Journal for History of Mathematics (한국수학사학회지)

The Korean Society for History of Mathematics (한국수학사학회)
권호 표지

History of Transcendental numbers and Open Problems

초월수의 역사와 미해결 문제

  • Park, Choon-Sung (Department of Mathematics & Information, Kyungwon University) ;
  • Ahn, Soo-Yeop (Department of Mathematics Education, Kunkuk University)
  • 박춘성 (경원대학교 수학정보학과) ;
  • 안수엽 (건국대학교 수학교육과)
  • Received : 2010.07.05
  • Accepted : 2010.08.13
  • Published : 2010.08.31
Download PDF
⟨ Previous Next ⟩

Abstract

Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.

초월수의 연구는 2000년 이상 수학자들을 괴롭혀 왔던 고대 그리스의 기하학 문제의 하나인 원적문제가 불가능하다는 것을 보여줌으로써 수학사의 중요한 분야임을 입증하였다. Liouville은 1844년에 처음으로 구체적인 초월수의 예를 제시하였고, 칸토어는 1874년에 초월수의 존재성을 증명하였다. Louville 정리는 많은 초월수를 만들어 낼 뿐 아니라 초월수의 존재성을 증명하는데 이용할 수 있다. 1873년에 Hermite가 자연로그의 밑수 e가 초월수임을 보이고, 1882년에 Lindemann이 원주율 $\pi$가 초월수임 증명하였다. 1934년에 Gelfond와 Schneider는 각각 힐버트의 7번째 문제에 대한 서로 다른 완전한 해를 찾았다. 1966년에 Baker는 Gelfond-Schneider 정리의 일반화된 결과를 증명하였다. 이 연구의 목적은 초월수의 개념과 발달과정을 살피고, 미해결 문제를 제시하여 초월수의 연구가 촉진되도록 후학들에게 연구 동기를 부여하고자 한다.

Keywords

References

  1. 고영미 . 이상욱, 구장산술의 방정식론의 교육학적 의미, 한국수학사학회지 23 (2010) No. 1, 25–40.
  2. 박창균, Cantor의 무한관, 한국수학사학회지 10 (1997) No. 2, 33–38.
  3. R. Apery, Irrationalite de $\zeta(2)$ et $\zeta(3)$, Asterisque 61 (1979), 11–13.
  4. A. Baker, Linear forms in the logarithms of algebraic numbers; (I), Mathematica, 13 (1966), 204–216.
  5. A. Baker, Effective methods in the theory of numbers, Actes du Congres International des Mathematiciens, Nice, 1970.
  6. A. Baker, Transcendental Number Theory, Cambridge University Press, 1975.
  7. K. Boehle,"Uber die Transzendenz von Potenzen mit algebraischen Exponenten." (Verallgemeinerung eines Satzes von A. Gelfond), Math. Ann., 108 (1933), 56-74. https://doi.org/10.1007/BF01452822
  8. Carl B. Boyer, "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. 1991.
  9. E. Burger, "Diophantine Inequalities and Irrationality Measures For Certain Transcendental Numbers," Indian J. Pure Appl. Math., 32, 2001.
  10. G. Cantor, "Uber eine Eigenschaft des Inbegriffers aller reellen algebraischen Zahlen," J. Reine Angew. Math. 77 (1874), 258–262.
  11. G. Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre," J. Reine Angew. Math. 84 (1878), 242–258.
  12. P. Erdos, "Representations of real numbers as sums and products of Liouville numbers," Michigan Math. J. Vol. 9 (1962), 59–60. https://doi.org/10.1307/mmj/1028998621
  13. P. Erdos, "Some Remarks and Problems in Number Theory Related to the Work of Euler," Mathematics Magazine 56 (1983), 292–298. https://doi.org/10.2307/2690369
  14. C. Hermite, "Sur la fonction exponentielle," C. R. Acad. Sci. Paris 77 (1873), 18–24.
  15. D. Hilbert, "Uber die Transcendenz der Zahlen e und $\pi$," Mathematische Annalen 43 (1893), 216–219.
  16. A. Gelfond, "Sur le septieme Probleme de D. Hilbert," Comptes Rendus Acad. Sci. URSS Moscou 2 (1934), 1–64.
  17. A. Gelfond, "Sur le septieme Probleme de Hilbert," Bull. Acad. Sci. URSS Leningrade 7 (1934), 623–634.
  18. P. Gordan, "Transcendenz von e und $\pi$," Math. Ann. 43 (1893), 222-224. https://doi.org/10.1007/BF01443647
  19. M. Kac & S. M. Ulam, Mathematics and Logic, Frederick A. Praeger, New York, 1968.
  20. J. Lambert, "Memoire sur quelques proprietes remarquables des quantites transcendentes circu-laires et logarithmiques," Histoire de l'Academie, (Berlin) XVII (1761), 265–322.
  21. F. Lindemann, "Uber die Zahl $\pi$," Mathematische Annalen 20 (1882), 213–225.
  22. F. Lindemann, "Uber die Ludolph'sche Zahl," Sitzungber. Konigl. Preuss. Akad. Wissensch. zu Berlin No. 2 (1882), 679–682.
  23. Y. Nesterenko, "Modular Functions and Transcendence Questions," Mat. Sbornik 187 (1996), 65–96.
  24. P. Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, 2000.
  25. T. Rivoal, La fonction zeta de Riemann prend une infnite de valuers irrationnelles aux entiers impairs, Comptes Rendus Acad. Sci. Paris Ser. I Math. 331 (2000), 267–270.
  26. M. Robinson, On certain transcendental numbers, Michigan Math. J. Vol. 31, Issue 1 (1984), 95–98.
  27. K. Roth, Rational Approximations to Algebraic Numbers, Mathematika 2 (1955), 1–20.
  28. T. Schneider, Transzendenzuntersuchungen periodischer Funktionen. I, J. reine angew. Math. 172 (1934), 65–69.
  29. T. Schneider, Transzendenzuntersuchungen periodischer Funktionen. II, J. reine angew. Math. 172 (1934), 70–74.
  30. Dan Sewell Ward, The Library of Halexandria, 2008. http://www.halexandria.org/ dward089.htm
  31. A. Thue, Uber Annaherungswerte algebraischer Zahlen, Journal fur die reine und angewandte Mathematik 135 (1909), 284–305.
  32. P. Wantzel, Recherches sur les moyens de reconnaitre si un Probleme de Geometrie peut se resoudre avec la regle et le compas, Journal de Mathematiques Pures et Appliquees 1 (1837), 366–372.
  33. K. Weierstrass, Zu Hrn. Lindemann's Abhandlung: 'Uber die Ludolph'sche Zahl', Sitzungber. Konigl. Preuss. Akad. Wissensch. zu Berlin No. 2 (1885), 1067–1086.
  34. Reader' s Digest Oxford Complete Wordfinder, Oxford University Press, Inc, Reader' s Digest, Pleasantville, New York, 1996.