Abstract
New operators are presented to introduce “arithmetic calculus”, where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements; and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a “Tree of Numbers” reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned; and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.
Highlights
Calculus plays a pivotal role in science and this is so successful that searching for other possible calculi seems irrelevant
This paper promotes new ways of thinking on 1) shifting the focus from sets/sequences of numbers to their elements; and 2) show that each element has a hidden structure that can be unearthed. This structure is handled by arithmetic calculus and conducemental algebra
Conventional algebra applied to sets/sequences of numbers based on natural number retards mathematics: the problems become unduly complicated; operations cannot be systematised similar to matrix-type operations; the potential for different methods of multiplication of two different sequences have not been transformed into classic operators
Summary
Calculus plays a pivotal role in science and this is so successful that searching for other possible calculi seems irrelevant. (2014) Introducing “Arithmetic Calculus” with Some Applications: New Terms, Definitions, Notations and Operators. The constituents of the new calculus are all floating within current mathematical practices but the aim is to collect them together and to brand them by a hallmark. In this process, the thinking is to change focus from the whole sequence to any of its single elements and express any single element as the product of building blocks and rules. New terms are italicised throughout the paper to inculcate their introduction. These are further collected in Glossary for the ease of reference
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