Pure and Applied Mathematics Journal

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Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution

Received: 17 April 2018    Accepted: 14 May 2018    Published: 08 August 2018
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Abstract

The division by zero has been a challenge over years, which is in two forms: one involves a non-zero numerator while the other involves a zero numerator. This work deals with the second form of division, with the aim of finding a solution to the equation obtained when the expression is equated to, say x, where x is not a quantity but the ‘number of times of one whole’. In this work, zero divided by itself has been exhausted using different approaches and methods to come to a conclusion; that this division has a unique solution, 1. Some of the methods employed include geometric series, logarithm, indices, reciprocals, factorials, self-operations, Euler’s number, binomial expansions, graphical method among others. The conclusion has been made that zero divided by zero is 1. The reverse of division by multiplication is not applicable because zero has been associated with two ‘abnormal’ properties or behaviour that’s not with other numbers.

DOI 10.11648/j.pamj.20180703.11
Published in Pure and Applied Mathematics Journal (Volume 7, Issue 3, June 2018)
Page(s) 20-36
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

Times of One Whole, Self-Operations, Shifting Method

References
[1] Dodge, C. W. (1996). Division by zero. The Journal of the Mathematics Teacher, 89(2), 148. Retrieved March 6, 2018, from the JSTOR database.
[2] Cajori, F. (1929). Absurdities due to division by zero, an historical note. The Journal of the Mathematics Teacher, 22(6), 336- 368. Retrieved March 6, 2018, from the JSTOR database.
[3] Boyer, C. B. (1943). An early reference to division by zero. The Journal of the American Mathematical Monthly, 50(8), 487- 491. Retrieved March 6, 2018, from the JSTOR database.
[4] Duncan, H. F. (1971). Division by zero. The Journal of the Arithmetic Teacher, 18(6), 381- 382. Retrieved March 6, 2018, from the JSTOR database.
[5] Hornsby Jr., E. J. & Cole, C. (1985). Division by zero. The Journal of the Mathematics Teacher, 78(8), 588- 589. Retrieved March 6, 2018, from the JSTOR database.
[6] Henry, B. (1969). Zero, the troublemaker. The Journal of the Arithmetic Teacher, 16(5), 365- 367. Retrieved March 6, 2018, from the JSTOR database.
[7] Sunder, V. K. (1990). Thou shalt not divide by zero. The Journal of the Arithmetic Teacher, 37(7), 50- 51. Retrieved March 6, 2018, from the JSTOR database.
[8] Lichtenberg, B. P. (1972). Zero is an even number. The Journal of the Arithmetic Teacher, 19(7), 535-538. Retrieved March 12, 2018, from the JSTOR database.
[9] Allinger, G. D. (1980). Johnny got a zero today. The Journal of the Mathematics Teacher, 73(3), 187-190. Retrieved March 12, 2018, from the JSTOR database.
[10] Ball L. D. (1990). Prospective Elementary and Secondary Teachers' Understanding of Division. Journal for Research in Mathematics Education, 21 (2), 132-144. Retrieved April 20, 2018, from the JSTOR database.
[11] Carnahan, H. W. (1926). Note on the Fallacy. The Mathematics Teacher, 19(8), 496-498. Retrieved April 20, 2018, from the JSTOR database.
[12] Sadi, A. (2007). Misconceptions in Numbers. UGRU Journal. 5(Fall 2007).
[13] Heid, M. K. & et al (2013). Sampler: Division Involving Zero.
[14] Matsuura, T. & Saitoh, S. (2016). Matrices and Division by Zero z/0 = 0. http://www.scirp.org/journal/alamt.
[15] Jan, P. B. & Ilija, B. (2016). Anti Aristotle- The Division of Zero by Zero. Journal of Applied Mathematics and Physics. http://www.scirp.org/journal/jamp.
Author Information
  • Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya

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  • APA Style

    Wangui Patrick Mwangi. (2018). Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution. Pure and Applied Mathematics Journal, 7(3), 20-36. https://doi.org/10.11648/j.pamj.20180703.11

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    ACS Style

    Wangui Patrick Mwangi. Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution. Pure Appl. Math. J. 2018, 7(3), 20-36. doi: 10.11648/j.pamj.20180703.11

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    AMA Style

    Wangui Patrick Mwangi. Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution. Pure Appl Math J. 2018;7(3):20-36. doi: 10.11648/j.pamj.20180703.11

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  • @article{10.11648/j.pamj.20180703.11,
      author = {Wangui Patrick Mwangi},
      title = {Mathematics Journal: Division of Zero by Itself - Division of Zero by Itself Has Unique Solution},
      journal = {Pure and Applied Mathematics Journal},
      volume = {7},
      number = {3},
      pages = {20-36},
      doi = {10.11648/j.pamj.20180703.11},
      url = {https://doi.org/10.11648/j.pamj.20180703.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.pamj.20180703.11},
      abstract = {The division by zero has been a challenge over years, which is in two forms: one involves a non-zero numerator while the other involves a zero numerator. This work deals with the second form of division, with the aim of finding a solution to the equation obtained when the expression is equated to, say x, where x is not a quantity but the ‘number of times of one whole’. In this work, zero divided by itself has been exhausted using different approaches and methods to come to a conclusion; that this division has a unique solution, 1. Some of the methods employed include geometric series, logarithm, indices, reciprocals, factorials, self-operations, Euler’s number, binomial expansions, graphical method among others. The conclusion has been made that zero divided by zero is 1. The reverse of division by multiplication is not applicable because zero has been associated with two ‘abnormal’ properties or behaviour that’s not with other numbers.},
     year = {2018}
    }
    

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