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Several New Type of Numbers Discovered to Upgrade the Existing ‘Theory of Numbers’ and a New Type of Geometry Introduced

This work (Part-2) is sequel to the first work (Part-1) on Region Mathematics (see the corresponding SciencePG Frontiers News Article). The subject Region Mathematics is a new direction in Mathematics providing a new shape to the existing super giant subject ‘Mathematics’ whatever volume of literature developed so far since the stone age of earth.

By Ranjit Biswas

Jun. 3, 2016

In Part-1, this new subject **Region Mathematics** was initiated with a new Algebra called by **‘Region Algebra’** and with a new Calculus called by **‘Region Calculus’**, as the beginning of a new era. In this work three more new family members of Region Mathematics are introduced which are : Theory of Objects, Theory of A-numbers and Region Geometry.

In Part-1, this new subject **Region Mathematics** was initiated with a new Algebra called by ‘Region Algebra’ and with a new Calculus called by **‘Region Calculus’**, as the beginning of a new era. In this work three more new family members of Region Mathematics are introduced which are : **Theory of Objects, Theory of A-numbers** and **Region Geometry**.

A new algebraic theory called by “Theory of Objects” is introduced which generates the notion of ‘prime/composite objects’ and then produces the existing notion of prime/composite numbers as a special case of ‘prime/composite objects’. The notion of ‘imaginary objects’ in a region is then introduced and it is observed that the classical ‘imaginary numbers’ (or, complex numbers) are just one particular instance of the ‘imaginary objects’. Although the birth of the particular instance ‘imaginary numbers’ took place in an independent way long before (i.e. long before the discovery of ‘imaginary objects’), but interestingly it happened out of a very particular ‘region’!, the fact which is unearthed and explained in this work. Neither Division Algebra nor any existing algebraic system alone can produce this theory on the development of prime numbers, composite numbers, imaginary numbers and compound numbers. The basic philosophy is that the status ‘imaginary’ is local with respect to the region concerned. One object could be imaginary with respect to one set, and may not be imaginary with respect to another set. For instance, one will agree that (2i+5) is an imaginary number for the set R and need not be so for another set. One very interesting topic introduced is the discovery of ‘compound numbers’.

Then introduced is another new giant direction in Number Theory. It is shown that every complete region A has its own ‘Theory of Numbers’ called by ‘Theory of A-numbers’, where the classical ‘Theory of Numbers’ is just one instance of it being the ‘Theory of RR-numbers’ corresponding to the particular complete region RR. It is claimed that the “Theory of Objects” will play a huge role to the Number Theorists in a new direction. In due time, the ‘Number Theorists’ may be re-designated with the new title ‘Object Theorists’ as they may need to cultivate the broad area ‘Theory of Objects’ in pursuance of cultivating the ‘Theory of Numbers’ in a much better style and fashion. In fact, one of the major contributions in this work on Region Mathematics is that several new type of numbers are discovered. Consequently the existing ‘Theory of Numbers’ needs to be updated, extended and viewed in a new style. All these new sets of numbers need to be studied further in the context of F-algebra, Associative Algebra and Division Algebra and ofcourse in the context of region algebra.

In Region Mathematics, the “Theory of Objects” then induces another new direction called by “Region Geometry”. The “Region Geometry” is interesting, being a generalization of our rich classical geometry of the existing notion.

The existing huge volume of mathematics is just a part of Region Mathematics; although apparently it seems that the existing volume of mathematics has been almost sufficiently supporting the demands of the world mathematician, world scientists, world statisticians, and world engineers in their all type of mathematical works and computations. Presently this new subject **Region Mathematics** is at its infant stage. But this new subject will certainly cause a huge multiplicity to the volume of mathematics in very near future.

Authors

Ranjit Biswas, Department of Computer Science & Engineering, Faculty of Engineering & Technology, JamiaHamdard University, Hamdard Nagar, New Delhi, India. (ranjitbiswas@yahoo.com)

Paper link:

http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=141&doi=10.11648/j.pamj.20160503.12