What is the identity of the irrational number


Not all numbers are the same before the Lord. The real numbers correspond to points on the number line and are similar in this sense, but there are still huge differences between them. Some of them were created by the early humans (1,2,3), others by the cultured Greeks (√2, π) and still others from the discoverers of differential and integral calculus (e). In addition to the elementary distinction between whole and non-whole numbers, there is a division into rational and irrational numbers or into algebraic and transcendent numbers. Other properties of certain - but not all - real numbers, such as normality and predictability in real time, are of more modern origin. In this chapter we present some unsolved problems about the properties of some famous numbers, and along the way we will encounter some interesting but lesser-known numbers, such as the Champernowne number (0.12345678910111213 ...) and the Liouville number (0.1100010000000000000001000 .. .).

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  1. G. Almkvist and B. Berndt, Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary, American Mathematical Monthly, 95 (1988) 585-608. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  2. A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, Mass., 1974. [§23] zbMATHGoogle Scholar
  3. T.M. Apostol, Another elementary proof of Euler’s formula for ζ(2n), American Mathematical Monthly, 80 (1973) 425-431. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  4. T.M. Apostol, A proof that Euler missed: Evaluating ζ(2) the easy way, The Mathematical Intelligencer, 5 (1983) 59-60. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  5. R. Ayoub, Euler and the zeta function, American Mathematical Monthly, 81 (1974) 1067-86. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  6. D. Bailey, Numerical results on the transcendence of constants involving π, e, and Euler’s constant, Mathematics of Computation, 50, (1988) 275-281. [§22] MathSciNetzbMATHGoogle Scholar
  7. D. Bailey, The computation of ir to 29,360,000 decimal digits using Borweins' quartically convergent algorithm, Mathematics of Computation, 50 (1988) 283-296. [§21] MathSciNetzbMATHGoogle Scholar
  8. A. Baker, Transcendental Number Theory, Cambridge University Press, London, 1975. [§22] CrossRefzbMATHGoogle Scholar
  9. P. Beckmann, A history of π, fifth ed., Golem Press, Boulder, 1982. [§21] Google Scholar
  10. B. Berndt, Elementary evaluation of ζ(2n), Mathematics Magazine, 48 (1975) 148-154. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  11. B. Berndt, Modular transformations and generalizations of several formulas of Ramanujan, Rocky Mountain Journal of Mathematics, 1 (1977) 147-189. [§24] MathSciNetCrossRefGoogle Scholar
  12. F. Beukers, A note on the irrationality of ζ(2) and ζ(3), Bulletin of the London Mathematical Society, 11 (1979) 268-272. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  13. J.M. Borwein and P.B. Borwein, On the complexity of familiar functions and numbers, SIAM review, 30 (1988) 589-601. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  14. J.M. Borwein and P.B. Borewine, Pi and the AGM, Wiley, New York, 1987. [§§23 and 24] zbMATHGoogle Scholar
  15. F. Burk, Euler's constant, The College Mathematics Journal, 16 (1985) 279. [§24] CrossRefGoogle Scholar
  16. J.W.S. Cassels, On a problem of Steinhaus about normal numbers, Colloquium Mathematicae, 7 (1959) 95-101. [§21] MathSciNetzbMATHGoogle Scholar
  17. D. Champernowne, The construction of decimals normal in the scale of ten, Journal of the London Mathematical Society, 8 (1933) 254-260. [§21] MathSciNetCrossRefGoogle Scholar
  18. A. Cobham, Uniform tag sequences, Mathematical Systems Theory, 6 (1972) 164-192. [§23] MathSciNetCrossRefzbMATHGoogle Scholar
  19. A. Copeland and P. Erdös, Note on normal numbers, Bulletin of the American Mathematical Society, 52 (1946) 857-860. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  20. M. Dekking, M. Mendes-France, and A. van der Poorten, Folds !, The Mathematical Intelligencer, 4 (1983) 130-138, 173-181, 190-195. [§23] Google Scholar
  21. M. Gardner, Slicing π into millions, Discover, January 6, 1985, 50-52. [§21] Google Scholar
  22. J. Hancl, A simple proof of the irrationality of π4, American Mathematical Monthly, 93 (1986) 374-75. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  23. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, fourth ed., Oxford, London, 1960. [§§21 and 22] zbMATHGoogle Scholar
  24. J. Hartmanis and R. Stearns, On the computational complexity of algorithms, Transactions of the American Mathematical Society, 117 (1965) 285-306. [§23] MathSciNetCrossRefzbMATHGoogle Scholar
  25. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York, 1982. [§24] CrossRefzbMATHGoogle Scholar
  26. Y. Canada, Y. Tamura, S. Yoshino, and Y. Ushiro, Calculation of π to 10,013,395 decimal places based on the Gauss-Legendre algorithm and Gauss arctangent relations, Computer Center, University of Tokyo, 1983. [§21] Google Scholar
  27. R. Kannan, A.K. Lenstra, and L. Loväsz, Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers, Mathematics of Computation, 50 (1988) 235-250. [§23] MathSciNetCrossRefzbMATHGoogle Scholar
