Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork

Lionel Bréhamet

doi:doi:10.11648/j.history.20170502.11

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Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork

Lionel Bréhamet

Published in History Research (Volume 5, Issue 2)

Received: 8 October 2016 Accepted: 17 February 2017 Published: 1 April 2017

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Abstract

For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1 /D₁ +... +1 /D_h, were given by using (h-1) divisors (d_i) of D₁. This ancient modus operandi, well recognized today, provides D_i=DD₁/d_i for i greater than 1. Decompositions selected (depending on d_i) have generally been studied by modern researchers through the intrinsic features of d_i itself. An unconventional method is presented here without considering the d_i properties but just the differences d_h-1- d_h. In contrast to widespread ideas about the last denominator like ‘D_h smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘D_h smaller or equal to 10D’, where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for h=3 or 4 that a detailed overview was possible in the past. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d₂ +... + d_h. Clearly the decisions fit with a minimal value of the differences d_h-1- d_h, independently of any d_i values.

Published in	History Research (Volume 5, Issue 2)
DOI	10.11648/j.history.20170502.11
Page(s)	17-29
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), 2017. Published by Science Publishing Group

Keywords

Rhind Papyrus, 2/n Table, Egyptian Fractions

References

[1]	T. E. PEET: The Rhind Mathematical Papyrus, British Museum 10057 and 10058, London: The University Press of Liverpool limited and Hodder - Stoughton limited (1923).
[2]	A. B. CHACE; l. BULL; H. P. MANNING; and R. C. ARCHIBALD: The Rhind Mathematical Papyrus, Mathematical Association of America, Vol.1 (1927), Vol. 2 (1929), Oberlin, Ohio.
[3]	G. ROBINS and C. SHUTE: The Rhind Mathematical Papyrus: An Ancient Egyptian Text, London: British Museum Publications Limited, (1987). [A recent overview].
[4]	R. J. GILLINGS: Mathematics in the Time of Pharaohs, MIT Press (1972), reprinted by Dover Publications (1982).
[5]	M. BRUCKHEIMER and Y. SALOMON: Some comments on R. J Gillings’s analysis of the 2/n table in the Rhind Papyrus, Historia Mathematica, Vol. 4, pp. 445-452 (1977).
[6]	E. R. ACHARYA: "Mathematics Hundred Years Before and Now", History Research, Vol. 3, No 3, pp. 41-47, (2015).
[7]	K. R. W. ZAHRT: Thoughts on Ancient Egyptian Mathematics Vol. 3, pp. 90-93 (2000), [available as pdf on the Denver site https://scholarworks.iu.edu/journals/index.php/iusburj/.../19842]
[8]	G. LEFEBVRE: In: Grammaire de L’Egyptien classique, Le Caire, Imprimerie de l’IFAO, 1954.
[9]	A. IMHAUSEN and J. RITTER: Mathematical fragments [see fragment UC32159]. (2004). In: The UCL Lahun Papyri, Vol. 2, pp. 71-96. Archeopress, Oxford, Eds M. COLLIER, S. QUIRKE.
[10]	A. ABDULAZIZ: On the Egyptian method of decomposing 2/n into unit fractions, Historia Mathematica, Vol. 35, pp. 1-18 (2008).
[11]	M. GARDNER: Egyptian fractions: Unit Fractions, Hekats and Wages - an Update (2013), available on the site of academia. edu. [Herein can be found an historic of various researches about the subject].
[12]	L. BREHAMET: Remarks on the Egyptian 2/D table in favor of a global approach (D composite number), arXiv:1404.0341 [math. HO] (2014).
[13]	L. FIBONACCI: Liber abaci (1202).
[14]	E. M. BRUINS: The part in ancient Egyptian mathematics, Centaurus, Vol. 19, pp. 241-251 (1975).

