A note on normality of $\sqrt{2}$ in base 2

TL;DR

This paper introduces a new geometric approach to test the normality of numbers in base 2, proving that an infinite class of numbers, including √2, are normal in base 2.

Contribution

It presents a novel method linking normality to geometric properties of associated vectors and proves √2's normality in base 2 for the first time.

Findings

An infinite class of numbers is normal in base 2.

√2 is proven to be normal in base 2.

A general approach for normality testing is developed.

Abstract

In this paper we study the property of normality of a number in base 2. A simple rule that associates a vector to a number is presented and the property of normality is stated for the vector associated to the number. The problem of testing a number for normality is shown to be equivalent to the test of geometrical properties of the associated vector. The paper provides a general approach for normality testing and then applies the proposed methodology to the study of particular numbers. The main result of the paper is to prove that an infinite class of numbers is normal in base 2. As a further result we prove that the irrational number $\sqrt{2}$ is normal in base 2.