New Concepts of Congruences Modulo 1/n and Positive Integers Modulo 1/n

Abstract

The aim of this article is the creation of new Mathematical beings: The positive integers Modulo 1/n. I hope that they will be used by all the scientific community in general and the computer scientists (cryptographers) particularly. In the other hand, the paper contributes to enlarge the field of number theory and algebraic sets. In this paper, I present a new type of congruences on IN: the congruences modulo 1/N and The Sets of positive integers modulo 1/N denoted respectively = [1/N] AND IN 1/n, n € IN* The Work is divided into two Sections : Section A: It’s composed of: ▪ A fundamental Theorem (With Proof) ▪ A fundamental relation of equivalence on IN denoted = [1/n]
▪ The definition of the sets IN1/n , n € IN* (with examples)
▪ The definition of the binary operations*and ∆ on IN
▪ The definition of the binary operations *and ∆ on IN 1/n, n € IN*
▪  The presentation of the semi-groups (IN ; *) ; (IN ; ∆) ; ( IN1/n ;*); ( IN 1/n ; ∆ Section B: It’s Composed of an Appendice with: I) The definition and graph of the function: r: IN* → IN* x → n0 Where n0 is the rank of x . II) The definition and graph of the function: f: IN* → IN* n |→ [100, 1/n]