An introduction to normal numbers

Abstract

This study is about normality of real numbers. In this study we will mainly look at the expansion of real numbers to any integer base b(b>1) and depending on that we will introduce the concept of normality. We will look at frequency of digit strings in the expansion of any real number to an integer base and if all possible digit strings of length k are equally frequent for each k in the former expansion, then we say the number is normal to the base b. While it is generally believed that many familiar irrational constants and algebraic irrationals are normal, normality has been proven only for numbers which are purposefully invented to be normal. In this study we will see different criteria for proving normality and also give an overview of the main results till the date. We will also give the complete proof of Borel's theorem i.e. Almost all real numbers are absolutely normal. Subsequently we will see some examples of normal numbers.