A computational criterion for the irrationality of some real numbers

TL;DR

This paper introduces a criterion based on the asymptotic average of decimal digits to determine the irrationality of certain real numbers, providing a new approach to understanding their decimal expansions.

Contribution

It establishes a novel irrationality criterion using asymptotic averages of decimal digits and characterizes the averages for simply normal numbers.

Findings

Numbers with zero asymptotic average of decimals are irrational.

Simply normal numbers have an asymptotic average of 9/2.

The criterion applies to non-finite decimal numbers.

Abstract

In this paper, we compute the asymptotic average of the decimals of some real numbers. With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is zero, then it is irrational. We also show that the asymptotic average of the decimals of simply normal numbers is 9/2.