Construction of the Transreal Numbers from Hyperreal Numbers

Reis, Tiago

doi:10.36285/tm.59

Cited by 6 publications

(13 citation statements)

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“…All four of the usual normed division algebras can be developed using the Cayley-Dickson Construction but this construction relies on the Cartesian form of complex numbers, which is degenerate for transnumbers, where a polar form must be used. Nonetheless we wonder if a polar form of this construction can be developed that constructs the transnumber systems, similar to transfields ( [6], Section IV).…”

Section: Discussionmentioning

confidence: 99%

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Transquaternions

Reis

Anderson²

2022

Transmath

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The quaternions extend the complex numbers and are used in physics and engineering. Division of quaternions by zero is not defined, which limits physical theories and engineering applications. We now introduce transquaternions, which totalise the arithmetical operations of quaternion addition, subtraction, multiplication, and both left and right division. In particular, division of quaternions by zero is allowed. The transquaternions are homeomorphic to the unit hypersphere or glome, including its interior, together with an isolated point. The 4D interior of the hypersphere is made up of the ordinary quaternions. The 3D surface of the hypersphere is made up of the infinite transquaternions, which are produced by dividing non-zero quaternions by zero. The isolated point, that lies outside the 4D space containing the hypersphere, is the transquaternion nullity, which is produced by dividing zero by zero. Transquaternions are a separable compact complete metric topological space.

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Section: Discussionmentioning

confidence: 99%

“…Transmathematics is a program that seeks to totalise the usual number systems by allowing division by zero. Thus the real numbers are totalised by the transreal numbers [1] [6] and the complex numbers are totalised by the transcomplex numbers [3] [5]. We now introduce the transquaternions as a totalisation of the quaternions.…”

Section: Introductionmentioning

confidence: 99%

Transquaternions

Reis

Anderson²

2022

Transmath

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“…Flags are commonly used to raise exceptions that need special attention; in arithmetical structures there are several. Inspired by floating point, in Anderson's transreal arithmetic [1,15] the corresponding entity is referred to as nullity, which acts like a non-signalling N aN . Inspired by exact computer arithmetic based on intervals, in Setzer's wheels [28,14] the absorptive element is denoted ⊥, which is used to control undesired properties of ∞.…”

Section: Totalisation and ⊥mentioning

confidence: 99%

Totalising Partial Algebras

Bergstra

Tucker

2022

Transmath

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We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers.

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“…As we stated in the introduction, there are a number of semantic methods for making division total in Q(÷). For example, as well as common meadows, there are wheels [21,16] and transreals [2,20], both of which add more than one peripheral elements. Our focus is on adding the absorbtive element ⊥ for division by zero.…”

Section: Extensions By Peripheral Elementsmentioning

confidence: 99%

Which Arithmetical Data Types Admit Fracterm Flattening?

Bergstra¹,

Tucker²

2022

SACS

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The formal theory of division in arithmetical algebras reconstructs fractions as syntactic objects called fracterms. Basic to calculation, is the simplification of fracterms to fracterms with one division operator, a process called fracterm attening. We consider the equational axioms of a calculus for calculating with fracterms to determine what is necessary and sufficient for the fracterm calculus to allow fracterm flattening. For computation, arithmetical algebras require operators to be total for which there are several semantical methods. It is shown under what constraints up to isomorphism, the unique total and minimal enlargement of a field Q(\div) of rational numbers equipped with a partial division operator \div has fracterm attening.

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Construction of the Transreal Numbers from Hyperreal Numbers

Abstract: We construct the transreal numbers and arithmetic from subsets of hyperreal numbers. In possession of this construction, we propose a contextual interpretation of the transreal arithmetical operations as vector transformations.

Cited by 6 publications

References 1 publication

Transquaternions

Transquaternions

Totalising Partial Algebras

Which Arithmetical Data Types Admit Fracterm Flattening?

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