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.
