The Mathematical Intelligencer

, Volume 35, Issue 1, pp 42–60 | Cite as

Walking on Real Numbers

  • Francisco J. Aragón ArtachoEmail author
  • David H. Bailey
  • Jonathan M. Borwein
  • Peter B. Borwein


Random Walk Rational Number Continue Fraction Mathematical Intelligencer Simple Random Walk 
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  1. [1]
    S. Albeverioa, M. Pratsiovytyie, and G. Torbine G, “Topological and fractal properties of real numbers which are not normal”. Bulletin des Sciences Mathématiques, 129 (2005), 615–630.Google Scholar
  2. [2]
    J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003.Google Scholar
  3. [3]
    D. H. Bailey and J. M. Borwein, “Normal numbers and pseudorandom generators,” Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday, Springer, 2012, in press.Google Scholar
  4. [4]
    D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, “An empirical approach to the normality of pi”. Experimental Mathematics, 2012; in press.Google Scholar
  5. [5]
    D. H. Bailey, J. M. Borwein, R. E. Crandall, and C. Pomerance. “On the binary expansions of algebraic numbers”. Journal of Number Theory Bordeaux, 16 (2004), 487–518.Google Scholar
  6. [6]
    M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.Google Scholar
  7. [7]
    D. H. Bailey, P. B. Borwein, and S. Plouffe, “On the rapid computation of various polylogarithmic constants”. Mathematics of Computation, 66, no. 218 (1997), 903–913.Google Scholar
  8. [8]
    D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques and applications”. Mathematics of Computation, 70, no. 236 (2000), 1719–1736.Google Scholar
  9. [9]
    D. H. Bailey and R. E. Crandall, “On the random character of fundamental constant expansions”. Experimental Mathematics, 10, no. 2 (2001), 175–190.Google Scholar
  10. [10]
    D. H. Bailey and R. E. Crandall, “Random generators and normal numbers,” Experimental Mathematics, 11 (2002), no. 4, 527–546.Google Scholar
  11. [11]
    D. H. Bailey and M. Misiurewicz, “A strong hot spot theorem,” Proceedings of the American Mathematical Society, 134 (2006), no. 9, 2495–2501.Google Scholar
  12. [12]
    G. Barat, R. F. Tichy, and R. Tijdeman, Digital blocks in linear numeration systems. Number theory in progress, 2 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin (1999), 607–631.Google Scholar
  13. [13]
    M. N. Barber and B. W. Ninham, Random and Restricted Walks: Theory and Applications, Gordon and Breach, New York, 1970.Google Scholar
  14. [14]
    A. Belshaw and P. B. Borwein, “Champernowne’s number, strong normality, and the X chromosome,” Proceedings of the Workshop on Computational and Analytical Mathematics in Honour of Jonathan Borwein’s 60th Birthday, Springer, 2012, in press.Google Scholar
  15. [15]
    L. Berggren, J. M. Borwein, and P. B. Borwein, Pi: a Source Book, Springer-Verlag, Third Edition, 2004.Google Scholar
  16. [16]
    J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd ed., A. K. Peters, Natick, MA, 2008.Google Scholar
  17. [17]
    J. Borwein, D. Bailey, N. Calkin, R. Girgensohn, R. Luke, V. Moll, Experimental Mathematics in Action. A. K. Peters, Natick, MA, 2007.Google Scholar
  18. [18]
    J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, John Wiley, New York, 1987, paperback 1998.Google Scholar
  19. [19]
    J. M. Borwein, P. B. Borwein, R. M. Corless, L. Jörgenson, and N. Sinclair, “What is organic mathematics?” Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997, 1–18.Google Scholar
  20. [20]
    P. B. Borwein, “On the irrationality of certain series.” Math. Proc. Cambridge Philos. Soc. 112 (1992) 141–146.Google Scholar
  21. [21]
    P. B. Borwein and L. Jörgenson, “ Visible structures in number theory,” Amer. Math. Monthly 108 (2001), no. 10, 897–910.Google Scholar
  22. [22]
    C. S. Calude, “Borel normality and algorithmic randomness,” in G. Rozenberg, A. Salomaa (eds.), Developments in Language Theory, World Scientific, Singapore, 1994, 113–129.Google Scholar
