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2013, № 12 (161)



Shukman E.V. NON-DECIMAL FRACTIONS IN PAPERS BY G.W. LEIBNIZ AND L. EULERThe article provides a comparative analysis between Leibniz's and Euler's results associated with non-decimal fractions. The study of papers, letters and unpublished manuscripts shows that Leibniz first explored binary fractions and tried to apply them to study transcendental numbers. However he had no significant progress because he did not use an effective method for converting fractional numbers into binary system. Euler knew the method for converting fractional numbers similar to the modern, and successfully applied binary fraction representations to calculate sums of series and infinite products.Key words: history of mathematica, non-decimal fractions, Gottfried Wilhelm Leibniz, Leonhard Euler.

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1. Leibniz G.W. Explication de l'arithmйtique binaire, qui se sert des seuls caractиres 0 et 1 avec des remarques sur son utilitй, et sur ce qu'elle donne le sens des anciennes figures Chinoies de Fohy /Die mathematische schriften von Gottfried Wilheim Leibniz, vol. VII C. I. Gerhardt (ed) — Halle:1863.– pp. 223-227.

2. Shirley J. W. Binary numeration before Leibniz // American Journal of Physics. –1951. –Vol. 19. — pp. 452-454.

3. Shukhman E.V., Shukhman A.E. Zametki o nedesyatichnykh sistemakh schisleniya v opublikovannykh rabotakh i zapisnykh knizhkakh Leonarda Eylera [Notes about nondecimal numerations in the published papers and notebooks by Leonard Euler]. // Matematika v vysshem obrazovanii. — 2008. — N 6.– pp. 143–146.(in Russian)

4. Shukhman E.V. O nedesyatichnykh predstavleniyakh drobnykh chisel v rabotakh matemetikov XVII-XVIII vv. [About non-decimal representations of fractional numbers in works by mathematicians in XVII–XVIII centuries]. // Istoriya nauki i tekhniki. — 2013. — N 1.– pp. 3–16.(in Russian)

5. Greve J., Gumin H., Hochsteller E. Herrn von Leibniz' Rechnung mit Null und Eins. — Berlin, Mьnchen: SiemensAG, 1966. — 62 p. 202

6. Lautz G. 300 Jahre leibnizsches dualzahlensystem // Biological Cybernetics. –1979. –Vol. 35. –No. 3. –pp. 175-181.

7. Zacher H.J. Die Hauptschriften zur Dyadik von G. W. Leibniz: Ein Beitr. zur Geschichte des binдren Zahlensystems — Frankfurt a. M.: Klostermann, 1973. — 384 s.

8. Lettres de M. de Leibnitz б M. Herman // Histoire de l'Academie royale des sciences et belles lettres, Tom 13. — Berlin:1759. — pp. 469-483.

9. Leibniz G.W. Pisma i esse o kitayskoy filosofii i dvoichnoy sisteme ischisleniya [Letters and essays on Chinese philosophy and the binary system]. Introd., transl., comment. by V.M. Yakovlev. — М.: Institut filosofii RAN, 2005– 404 p. (in Russian)

10. Bernoulli J. Der Briefwechsel von Jacob Bernoulli. — Basel: Birkhauser, 1993. — 309 p.

11. Der Online-Ritterkatalog. — URL: http://leibniz.bbaw.de/ritter/

12. Leibniz G.W. Leibniz-Handschriften. — XXXV.– Niedersachsen Landesbibliothek Hannover.

13. Bodemann E. Die Leibniz-Handschriften der Kцniglichen цffentlichen Bibliothek zu Hannover. — Hannover, 1895. — 383 s.

14. Bellustin V.K. Kak postepenno lyudi doshli do nastoyatshej arifmetiki. [How did people progressively come to present arithmetic?] — M.: Uchpedgiz, 1941. — 200 p. (in Russian)

15. Matvievskaya G.P. On Euler's surviving manuscripts and notebooks. // Euler and modern science. — Washington DC: MAA, 2007. — pp. 119–126.

16. Sankt-Petersburg's branch of Archive Russian Science Academy, collection 136, list 1, documents 129–140.

17. Shukhman E.V. Priblizhennoe vychislenie nekotorykh matematicheskikh konstant v opublikovannykh i neopublikovannykh rabotakh L. Eulera [Approximate calculation of some mathematics constants in published and unpublished works by L. Euler]// Vestnik Orenburgskogo gosudarstvennogo universiteta. — 2010. –N 9. –pp. 74–80. (in Russian)

18. Juskevic A.P., Winter E. Leonhard Euler und Christian Goldbach. Briefwechsel 1729-1764. — Berlin:Akademie-Verlag, 1965. — 420 p.

19. Ozhigova E.P. Razvitie teorii chisel v Rossii [The development of the theory of numbers in Russia]. –Мoscow, 2003. — 360 p. (in Russian)


About this article

Author: Shuhman E.V.

Year: 2013


Editor-in-chief
Sergey Aleksandrovich
MIROSHNIKOV

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