Real Number Walks versus Algorithmic Random Walks

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This Demonstration compares an irrational number walk (based on its digital expansion) with algorithmic random walks. The irrational number walks are mathematical constants , , (the golden ratio), (Euler–Mascheroni constant), , log(2) and , where is a prime; they seem to be indistinguishable from algorithmic random walks.


Walks based on the digits of Liouville's constant and the like, which omit some digits entirely, clearly cannot be considered random at all.


Contributed by: Khoa Tran and Laila Zhexembay  (January 2017)
(Illinois Mathematics Summer REU Program 2016)
Open content licensed under CC BY-NC-SA



This Demonstration is based on [1], where walks are constructed based on real numbers as follows: the step of the walk is a unit step in direction , where is the digit of the expansion in base of the number. For example, in the case of base 2, the random walk is a one-dimensional walk that moves by for the digit 1 and by for the digit 0. Such real number walks can help visualize the randomness in the digits of famous irrational constants such as .

In the 3D case, the base is six, to match the six directions of the 3D axes.


[1] F. J. A. Artacho, D. H. Bailey, J. M. Borwein and P. B. Borwein, "Walking on Real Numbers," The Mathematical Intelligencer, 35(1), 2013 pp. 42–60. doi:10.1007/s00283-012-9340-x.

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