Corpus ID: 201023714

Division by Zero : A Survey of Options

@inproceedings{Bergstra2019DivisionBZ,
title={Division by Zero : A Survey of Options},
author={Jan A. Bergstra},
year={2019}
}
The idea that, as opposed to the conventional viewpoint, division by zero may produce a meaningful result, is long standing and has attracted inter- est from many sides. We provide a survey of some options for defining an outcome for the application of division in case the second argument equals zero. The survey is limited by a combination of simplifying assumptions which are grouped together in the idea of a premeadow, which generalises the notion of an associative transfield.
8 Citations

Topics from this paper

Review of Suppes 1957 Proposals For Division by Zero
• James Anderson
• Mathematics
• 2021
We review the exposition of division by zero and the definition of total arithmetical functions in "Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. ThisExpand
Most General Algebraic Specifications for an Abstract Datatype of Rational Numbers
The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced. Three examples of such specifications are given. A preference is formulated forExpand
The Transrational Numbers as an Abstract Data Type
• Mathematics
• Transmathematica
• 2020
In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a newExpand
Fractions in Transrational Arithmetic
In transrational arithmetic each closed fraction may be written as a simple fraction. It is shown that unlike in involutive meadows, in transrational arithmetic open fractions cannot be written as aExpand
Sumterms, Summands, Sumtuples, and Sums and the Meta-arithmetic of Summation
Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) aExpand
Sumterms, Summands, Sumtuples, and Sums and the Meta-Arithmetic of Summation
• J. Bergstra
• Computer Science, Mathematics
• Sci. Ann. Comput. Sci.
• 2020
Sumterms and sumtuples are introduced as syntactic entities and a new description is obtained of the notion of a sum as a role which can be played by a number. Expand
Arithmetical Datatypes, Fracterms, and the Fraction Definition Problem
The main contribution of the paper is to provide a detailed terminology of fracterms and an outline of a survey of both forms of definitions of the notion of a fraction is given. Expand
Promise Theory as a Tool for Informaticians
The requirement of determinacy, the requirement of decomposition of aggregate agents, and the feature of a promise bias are introduced in the account of promises by providing more detailed requirements on promises. Expand

References

SHOWING 1-10 OF 22 REFERENCES
Who Did Derive First the Division by Zero $1/0$ and the Division by Zero Calculus $\tan(\pi/2)=0, \log 0=0$ as the Outputs of a Computer?
In this short paper, we will introduce an essence of the division by zero calculus and the situation from the viewpoint of computers that will contain a surprising news on the division by zeroExpand
Division by zero in non-involutive meadows
• Computer Science, Mathematics
• J. Appl. Log.
• 2015
This paper studies ‘non-involutive meadows’, i.e. variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-inVolutive meadow inWhich the multiplier inverse ofzero is one. Expand
Meadows and the equational specification of division
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 2009
A new axiomatic concept for number systems with division that uses only equations is studied: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0^-^1=0, and a general representation theorem is given for meadows. Expand
The Axiomatic Construction Of A New Algebraic Structure In Order To Extend A Field And Define Division By Zero
In order to define the operation of Division By Zero, a new algebraic structure will be created as an extension to a Field in such a way that for every non-zero $x$ which is an element of a Field,Expand
Division by Zero in Common Meadows
• Computer Science, Mathematics
• Software, Services, and Systems
• 2015
This work provides a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows that admit a certain cancellation law. Expand
Transformation of fractions into simple fractions in divisive meadows
• Computer Science, Mathematics
• J. Appl. Log.
• 2016
This work investigates which divisive meadows admit transformation of fractions into simple fractions, i.e. fractions without proper subterms that are fractions. Expand
Reality of the Division by Zero z / 0 = 0
• 2016
In this paper, we will give some clear evidences of the reality of the division by zero z/0 = 0 with a fundamental algebraic theorem, and physical and geometrical examples; that is, 1) a fieldExpand
Fracpairs and fractions over a reduced commutative ring
• Mathematics
• 2014
In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce "fracpairs" as pairs subject to laws consistent with the use of the pair asExpand