Digits Walks of Transcendental Numbers

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A "walk" on the digits of a number can be used to evaluate the normality of a transcendental number in base 2 [1].


In this Demonstration a two-dimensional walk with unit steps is created by blocking the binary digits into pairs and assigning each of the four possible pairs to one of the four directions: northeast, southeast, southwest, and northwest. The walk of a base-2 normal number is expected to resemble a random walk and should not wander increasingly far from the origin.


Contributed by: Erik Mahieu (February 2012)
Open content licensed under CC BY-NC-SA



Normality of numbers is the subject of ongoing studies and a walk may only be a first indication of whether a number is normal.

Snapshot 1: was proved to be 2-normal and its walk wanders close to the origin

Snapshot 2: has a completely different patterned walk and was nevertheless proven to be 2-normal

Snapshot 3: has a walk similar to but has not proven to be normal yet


[1] A. Belshaw and P. Borwein. "Champernowne’s Number, Strong Normality, and the X Chromosome." Centre for Experimental and Constructive Mathematics. www.cecm.sfu.ca/personal/pborwein/PAPERS/P236.pdf.

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