‘ALGEBRA’ to ‘NUMBER THEORY’ to ‘GEOMETRY’ to ‘CALCULUS’: Recent Developments in Pure Mathematics at its Fundamental Level, Opening Scope for its Advanced Roles in Application Areas

Ranjit Biswas^1,

¹Department of Computer Science & Engineering Jamia Hamdard University Hamdard Nagar, New Delhi-110062, India

Cite this paper:
Ranjit Biswas. ‘ALGEBRA’ to ‘NUMBER THEORY’ to ‘GEOMETRY’ to ‘CALCULUS’: Recent Developments in Pure Mathematics at its Fundamental Level, Opening Scope for its Advanced Roles in Application Areas. Turkish Journal of Analysis and Number Theory. 2026; 14(1):29-48. doi: 10.12691/tjant-14-1-4

Abstract

It is a review work, the journey being from the birth of an algebraic structure ‘Region” (the minimal but most important algebraic structure in Algebra), then to the “Theory of Objects” being a generalized concept of the Theory of Numbers, and then introducing “A-numbers”, and then an application of A-numbers to introduce “Region Geometry” and finally about “Region Calculus”. The work reports the recent developments in Pure Mathematics at its fundamental level. On making a quick visit to the notion of the minimal but most important algebraic structure Region, and then on making a quick visit to the ‘Theory of Objects”, the author here develops a new theory called by “Theory of A-numbers” corresponding to a complete region A. The classical “Number Theory” is also an instance of “Theory of A-numbers” when the complete region A is the particular complete region RR corresponding to the region R. The new notion of im-numbers and compound numbers introduced in Number Theory will play huge role in future research work in the STEM subjects. The notion of “im-number” is a generalized concept of the ‘imaginary number i’, and the notion of “compound numbers” is a generalized concept of complex numbers. And then an application of the notion of A-numbers is done to introduce a new geometry called by ‘Region Geometry’ in Pure Mathematics. Finally, a new concept of calculus is developed called by “region Calculus” in the style of Newton’s calculus. Region calculus is a unified model of all possible calculus, and Newton calculus can be viewed as a particular instance of ‘Region Calculus’. The complete work reported here launches a new topic “Region Mathematics” in the giant subject Pure Mathematics. And the work will certainly cater to the future research activities in Pure Mathematics, in all the STEM subjects, including Data Science, AI and modern Statistics.

Keywords:
region region space signed object im-number compound number unit length A-number onteger object linear continuum line object line object circle region geometry limit continuity region calculus

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