  28. R. Kannan and L.A. McGeoch, Basis reduction and evidence for transcendence of certain numbers, in Sixth Conference on Foundations of Software Technology and Theoretical Computer Science Conference, Lecture Notes in Computer Science, No. 241, Springer, Berlin 1986 263-269. [§22] CrossRefGoogle Scholar
  29. K. Knopp, Theory and Application of Infinite Series, 6th edition, Springer, Berlin, 1996. [§24] CrossRefzbMATHGoogle Scholar
  30. D.E. Knuth, The Art of Computer Programming, vol. 1, Addison-Wesley, Reading, Mass., 1968. [§24] zbMATHGoogle Scholar
  31. D.E. Knuth, The Art of Computer Programming, vol. 2, Addison-Wesley, Reading, Mass., 1971. [§§23 and 24] Google Scholar
  32. N. Koblitz, P-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed., Springer, New York, 1984. [§24] CrossRefGoogle Scholar
  33. D.H. Lehmer, Interesting series involving the central binomial coefficient, American Mathematical Monthly, 92 (1985) 449-457. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  34. J.H. Loxton and A.J. van der Poorten, Arithmetic properties of the solutions of a class of functional equations, Journal for Pure and Applied Mathematics, 330 (1982) 159-172. [§23] zbMATHGoogle Scholar
  35. J.H. Loxton and A.J. van der Poorten, Arithmetic properties of automata: regular sequences, Journal for Pure and Applied Mathematics, 392 (1988) 57-69. [§23] zbMATHGoogle Scholar
  36. K. Mahler, Arithmetical properties of the digits of the multiples of an irrational number, Bulletin of the Australian Mathematical Society, 8 (1973) 191-203. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  37. K. Mahler, Fifty years as a mathematician, Journal of Number Theory, 14 (1982) 121-155. [§23] MathSciNetCrossRefzbMATHGoogle Scholar
  38. Z.A. Melzak, Companion to Concrete Mathematics, Wiley, New York, 1973. [§24] zbMATHGoogle Scholar
  39. M. Mendes-France and A.J. van der Poorten, Arithmetic and analytic properties of paperfolding sequences, Bulletin of the Australian Mathematical Society, 24 (1981) 123-131. [§23] MathSciNetCrossRefzbMATHGoogle Scholar
  40. I. Niven, A simple proof that π is irrational, Bulletin of the American Mathematical Society, 53 (1947) 509. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  41. I. Nives, Numbers: Rational and Irrational, New Mathematical Library, vol. 1, Random House, New York, 1961. [§23] Google Scholar
  42. I. Nives, Irrational Numbers, Carus Mathematical Monographs, No. 11, The Mathematical Association of America. Wiley, New York, 1967. [§§21, 22, and 23] Google Scholar
  43. I. Papadimitriou, A simple proof of the formula Σk = 1k−2 = π2/6, American Mathematical Monthly, 80 (1973) 424-425. [§24] MathSciNetCrossRefzbMATHGoogle Scholar
  44. C. Reid, Hilbert, Springer, New York, 1970. [§22] CrossRefzbMATHGoogle Scholar
  45. P. Ribenboim, Consecutive Powers, Expositiones Mathematicae, 2 (1984) 193-221. [§21] MathSciNetzbMATHGoogle Scholar
  46. H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1985. [§21] CrossRefzbMATHGoogle Scholar
  47. M.L. Robinson, On certain transcendental numbers, Michigan Mathematical Journal , 31 (1984) 95-98. [§22] MathSciNetCrossRefzbMATHGoogle Scholar
  48. E. Salamin, Computation of π using arithmetic-geometric mean, Mathematics of Computation, 30 (1976) 565-570. [§21] MathSciNetzbMATHGoogle Scholar
  49. W. Schmidt, On normal numbers, Pacific Journal of Mathematics, 10 (1960) 661-672. [§21] MathSciNetCrossRefzbMATHGoogle Scholar
  50. C.L. Seal, Transcendental Numbers, Annais of Mathematics Studies, No. 16, Princeton University Press, Princeton, 1949. [§22] Google Scholar
  51. S.B. Smith, The Great Mental Calculators, Columbia University Press, New York, 1983. [§21] zbMATHGoogle Scholar
  52. E.L. Stark, The series Σk = 1k−ss = 2,3,4, ..., once more, Mathematics Magazine, 47 (1974) 197-202. [§24] CrossRefzbMATHGoogle Scholar
  53. R.G. Stoneham, On the uniform e-distribution of residues within the periods of rational fractions with applications to normal numbers, Acta Arithmetica, 22 (1973) 371-389. [§21] MathSciNetzbMATHGoogle Scholar
  54. E. Thorp and R. Whitley, Poincare’s conjecture and the distribution of digits in tables, Compositio Mathematica, 23 (1971) 233-250. [§21] MathSciNetzbMATHGoogle Scholar
  55. R. Tijdeman, Hilbert’s seventh problem: on the Gelfond-Baker method and its applications, in Mathematical Developments Arising From Hilbert Problems, Proceedings of Symposia in Pure Mathematics, 28, Part 1, American Mathematical Society, Providence, 1976. [§22] Google Scholar
  56. E.C. Titchmarsh, The Theory of Functions, 2nd edition, Oxford University Press, London, 1939. [§24] zbMATHGoogle Scholar
  57. A. van der Poorten, A proof that Euler missed, The Mathematical Intelligencer, 1 (1979) 195-203. [§24] CrossRefzbMATHGoogle Scholar
  58. S. Wagon, The evidence: Is π normal?, The Mathematical Intelligencer, 1: 3 (1985) 65-67. [§21] MathSciNetGoogle Scholar
  59. A. Because Number Theory, An approach through history from Hammurapi to Legendre, Birkhäuser, Boston, 1984. [§24] zbMATHGoogle Scholar
  60. AT THE. Yaglom and I.M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Vol. II, Holden-Day, San Francisco, 1967. [§24] zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1. University of Washington USA
  2. 2.Macalester CollegeUSA