Množica transcedentnih realnih števil : diplomsko delo

Miklavec, Tilen (author), Šparl, Primož (mentor)

Realna števila delimo na racionalna in iracionalna števila. To delitev učenci spoznajo že v osnovni šoli, nekoliko bolj pa v srednji šoli. Precej manj znano je, da realna števila delimo tudi na algebraična in transcendentna števila. Realna števila, ki so ničle kakšnega polinoma z racionalnimi koeficienti, imenujemo algebraična števila. Po drugi strani realna števila, ki niso ničle nobenega takšnega polinoma, imenujemo transcendentna števila. Ta delitev realnih števil na algebraična in transcendentna števila predstavlja glavno temo diplomskega dela. Tako algebraičnih kot transcendentnih realnih števil je neskončno, a kljub neskončnosti enih in drugih lahko določimo, katerih je več. Glavni cilj diplomskega dela je predstavitev dokaza, da je transcendentnih realnih števil bistveno več kot algebraičnih, kar se morda zdi presenetljivo, saj v osnovni in srednji šoli večinoma operiramo le z algebraičnimi realnimi števili. Ce nekoliko poenostavimo, sta namreč edini transcendentni realni števili, ki ju spoznamo v petnajstih letih šolanja, razmerje med obsegom in premerom kroga, torej število π, in osnova naravnega logaritma, število e.

• Language: Slovenian
• Keywords: algebraična števila, neskončne množice
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [T. Miklavec]
• Year: 2017
• Number of pages: 27 str.
• PID: 20.500.12556/RUL-95169
• UDC: 51(043.2)
• COBISS.SI-ID: 11696457
• Publication date in RUL: 19.09.2017

MIKLAVEC, Tilen, 2017, Množica transcedentnih realnih števil : diplomsko delo [online]. Bachelor’s thesis. T. Miklavec. [Accessed 8 April 2025]. Retrieved from: http://pefprints.pef.uni-lj.si/id/eprint/4659

Secondary language

• Language: English
• Title: The set of transcendental real numbers

Abstract

Real numbers are divided into rational and irrational numbers. Students learn about this division already in elementary school, but they become more familiar with it in secondary school. It is not so well known that real numbers are also divided into algebraic and transcendental numbers. Real numbers which are zeros of some polynomial with rational coefficients are called algebraic numbers. On the other hand, real numbers that are not zeros of any such polynomial are called transcendental numbers. This division of real numbers into algebraic and transcendental numbers represents the main topic of this diploma thesis. The set of algebraic real numbers and the set of transcendental real numbers are both infinite, but despite their infinity, we can still determine which is larger. The main goal of this diploma thesis is to present the proof that there are significantly more transcendental real numbers than algebraic real numbers, what may seem surprising, since we mostly operate with algebraic real numbers in elementary and secondary schools. Namely, almost the only transcendental real numbers that we learn about at school are the ratio between the circumference and the diameter of the circle, that is, the number π, and the basis of the natural logarithm, the number e.

• Keywords: mathematics, matematika