Statistical Comparison of Decimal Expansions of Irrational Numbers with π and e

It is indeed possible for the decimal expansion of an irrational number to differ statistically from that of well-known constants such as π (pi) and e (Euler's number). This is a fascinating and complex area in the realm of mathematics, particularly number theory. In this article, we will delve into the statistical distribution of digits in irrational numbers, compare the expansions of different irrationals, and discuss the current understanding based on empirical evidence and theoretical underpinnings.

Statistical Distribution of Digits

Mathematically, the decimal expansions of irrational numbers like π and e are believed to be uniformly distributed. This property means that each of the digits from 0 to 9 appear with equal probability in the long run, and the sequences of digits are considered random. This randomness is often referred to as a normal number.

In simpler terms, the frequency of each digit in the long decimal expansion of π or e should approach 10% as the number of digits increases. For example, in the expansion of π, the digit '2' should appear approximately 10% of the time, just like the digit '5', and so on for all other digits from 0 to 9. This concept is not restricted to π and e alone, but applies to other irrational numbers as well, as long as they are proven to be normal numbers.

Comparison Among Irrationals

From a statistical viewpoint, there is no fundamental reason to expect the decimal expansion of any irrational number to differ significantly from that of π or e, unless specific properties of the number suggest otherwise. For instance, some irrational numbers are constructed with specific digit patterns, such as the Champernowne Constant, which is formed by concatenating whole numbers. This specific construction makes the Champernowne Constant known to be a normal number, but it is not to be confused with π or e in terms of digit distribution.

Empirical Evidence

Empirical investigations into the decimal expansions of π, e, and other irrationals like √2 or the golden ratio (φ) often look for anomalies in digit frequencies, the occurrence of specific sequences, or other statistical properties. So far, extensive computational evidence has not revealed significant deviations in the statistical behavior of these constants from what would be expected if they were truly normal. In other words, the digits in π and e appear random and have the frequency distribution expected for a normal number.

Theoretical Underpinnings

Theoretically, if π and e are indeed normal, as strongly suspected but not yet proven, their decimal expansions would not only appear random but would contain every possible finite sequence of digits. This would mean that their expansions would be no different in a statistical sense from any other normal irrational number's expansion. For instance, in the limit, the sequence "58291437" should appear in the expansion of π just as frequently as any other sequence of the same length.

Mathematicians continue to explore these properties, making the study of the digit expansions of irrational numbers an active area of research. Statistical methods and computational tools are continuously being developed to test the hypothesis of normality and to explore the nature of these fascinating constants further.

In conclusion, based on current understanding and computational evidence, the decimal expansions of irrational numbers like π and e do not differ statistically in a significant way from the expansions of other irrational numbers, assuming all are normal. Each of these numbers, if normal, should exhibit the property that every finite sequence of digits appears with the frequency expected from a truly random sequence.