Experimental Analysis on the Normality of Using Advanced Data-Mining Techniques (Experimental Analysis on the Normality of pi, e, phi, and square root of 2 Using Advanced Data-Mining Techniques)

Abstract

The main focus of the work described in this paper is to examine whether the famous mathematical constants are normal numbers. We have conducted extensive experiments with different attributes for each constant using advanced data-mining techniques, and we have tried to express a theoretical model that can help to determine with high probability whether the numbers are normal in base ten. We have expanded and generalized the experimental results so as to formulate conjectures about the attributes of a normal number, and we have presented conjectures that can lead to determining whether a number is normal. The experimental results and analysis have shown that indeed, satisfy the definition of a normal number. Not only does the distribution of each of the base-10 digits occur with frequency approximately one-tenth, as is known already for very large sequences, but we have also shown for the first time that all arrangements with repetition of digits up to a specific length, depending on the number of decimals examined, occur also with the expected frequencies. As a result, we establish a new process to examine not only whether these constants are just simply normal but normal as well.

2000 AMS Subject Classification::

• LERP
• 11A41
• 11A55
• 68T01
• 65Y99

Keywords:

• Normal number
• random string
• irrational numbers
• pi
• Euler number
• phi
• square root of 2
• longest expected repeated pattern

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Corrigendum

Notes

¹On 28 December 2013, after this article was submitted, Alexander J. Yee and Shigeru Kondo announced the computation of 12.1 trillion digits of π; see http://www.numberworld.org/misc_runs/pi-12t/.