The Irrationality via Diophantine Approximation and Mathematica (Study of Mathematical Software and Its Effective Use for Mathematics Education)

Abstract

In the present notes, we study the irrationality of a real number. We explain here Weyl's Criterion: whenever a real number $theta$ is irrational, then the fractional part {n $theta$} (n = 1, 2, 3, cdots) of n $theta$ is uniformly distributed mod 1. We try to adapt it to visualize the sequence for the purpose to observe what happens for certain real number $theta$ whose irrationality is expected but not yet proven. We investigate the behavior of the sequence from the viewpoint of Diophantine approximation. We use Mathematica software to see the phenomena.