Abstract
Linked with existence of the almighty, the operation of division by zero which is considered as undefined or indeterminate or infinite sometimes, has been a topic of serious altercation among mathematicians and philosophers for so long. History is evident of the various attempts made to clearly define the algebra of zero, including the idea of division by zero. This includes the evolution of the idea of zero division and various insights from mathematicians like Euler, Craig and more. The realm of the paper contains all such significant attempts and it deals with the idea of the history that wishes to explore the evolution of infinity and undefined. The prime aim of this paper is to clearly distinguish the three concepts of Undefined, Indeterminate and Infinity, along with the concept of division by zero. It strives to provide a notation to undefined expressions, along with citing justifications and results with the help of unpretentious examples from key branches.
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Notes
Division by zero was featured long ago in ancient Indian mathematics, titled as Zero arithmetic (sunya ganita).
This believed mystery was soon explained by John Bernoulli.
Zero to the power of zero (\(0^0\)) is an ambiguous case, as its value is not agreed upon by all mathematicians. In algebra, the value is considered to be 1, while in analysis, it is considered as undefined (This paper considers it to be undefined for further use).
\(\liminf \) of a sequence \({<}x_n{>}\) is defined as—\(\liminf {x_n}{:=}sup\{inf\{x_m:m \ge n\}:n \ge 0\}.\)
\(\limsup \) of a sequence \({<}x_n{>}\) is defined as—\(\limsup {x_n}{:=}inf\{sup\{x_m:m \ge n\}:n \ge 0\}.\)
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Chaudhary, A., Batra, L. Defining the Undefined:
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Axiomathes 32, 1401–1413 (2022). https://doi.org/10.1007/s10516-021-09585-0
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DOI: https://doi.org/10.1007/s10516-021-09585-0