This is a preview of subscription content, access via your institution.

## References

- [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. - [2]
J.-P. Allouche and J. Shallit,

*Automatic Sequences: Theory, Applications, Generalizations*. Cambridge University Press, Cambridge, 2003. - [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. - [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. - [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. - [6]
M. Barnsley,

*Fractals Everywhere*, Academic Press, Inc., Boston, MA, 1988. - [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. - [8]
D. H. Bailey and D. J. Broadhurst, “Parallel integer relation detection: Techniques and applications”.

*Mathematics of Computation*,**70**, no. 236 (2000), 1719–1736. - [9]
D. H. Bailey and R. E. Crandall, “On the random character of fundamental constant expansions”.

*Experimental Mathematics*,**10**, no. 2 (2001), 175–190. - [10]
D. H. Bailey and R. E. Crandall, “Random generators and normal numbers,”

*Experimental Mathematics*,**11**(2002), no. 4, 527–546. - [11]
D. H. Bailey and M. Misiurewicz, “A strong hot spot theorem,”

*Proceedings of the American Mathematical Society*,**134**(2006), no. 9, 2495–2501. - [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. - [13]
M. N. Barber and B. W. Ninham,

*Random and Restricted Walks: Theory and Applications*, Gordon and Breach, New York, 1970. - [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. - [15]
L. Berggren, J. M. Borwein, and P. B. Borwein,

*Pi: a Source Book*, Springer-Verlag, Third Edition, 2004. - [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. - [17]
J. Borwein, D. Bailey, N. Calkin, R. Girgensohn, R. Luke, V. Moll,

*Experimental Mathematics in Action*. A. K. Peters, Natick, MA, 2007. - [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. - [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. - [20]
P. B. Borwein, “On the irrationality of certain series.”

*Math. Proc. Cambridge Philos. Soc.***112**(1992) 141–146. - [21]
P. B. Borwein and L. Jörgenson, “ Visible structures in number theory,”

*Amer. Math. Monthly***108**(2001), no. 10, 897–910. - [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. - [23]
C.S. Calude,

*Information and Randomness: An Algorithmic Perspective*, 2nd ed., Revised and Extended, Springer-Verlag, Berlin, 2002. - [24]
D. G. Champernowne, “The construction of decimals normal in the scale of ten.”

*Journal of the London Mathematical Society*,**8**(1933) 254–260. - [25]
M. Coons, “(Non)automaticity of number theoretic functions,”

*J. Théor. Nombres Bordeaux*,**22**(2010), no. (2), 339–352. - [26]
A. H. Copeland and P. Erdős, “Note on normal numbers,”

*Bulletin of the American Mathematical Society*,**52**(1946), 857–860. - [27]
R. E. Crandall, “The googol-th bit of the Erdős–Borwein constant,”

*Integers*, A23, 2012. - [28]
M. Dekking, M. Mendès France, and A. van der Poorten, “Folds,”

*Math. Intelligencer***4**(1982), no. 3, 130–138. - [29]
M. Dekking, M. Mendès France, and A. van der Poorten, “Folds II,”

*Math. Intelligencer***4**(1982), no. 4, 173–181. - [30]
M. Dekking, M. Mendès France, and A. van der Poorten, “Folds III,”

*Math. Intelligencer***4**(1982), no. (4), 190–195. - [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. - [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. - [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. - [34]
P. Hertling, “Simply normal numbers to different bases,”

*Journal of Universal Computer Science*,**8**, no. 2 (2002), 235–242. - [35]
H. J. Jeffrey, Chaos game representation of gene structure,

*Nucl. Acids Res.***18**no 2, (1990) 2163–2170. - [36]
B. D. Hughes,

*Random Walks and Random Environments*,*Vol. 1. Random Walks*, Oxford Science Publications, New York, (1995). - [37]
H. Kaneko, “On normal numbers and powers of algebraic numbers,”

*Integers*,**10**(2010), 31–64. - [38]
D. Khoshnevisan, “Normal numbers are normal,”

*Clay Mathematics Institute Annual Report*(2006), 15 & 27–31. - [39]
G. Marsaglia, “On the randomness of pi and other decimal expansions,” preprint, 2010.

- [40]
G. Martin, “Absolutely abnormal numbers,”

*Amer. Math. Monthly*,**108**(2001), no. 8, 746-754. - [41]
J. Mah and J. Holdener, “When Thue–Morse meets Koch,”

*Fractals*,**13**(2005), no. 3, 191–206. - [42]
S. M. Ross,

*Stochastic Processes*. John Wiley & Sons, New York, 1983. - [43]
R. Stoneham, “On absolute \((j, \varepsilon)\)-normality in the rational fractions with applications to normal numbers,”

*Acta Arithmetica*,**22**(1973), 277–286. - [44]
M. Queffelec, “Old and new results on normality,”

*Lecture Notes – Monograph Series*,**48**,*Dynamics and Stochastics*, 2006, Institute of Mathematical Statistics, 225–236. - [45]
W. Schmidt, “On normal numbers,”

*Pacific Journal of Mathematics*,**10**(1960), 661–672. - [46]
A. J. Yee, “y-cruncher-multi-threaded pi program,” http://www.numberworld.org/y-cruncher, 2010.

- [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 http://hdl.handle.net/2142/28348.

## Author information

### Affiliations

### Corresponding author

## Additional information

Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231.

## Rights and permissions

## About this article

### Cite this article

Aragón Artacho, F.J., Bailey, D.H., Borwein, J.M. *et al.* Walking on Real Numbers.
*Math Intelligencer* **35, **42–60 (2013). https://doi.org/10.1007/s00283-012-9340-x

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Random Walk
- Rational Number
- Continue Fraction
- Mathematical Intelligencer
- Simple Random Walk