  23. [23]
    C.S. Calude, Information and Randomness: An Algorithmic Perspective, 2nd ed., Revised and Extended, Springer-Verlag, Berlin, 2002.Google Scholar
  24. [24]
    D. G. Champernowne, “The construction of decimals normal in the scale of ten.” Journal of the London Mathematical Society, 8 (1933) 254–260.Google Scholar
  25. [25]
    M. Coons, “(Non)automaticity of number theoretic functions,” J. Théor. Nombres Bordeaux, 22 (2010), no. (2), 339–352.Google Scholar
  26. [26]
    A. H. Copeland and P. Erdős, “Note on normal numbers,” Bulletin of the American Mathematical Society, 52 (1946), 857–860.Google Scholar
  27. [27]
    R. E. Crandall, “The googol-th bit of the Erdős–Borwein constant,” Integers, A23, 2012.Google Scholar
  28. [28]
    M. Dekking, M. Mendès France, and A. van der Poorten, “Folds,” Math. Intelligencer 4 (1982), no. 3, 130–138.Google Scholar
  29. [29]
    M. Dekking, M. Mendès France, and A. van der Poorten, “Folds II,” Math. Intelligencer 4 (1982), no. 4, 173–181.Google Scholar
  30. [30]
    M. Dekking, M. Mendès France, and A. van der Poorten, “Folds III,” Math. Intelligencer 4(1982), no. (4), 190–195.Google Scholar
  31. [31]
    D. Y. Downham and S. B. Fotopoulos, “The transient behaviour of the simple random walk in the plane,” J. Appl. Probab. 25 (1988), no. 1, 58–69.Google Scholar
  32. [32]
    D. Y. Downham and S. B. Fotopoulos, “A note on the simple random walk in the plane,” Statist. Probab. Lett., 17 (1993), no. 3, 221–224.Google Scholar
  33. [33]
    A. Dvoretzky and P. Erdős, “Some problems on random walk in space,” Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, (1951), 353–367.Google Scholar
  34. [34]
    P. Hertling, “Simply normal numbers to different bases,” Journal of Universal Computer Science, 8, no. 2 (2002), 235–242.Google Scholar
  35. [35]
    H. J. Jeffrey, Chaos game representation of gene structure, Nucl. Acids Res. 18 no 2, (1990) 2163–2170.Google Scholar
  36. [36]
    B. D. Hughes, Random Walks and Random Environments, Vol. 1. Random Walks, Oxford Science Publications, New York, (1995).Google Scholar
  37. [37]
    H. Kaneko, “On normal numbers and powers of algebraic numbers,” Integers, 10 (2010), 31–64.Google Scholar
  38. [38]
    D. Khoshnevisan, “Normal numbers are normal,” Clay Mathematics Institute Annual Report (2006), 15 & 27–31.Google Scholar
  39. [39]
    G. Marsaglia, “On the randomness of pi and other decimal expansions,” preprint, 2010.Google Scholar
  40. [40]
    G. Martin, “Absolutely abnormal numbers,” Amer. Math. Monthly, 108 (2001), no. 8, 746-754.Google Scholar
  41. [41]
    J. Mah and J. Holdener, “When Thue–Morse meets Koch,” Fractals, 13 (2005), no. 3, 191–206.Google Scholar
  42. [42]
    S. M. Ross, Stochastic Processes. John Wiley & Sons, New York, 1983.Google Scholar
  43. [43]
    R. Stoneham, “On absolute \((j, \varepsilon)\)-normality in the rational fractions with applications to normal numbers,” Acta Arithmetica, 22 (1973), 277–286.Google Scholar
  44. [44]
    M. Queffelec, “Old and new results on normality,” Lecture Notes – Monograph Series, 48, Dynamics and Stochastics, 2006, Institute of Mathematical Statistics, 225–236.Google Scholar
  45. [45]
    W. Schmidt, “On normal numbers,” Pacific Journal of Mathematics, 10 (1960), 661–672.Google Scholar
  46. [46]
    A. J. Yee, “y-cruncher-multi-threaded pi program,”, 2010.
  47. [47]
    A. J. Yee and S. Kondo, “10 trillion digits of pi: A case study of summing hypergeometric series to high precision on multicore systems,” preprint, 2011, available at

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
    Email author
  • David H. Bailey
    • 2
  • Jonathan M. Borwein
    • 1
  • Peter B. Borwein
    • 3
  1. 1.Centre for Computer Assisted Research Mathematics and its Applications (CARMA)University of NewcastleCallaghanAustralia
  2. 2.Lawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.IRMACSSimon Fraser UniversityBurnabyCanada

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