Me, Math and the Internet (4)

PART FOUR:
HIDDEN FIELD OF REAL NUMBERS.

The story of crisis like this. In my search of new physics, I finally found a site of Doctor Rugerro Maria Santili quantum mechanics that have been revised into what he called hadrons mechanics. What is the basis of the revision? Apparently he changed his field of real numbers with unit = 1 be a real number field with unit = -1. How can that be? The answer is simple, try to redefine multiplication of real numbers with multiplication a ** b == - a * b, then clearly this new product meets the properties of commutative, associative and distributive for + and **. And easily proved that this new unit number field is -1. 

 

Thus, with the set of all real numbers we can create a new field that is isomorphic to the usual field of real numbers. Old numbers defined the new so-called isonumbers. Hadrons mechanics is based on the new mathematical construct of  isonumbers where he defines an all new calculus and analysis with the new multiplication.  

 

Well, when I saw a couple iso for the field of real numbers, I'm thinking of looking for another couple to the field of real numbers. I also generalize the multiplication multiplication * where ** a ** b == a * r * b where r is any real number not equal to zero. This new multiplication obviously also commutative, associative and distributive towards +. It can easily be proved that the new unit for multiplication is 1 / r. In other words, for a given r we can make a real number field or a field with a new multiplication ** == * r *. I call this new number alti-real numbers in the hypercomplex eGroup  owned by Jens Koeplinger .

Even in eGroup I redefine real numbers as omni-real numbers in which addition + is redefined ++ == +s+ and the multiplication  * is redefined to ** == *r* . In this case, the new addition is both commutative, and associative and have -s as a unit. It turns out that the two new defined operation are no longer distributive ** to ++. So all omni-real numbers form a quasi-field. As a consequence the field of alti-real numbers is a special case of quasi-field of omni-real numbers with s = 0. Thus we can build again hypercomplex numbers on the basis of quasi-field of omni-real numbers or the field of alti-real numbers. This is the logical consequence of the expansion of the field of real numbers and a huge work is waiting for the examination of it. The work I am facing is gigantic in the future.

Because of this discovery, I have to rewrite the n-color numbers part 3 and I found that the two numbers are the same color can be multiplied with each other if we define how multiplication of two 1 in the same color. Usually we choose the theoretical colored 1 to itself is equal to same colored 1. But it turns out with our new knowledge about the number alti-real, then I can multiply a colored 1 with itself to get colored r which is not equal to zero. Because color is none other than the unit vector in a given direction, then we can define a one-dimensional vector space with a multiplication of two unit vectors is r* unit vector. So I found the alti-vector number.

Later, after reading a paper by a follower of Rugerro Santilli in China, I discovered that the alti-real numbers and alti-vectors numbers, I think I found it, each of which is none other than the type one and type two Santilli numbers. Santilli basically never limit the number r of alti-real multiplication with -1. Even a mathematics professor from China later developed the theory of Numbers to Santillian Number Theory with the new multiplication. He even developed a new theory of cryptography by using two types of field of Santillian finite natural numbers.

But if you read his paper carefully, he seems to restrict to linear codes. I think if the theory be extended to the Marek-Santillian finite polyplex ring, then we can also make more complex Santillian cryptograpy. This is my homework now. I had to rewrite my fourth dialogue about polynomial finite numbers which I will refer to it as popularized by n-colored numbers. Sketch for it is already written, but now the third series of Mathematical Dialogue only new part two, and then jammed. Hopefully I will have the strength to break the deadlock.