Hyperreal Numbers for Infinite Divergent Series

Bartlett, Jonathan; Gaastra, Logan; Nemati, David

doi:10.33014/issn.2640-5652.2.1.bartlett-et-al.1

“…The principal part function, ptðÞ, will yield the most significant component of a hyperreal expression [7]. In a hyperreal expression, imagine ω representing a benchmark infinite value, with ε ¼ 1 ω representing an associated benchmark infinitesimal.…”

Section: The Standard and Principal Part Functionsmentioning

confidence: 99%

Total and Partial Differentials as Algebraically Manipulable Entities

Fite¹,

Bartlett²

2023

Operator Theory - Recent Advances, New Perspectives and Applications

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Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to prevent contradictions. However, this is primarily a notational and conceptual problem. The work of Abraham Robinson has shown that there is nothing contradictory about the concept of an infinitesimal differential operating in isolation. In order to make this system extend to all of calculus, however, some tweaks to standard calculus notation are required. Understanding differentials in this way actually provides a more straightforward understanding of all of calculus for students, and minimizes the number of specialized theorems students need to remember, since all terms can be freely manipulated algebraically.

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“…The principal part function, pt(), will yield the most significant component of a hyperreal expression [7]. In a hyperreal expression, imagine ω representing a benchmark infinite value, with ǫ = 1 ω representing an associated benchmark infinitesimal.…”

Section: The Standard and Principal Part Functionsmentioning

confidence: 99%

Total and Partial Differentials as Algebraically Manipulable Entities

Fite,

Bartlett

2022

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Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to prevent contradictions. However, this is primarily a notational and conceptual problem. The work of Abraham Robinson has shown that there is nothing contradictory about the concept of an infinitesimal differential operating in isolation. In order to make this system extend to all of calculus, however, some tweaks to standard calculus notation are required. Understanding differentials in this way actually provides a more straightforward understanding of all of calculus for students, and minimizes the number of specialized theorems students need to remember, since all terms can be freely manipulated algebraically.

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Sizes of Countable Sets

Trlifajová

2023

Philosophia Mathematica

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The paper introduces the notion of size of countable sets, which preserves the Part-Whole Principle. The sizes of the natural and the rational numbers, their subsets, unions, and Cartesian products are algorithmically enumerable as sequences of natural numbers. The method is similar to that of Numerosity Theory, but in comparison it is motivated by Bolzano’s concept of infinite series, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree, but some differ, such as the size of rational numbers. However, set sizes are only partially, not linearly, ordered. Quid pro quo.

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Hyperreal Numbers for Infinite Divergent Series

Cited by 3 publications

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Total and Partial Differentials as Algebraically Manipulable Entities

Total and Partial Differentials as Algebraically Manipulable Entities

Total and Partial Differentials as Algebraically Manipulable Entities

Sizes of Countable Sets

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