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Lionel Bréhamet. (2017). Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. History Research, 5(2), 17-29. https://doi.org/10.11648/j.history.20170502.11

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Lionel Bréhamet. Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. Hist. Res. 2017, 5(2), 17-29. doi: 10.11648/j.history.20170502.11

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Lionel Bréhamet. Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. Hist Res. 2017;5(2):17-29. doi: 10.11648/j.history.20170502.11

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@article{10.11648/j.history.20170502.11,
  author = {Lionel Bréhamet},
  title = {Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork},
  journal = {History Research},
  volume = {5},
  number = {2},
  pages = {17-29},
  doi = {10.11648/j.history.20170502.11},
  url = {https://doi.org/10.11648/j.history.20170502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.history.20170502.11},
  abstract = {For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1 /D1 +... +1 /Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences dh-1- dh. In contrast to widespread ideas about the last denominator like ‘Dh smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘Dh smaller or equal to 10D’, where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for h=3 or 4 that a detailed overview was possible in the past. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2 +... + dh. Clearly the decisions fit with a minimal value of the differences dh-1- dh, independently of any di values.},
 year = {2017}
}

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TY  - JOUR
T1  - Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork
AU  - Lionel Bréhamet
Y1  - 2017/04/01
PY  - 2017
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DO  - 10.11648/j.history.20170502.11
T2  - History Research
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AB  - For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1 /D1 +... +1 /Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences dh-1- dh. In contrast to widespread ideas about the last denominator like ‘Dh smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘Dh smaller or equal to 10D’, where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for h=3 or 4 that a detailed overview was possible in the past. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2 +... + dh. Clearly the decisions fit with a minimal value of the differences dh-1- dh, independently of any di values.
VL  - 5
IS  - 2
ER  -

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Author Information

Lionel Bréhamet

Independent Scholar, Bordeaux, France

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Plain Text BibTeX RIS

APA Style

Lionel Bréhamet. (2017). Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. History Research, 5(2), 17-29. https://doi.org/10.11648/j.history.20170502.11

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ACS Style

Lionel Bréhamet. Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. Hist. Res. 2017, 5(2), 17-29. doi: 10.11648/j.history.20170502.11

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AMA Style

Lionel Bréhamet. Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork. Hist Res. 2017;5(2):17-29. doi: 10.11648/j.history.20170502.11

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@article{10.11648/j.history.20170502.11,
  author = {Lionel Bréhamet},
  title = {Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork},
  journal = {History Research},
  volume = {5},
  number = {2},
  pages = {17-29},
  doi = {10.11648/j.history.20170502.11},
  url = {https://doi.org/10.11648/j.history.20170502.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.history.20170502.11},
  abstract = {For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1 /D1 +... +1 /Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences dh-1- dh. In contrast to widespread ideas about the last denominator like ‘Dh smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘Dh smaller or equal to 10D’, where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for h=3 or 4 that a detailed overview was possible in the past. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2 +... + dh. Clearly the decisions fit with a minimal value of the differences dh-1- dh, independently of any di values.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Egyptian 2/D Table (D Prime Number): An Entirely New Analysis Consistent with the Idea of a Progressive Teamwork
AU  - Lionel Bréhamet
Y1  - 2017/04/01
PY  - 2017
N1  - https://doi.org/10.11648/j.history.20170502.11
DO  - 10.11648/j.history.20170502.11
T2  - History Research
JF  - History Research
JO  - History Research
SP  - 17
EP  - 29
PB  - Science Publishing Group
SN  - 2376-6719
UR  - https://doi.org/10.11648/j.history.20170502.11
AB  - For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1 /D1 +... +1 /Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1. Decompositions selected (depending on di) have generally been studied by modern researchers through the intrinsic features of di itself. An unconventional method is presented here without considering the di properties but just the differences dh-1- dh. In contrast to widespread ideas about the last denominator like ‘Dh smaller than 1000’, it is more appropriate to adopt a global boundary of the form ‘Dh smaller or equal to 10D’, where 10 comes from the Egyptian decimal system. Singular case 2/53 (with 15 instead of 10) is explained. The number of preliminary alternatives before the final decisions is found to be so low (71) for h=3 or 4 that a detailed overview was possible in the past. A simple additive method of trials, independent of any context, can be carried out, namely 2n+1= d2 +... + dh. Clearly the decisions fit with a minimal value of the differences dh-1- dh, independently of any di values.
VL  - 5
IS  - 2
ER  -

Copy | Download