Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero

Peter Jean-Paul; Shanaz Wahid

doi:doi:10.11648/j.pamj.20211002.12

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Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero

Peter Jean-Paul, Shanaz Wahid

Published in Pure and Applied Mathematics Journal (Volume 10, Issue 2)

Received: 16 April 2021 Accepted: 30 April 2021 Published: 8 May 2021

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Abstract

Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.

Published in	Pure and Applied Mathematics Journal (Volume 10, Issue 2)
DOI	10.11648/j.pamj.20211002.12
Page(s)	49-61
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Unstructured Numbers, Semi-structured Complex Number, Zero

References

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[2]	"Division by zero," [Online] Avaliable at: https://en.wikipedia.org/wiki/Division_by_zero#cite_ref-2. [Retirved 31 December 2020].
[3]	H. Okumura and S. Saitoh, "Wasan Geometry and Division by Zero Calculus," Sangaku Journal of Mathematics (SJM), vol. 2, pp. 57-73, 2018.
[4]	T. S. dos Reis, W. Gomide, and J. A. Anderson, "Construction of the Transreal Numbers and Algebraic Transfields," International Journal of Applied Mathematics, vol. 46, no. 1, 2016.
[5]	C. Dunne, J. B. Wells, and F. Kamareddine, "Adding an Abstraction Barrier to ZF Set Theory," in International Conference on Intelligent Computer Mathematics, 2020, pp. 89-104: Springer.
[6]	H. Okumura and S. Saitoh, "The Descartes circles theorem and division by zero calculus," arXiv preprint arXiv: 1711.04961, 2017.
[7]	S. SAITOH, Introduction to the Division by Zero Calculus. Scientific Research Publishing, Inc. USA, 2021.
[8]	UFUOMA, O. (2018). The differential and integral calculus in Bhaskara's framework: Exploration of zero and infinity. Asian Journal of Mathematics and Computer Research, 511-550.
[9]	J. Czajko, "Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces," World Scientific News, vol. 92, no. 2, pp. 171-197, 2018.
[10]	J. Czajko, "Quaternionic division by zero is implemented as multiplication by infinity in 4D hyperspace," World Scientific News, vol. 94, no. 2, pp. 190-216, 2018.
[11]	H. Okumura, "An Analogue to Pappus Chain theorem with Division by Zero," in Forum Geom, 2018, vol. 18, pp. 409-412.
[12]	H. Okumura, "Wasan geometry with the division by 0," arXiv preprint arXiv: 1711.06947, 2017.
[13]	H. Okumura and S. Saitoh, "Applications of the division by zero calculus to Wasan geometry," GLOBAL JOURNAL OF ADVANCED RESEARCH ON CLASSICAL AND MODERN GEOMETRIES”(GJARCMG), vol. 7, no. 2, pp. 44-49, 2018.
[14]	T. Matsuura, H. Okumura, and S. Saitoh, "Probability and Stochastic Analysis in Reproducing Kernels and Division by Zero Calculus," 2020.
[15]	W. P. Mwangi, "Definite Probabilities from Division of Zero by Itself Perspective," Asian Journal of Probability and Statistics, pp. 1-26, 2020.
[16]	Y. Zhang, J. Guo, D. Zhang, B. Qiu, and Z. Yang, "Output tracking of time-varying linear system using ZD controller with pseudo division-by-zero phenomenon illustrated," in IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society, 2017, pp. 3075-3080: IEEE.

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Peter Jean-Paul, Shanaz Wahid. (2021). Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure and Applied Mathematics Journal, 10(2), 49-61. https://doi.org/10.11648/j.pamj.20211002.12

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ACS Style

Peter Jean-Paul; Shanaz Wahid. Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure Appl. Math. J. 2021, 10(2), 49-61. doi: 10.11648/j.pamj.20211002.12

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AMA Style

Peter Jean-Paul, Shanaz Wahid. Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure Appl Math J. 2021;10(2):49-61. doi: 10.11648/j.pamj.20211002.12

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@article{10.11648/j.pamj.20211002.12,
  author = {Peter Jean-Paul and Shanaz Wahid},
  title = {Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {2},
  pages = {49-61},
  doi = {10.11648/j.pamj.20211002.12},
  url = {https://doi.org/10.11648/j.pamj.20211002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211002.12},
  abstract = {Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.},
 year = {2021}
}

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TY  - JOUR
T1  - Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero
AU  - Peter Jean-Paul
AU  - Shanaz Wahid
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PY  - 2021
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JF  - Pure and Applied Mathematics Journal
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AB  - Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.
VL  - 10
IS  - 2
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Author Information

Peter Jean-Paul

School of Engineering, Computer and Mathematical Sciences, Design and Creative Technologies, Auckland University of Technology, Auckland, New Zealand
Shanaz Wahid

School of Engineering, Computer and Mathematical Sciences, Design and Creative Technologies, Auckland University of Technology, Auckland, New Zealand

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APA Style

Peter Jean-Paul, Shanaz Wahid. (2021). Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure and Applied Mathematics Journal, 10(2), 49-61. https://doi.org/10.11648/j.pamj.20211002.12

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ACS Style

Peter Jean-Paul; Shanaz Wahid. Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure Appl. Math. J. 2021, 10(2), 49-61. doi: 10.11648/j.pamj.20211002.12

Copy | Download

AMA Style

Peter Jean-Paul, Shanaz Wahid. Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero. Pure Appl Math J. 2021;10(2):49-61. doi: 10.11648/j.pamj.20211002.12

Copy | Download

@article{10.11648/j.pamj.20211002.12,
  author = {Peter Jean-Paul and Shanaz Wahid},
  title = {Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero},
  journal = {Pure and Applied Mathematics Journal},
  volume = {10},
  number = {2},
  pages = {49-61},
  doi = {10.11648/j.pamj.20211002.12},
  url = {https://doi.org/10.11648/j.pamj.20211002.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20211002.12},
  abstract = {Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Unstructured and Semi-structured Complex Numbers: A Solution to Division by Zero
AU  - Peter Jean-Paul
AU  - Shanaz Wahid
Y1  - 2021/05/08
PY  - 2021
N1  - https://doi.org/10.11648/j.pamj.20211002.12
DO  - 10.11648/j.pamj.20211002.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 49
EP  - 61
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20211002.12
AB  - Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered field” and hence prove its existence. Examples of how these semi-structured complex numbers are used algebraically are provided. The successful development of this proposed number set has implications not just in the field of mathematics but in other areas of science where division by zero is essential.
VL  - 10
IS  - 2
ER  -

Copy | Download