In this diploma thesis we first present real numbers and the two divisions of the set of real numbers. It is well known that real numbers are divided into rational and irrational numbers, yet not so commonly known is the division into algebraic and transcendental numbers. We will focus on the last division and first take a look at how accurately the rational numbers can approximate the algebraic numbers. It was these estimates that later led the French mathematician Joseph Liouville to the construction of the Liouville numbers, which have been very influential in the history of mathematic and the development of the number theory, because they were the first numbers proven to be transcendental. In the thesis it will be proved all Liouville numbers are irrational, transcendental and can be represented also as infinite continued fraction.We will also show that any real number can be represented as a sum and a product of two Liouville numbers. In the last part we will show some important characteristics of the set of Liouville numbers, such as countability, density in R and Lebesgue measure. The main goal of this thesis is to present the Liouville numbers to general public and to present some important characteristics of the Liouville numbers and the set of the Liouville numbers.
|
