Decimalni zapis realnog broja

master's thesis

• Title (english): Decimal notation of a real number
• Author: Nataša Gložinić
• Mentor: Andrej Dujella (mentor)
• Committee member: Andrej Dujella (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Decimalni zapis realnog broja često je prisutan u svakodnevnim situacijama. Učenici se u nastavi s decimalnim zapisom prvi puta susreću u petom razredu, a u kasnijim godinama upoznaju neka svojstva decimalnog zapisa realnih brojeva. U prvom poglavlju opisan je decimalni zapis realnog broja. Dokazano je da svaki realan broj ima barem jedan zapis u decimalnom obliku te je dan algoritam za određivanje decimalnog zapisa. Drugo poglavlje sadrži decimalni zapis prirodnih i cijelih brojeva te dokaz da svaki prirodan broj ima decimalni prikaz. U trećem je poglavlju razrađen decimalni zapis racionalnih brojeva. Iskazana je i dokazana karakterizacija racionalnih brojeva promatranjem perioda decimalnog zapisa. Također su iskazana i dokazana te primjerima potkrijepljena svojstva konačnosti i periodičnosti decimalnog zapisa racionalnih brojeva. U kratkim je crtama razmatrana duljina perioda i pretperioda. Četvrto poglavlje bavi se decimalnim zapisom iracionalnih brojeva pa je iskazana karakterizacija iracionalnih brojeva promatranjem decimalnog zapisa te je dokazana iracionalnost Champernowneove konstante. U petom su poglavlju dane odabrane zanimljivosti o decimalnom zapisu realnih brojeva poput cikličkih permutacija decimala te Midyjev teorem. Šesto, ujedno i posljednje poglavlje, bavi se decimalnim zapisom realnih brojeva u nastavi matematike. Ukratko su opisani propisani ishodi učenja te riješeni odabrani primjeri zadataka korištenih u nastavi matematike u osnovnoškolskom i srednjoškolskom obrazovanju.

Abstract (english)

The decimal notation of a real number is often present in everyday situations. Students encounter decimal notation for the first time in the fifth grade and in later years they get to know some properties of the decimal notation of real numbers. The decimal notation of a real number is described in the first chapter. It is proved that every real number has at least one notation in decimal form and an algorithm for determining the decimal notation is given as well. The second chapter contains the decimal notation of natural numbers and integers and a proof that every natural number has a decimal representation. In the third chapter, the decimal notation of rational numbers is elaborated. The characterization of rational numbers is presented and proven by observing the period of the decimal notation. The properties of finiteness and periodicity of the decimal notation of rational numbers are also stated and proved and supported by examples. The length of the period and the preperiod was briefly considered. The fourth chapter deals with the decimal notation of irrational numbers, so the characterization of irrational numbers is shown by observing the decimal notation and the irrationality of Champernowne’s constant is proven. In the fifth chapter, selected interesting facts about the decimal notation of real numbers are given, such as cyclic permutations of decimals and Midy’s theorem. The sixth, also the last chapter, deals with the decimal notation of real numbers in mathematics lessons. Prescribed learning outcomes are briefly described and selected examples of tasks used in mathematics teaching in primary and secondary education are solved.

• Keywords: algoritam za određivanje decimalnog zapisa; decimalni zapis prirodnih i cijelih brojeva; decimalni zapis racionalnih brojeva; decimalnim zapisom iracionalnih brojeva; iracionalnost Champernowneove konstante; Midyjev teorem
• Keywords (english): algorithm for determining the decimal notation; decimal notation of natural and integers; decimal notation of rational numbers; decimal notation of irrational numbers; irrationality of Champernowne's constant; Midy's theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:877473
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-12-14 13:37:01