Konstrukcija realnih števil

Abstract

Keywords

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Language:	Slovenian
Year of publishing:	2013
Source:	Ljubljana
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[R. Grešak]
UDC:	51(043.2)
COBISS:	9801801
Views:	1243
Downloads:	195
Average score:	0 (0 votes)
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Secondary language:	English
Secondary title:	Construction of the real numbers
Secondary abstract:	In this thesis, there are described two standard constructions of the real numbers, these are the construction of real numbers via Dedekind cuts and the construction with metric fill of the rational numbers. Rational numbers are already a linearly ordered commutative field, so we first list the axioms of a linearly ordered commutative field. Then we take a look to the Dedekind's axiom, which only applies to real numbers and distinguishes between real and rational numbers. In thesis, there are proven some of these axioms. Beside that we prove the uniqueness theorem, which says that the set that satisfies axioms in linearly ordered commutative field, is unique up to isomorphism. Because these two constructions are quite complicated, we end thesis with method, how to explain construction of the real numbers to the secondary school students.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. Ljubljana, Pedagoška fak., Matematika-računalništvo
Pages:	23 str.
Type (ePrints):	thesis
Title (ePrints):	Construction of the real numbers
Keywords (ePrints):	realna števila
Keywords (ePrints, secondary language):	real numbers
Abstract (ePrints):	V diplomskem delu sta opisani dve standardni vpeljavi realnih števil, to je vpeljava s pomočjo Dedekindovih rezov in vpeljava z metrično napolnitvijo racionalnih števil. Že racionalna števila so linearno urejen komutativen obseg, zato najprej naštejemo aksiome v linearno urejenem komutativnem obsegu. Nato si ogledamo Dedekindov aksiom, ki velja le za realna števila in tako loči med realnimi in racionalnimi števili. V diplomskem delu je dokazanih nekaj teh aksiomov. Poleg tega pa dokažemo še izrek o enoličnosti realnih števil, ki pove, da je množica, ki zadosti aksiomom za linearno urejen komutativni obseg, do izomorfizma natančno ena sama. Ker sta ti dve vpeljavi precej zapleteni, si na koncu ogledamo še način, kako razložiti vpeljavo realnih števil dijakom v srednjih šolah.
Abstract (ePrints, secondary language):	In this thesis, there are described two standard constructions of the real numbers, these are the construction of real numbers via Dedekind cuts and the construction with metric fill of the rational numbers. Rational numbers are already a linearly ordered commutative field, so we first list the axioms of a linearly ordered commutative field. Then we take a look to the Dedekind's axiom, which only applies to real numbers and distinguishes between real and rational numbers. In thesis, there are proven some of these axioms. Beside that we prove the uniqueness theorem, which says that the set that satisfies axioms in linearly ordered commutative field, is unique up to isomorphism. Because these two constructions are quite complicated, we end thesis with method, how to explain construction of the real numbers to the secondary school students.
Keywords (ePrints, secondary language):	real numbers
ID:	8311800