Liouvilleova števila

diplomsko delo

Tina Perše (Author), Boštjan Kuzman (Mentor), Tadej Starčič (Co-mentor)

Abstract

V diplomskem delu se bomo najprej spoznali z realnimi števili in dvema delitvama te množice. Dobro poznana je delitev realnih števil na racionalna in iracionalna števila, omenili pa bomo tudi delitev na algebraična in transcendentna števila. V nadaljevanju se bomo osredotočili na zadnjo delitev in najprej pogledali, kako natančno lahko z racionalnimi števili aproksimiramo algebraična števila. Prav te ocene so namreč francoskega matematika Josepha Liouvillea pripeljale do konstrukcije Liouvilleovih števil, ki jih bomo definirali v nadaljevanju dela. Ta števila imajo v zgodovini matematike in razvoju teorije števil pomemben vpliv, saj so bila prva števila, za katera je bila dokazana transcendentna narava le-teh. Zatem bomo navedli dokaz o iracionalnosti in transcendentnosti Liouvilleovh števil. Pokazali bomo tudi, da lahko Liouvilleova števila zapišemo v obliki neskončnega verižnega ulomka. Sledi še dokaz, da lahko poljubno realno število zapišemo kot seštevek dveh Liouvilleovih števil. V zadnjem delu pa bomo Liouvilleova števila pogledali kot množico in pokazali nekaj pomembnih lastnosti. To so števnost, gostost v R in Lebesgueova mera. Glavni cilj tega dela je predstaviti Liouvilleova števila in podati lastnosti števil ter množice teh števil.

Keywords

Liouvilleovo število;transcendentno število;algebraično število;števnost;gostost;Lebesgueova mera;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[T. Perše]
UDC:	511.11(043.2)
COBISS:	12151625
Views:	498
Downloads:	108
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Liouville numbers
Secondary abstract:	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.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj: Matematika-tehnika
Pages:	31 str.
ID:	10973419