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. 

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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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
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
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
Equations for formally real meadows
TLDR
Two complete axiomatizations of the equational theories of the real numbers are given with respect to signatures of meadows with a single axiom scheme expressing formal realness. Expand
The rational numbers as an abstract data type
TLDR
An equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations is given, which results in 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. Expand
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