in that moment, got an unprecedented spatial scale. We gain understanding that any number, though simple, is always a couple of real and imaginary values.
So, since the last centuries of the second millennium a.C., we have learned to live with the idea that, in the universe, there were numbers whose value was totally real, and whose numbers could also contain an imaginary percentage.
However, the humanity began to behold and to pursue its own limits! It didn't take time to we perceived that the whole thing would not stop there. In the Age of Reason who would be satisfied in a Numerical System on the plan, since we already had the domain of Geometry? Moreover, new technologies require knowledges beyond the simple imagination.
The new beat came from the possibility to raise any number into an imaginary power. What would mean, for example, a number one raised into an imaginary power? This problem come from the rotation of vectors, in computer graphics. The Quaternions ( ) emerged as an alternative to the first Set of Numbers called HyperComplex.
And hardly our Plan of Complex Numbers had become the Space of Quaternions, it already spoke about the hyperspace of the HyperComplex Numbers [H]. The numerical dimensions exceeded even the fourth dimension... They went so beyond that mixed up with own numeric infinite! The graph below, with a sphere in the center, is merely illustrative! There is no way to represent correctly the fourth, the fifth and the n-th dimensions in a plan or even into space!
Yep... now it looks like that we can live in peace in our Mathematical Cosmo... Our reason overcame itself. Apparently, nothing more is necessary to expand in our Numerical System.
But we need to fill the gaps of discoveries that have been left behind. Our simple Arithmetic, for example, dances, jumps up, does zip-zap, does twist, makes handstand, and so on when it is applied within all these sets!
The base is well established! Now we must continue to build the Mathematics!