복소수 개념의 발달과 교육적 함의

Development of the concept of complex number and it's educational implications

  • Lee, Dong-Hwan (Korea Foundation for the Advancement of Science and Creativity)
  • 투고 : 2012.07.03
  • 심사 : 2012.08.16
  • 발행 : 2012.08.30

초록

본 논문은 복소수 개념이 정당화되는 과정에서 실수와 허수 사이의 관계가 어떻게 변화했는지를 살펴보았다. 허수가 처음 등장한 16세기에 수학자들은 현재와 동일하게 허수를 계산할 수 있었지만 허수를 수학적 대상으로 인정하기까지는 200여년의 시간이 필요했다. 수학이 발달하면서 나타나는 새로운 문제 상황이 실수와 허수의 조화를 요구하였고, 그 결과 복소수의 개념이 점차 명확해졌다. 복소수 개념 발달의 역사는 실수와 허수의 대립이 해소되어 실수와 허수를 복소수로 포괄할 수 있는 관점을 찾아가는 과정이었다. 실수와 허수가 어떤 점에서 대립을 하였고, 수학자들은 이러한 대립에 어떻게 대처하였는가에 분석의 초점을 두고, 실수와 허수의 관계를 정립하는 과정에서 나타난 새로운 사고방식이나 관점을 확인하고 그 영향을 살펴본다. 그리고 이러한 분석결과가 보여주는 교육적 함의를 기술하였다.

When imaginary numbers were first encountered in the 16th century, mathematicians were able to calculate the imaginary numbers the same as they are today. However, it required 200 years to mathematically acknowledge the existence of imaginary numbers. The new mathematical situation that arose with a development in mathematics required a harmony of real numbers and imaginary numbers. As a result, the concept of complex number became clear. A history behind the development of complex numbers involved a process of determining a comprehensive perspective that ties real numbers and imaginary numbers in a single category, complex numbers. This came after a resolution of conflict between real numbers and imaginary numbers. This study identified the new perspective and way of mathematical thinking emerging from resolving the conflicts. Also educational implications of the analysis were discussed.

키워드

참고문헌

  1. Birkhoff & MacLane, A survey of modern algebra(4th ed.), NY: Macmillan, 1977.
  2. J, Bos, Redefining Geometrical Exactness: Descartes' transformation of the early modern concept of construction, NY: Springer-Verlag, 2001.
  3. Bourbaki, Elements of the history of mathematics, 1984.
  4. G, Cardano, Ars magna, 1545. English translation: R. T. Witmer, Ars magna or the rules of algebra (M.I.T. Press, Cambridge MA, 1968), Reprinted by Dover (New York, 1993).
  5. T, Dantzig, Number : The language of science(4th ed.), NY: The Free press, 1954.
  6. R, Descartes, The geometry of Rene Descartes : with a facsimile of the first edition(translated by David E. Smith & Marcia L. Latham), NY: Dover Publications, 1954.
  7. Dieudonne, Mathematics-The Music of Reason, Springer, 1926.
  8. L, Euler, On the controversy between Leibniz and Bernoulli concerning logarithms of negative and imaginary numbers, 1749. (Translated by Stacy G. Langton) [http://home.sandiego. edu/-langton/elog.pdf]
  9. E, Glas,"Fallibilism and the Use of History in Mathematics Education." Science & Education, Vol. 7, Issue 4(1998), pp. 361-379. https://doi.org/10.1023/A:1008695214877
  10. D, Hilbert, "Uber das Unendliche" [On the infinite], Mathematische Annalen 95, (1926) (Lecture given Munster, 4 June 1925. English translation in van Heijenoort 1967, pp. 367-92)
  11. I, Kleiner, "Thinking the unthinkable: the story of complex numbers" (with a moral), Mathematics Teacher, Vol. 81, Issue 7(1988), pp. 583-592.
  12. F, Klein, Elementary Mathematics from an Advanced Standpoint: Arithmetic.Algebra.Analysis, NY: Dover Publications, 1968.
  13. D, Laugwitz, Bernhard Riemann, 1826-1866 : turning points in the conception of mathematics, 1996. (translated by Abe Shenitzer)
  14. S, Mac Lane, Mathematics, form and function, NY: Springer-Verlag, 1986
  15. A, Macfarlane, Ten British Mathematicians of the 19th Century, 1916.
  16. De Morgan, Trigonometry and Double Algebra, London: Taylor, Walton and Maberly, 1849.
  17. De Morgan, On the study and difficulties of mathematics(2nd ed), Chicago: Open Court Publishing, 1898.
  18. E, Nagel, "Impossible Numbers: A Chapter in the History of Modern Logic." Studies in the History of Ideas, 3,(1935), pp. 429-479.
  19. J, Stillwell, Numbers and geometry, NY: Springer, 1998.
  20. H. J. S, Smith, The Collected mathematical papers of Henry John Stephen Smith(Edited by J.W.L. Glaisher), The Clarendon Press, Oxford, 1894.
  21. Tignol, J. (2001). Galois' Theory of Algebraic Equations, NJ: World Scientific.
  22. C, Wessel, On the Analytical Representation of Direction(translated by Flemming Damhus(1999)). Copenhagen: Kort & Matrikelstryrelsen, 1797.