Elipsa pridružena broju 1/7

master's thesis

• Title (english): Ellipse associated with number 1/7
• Author: Mateja Razum
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu prikazana je i objašnjena sljedeća zanimljiva činjenica. Kad se od susjednih znamenaka perioda 142857 u decimalnom zapisu racionalnog broja 1=7 formira šest uređenih parova, uzimajući ciklički poredak te se uređeni parovi shvate kao koordinate točaka u pravokutnom kartezijevom sustavu, dobivenih šest točaka nalazi se na jednoj elipsi. Analogni rezultat dobiva se uzimanjem po dviju susjednih znamenaka navedenog perioda. Poopćenje ove pojave dovodi i do postupka za generiranje niza daljnjih primjera. U tu svrhu potrebno je uočiti simetriju promatranih skupova od po šest točaka i povezati je sa centralnosimetričnim krivuljama 2. reda, dakle centralnim konikama. U završnom poglavlju dan je pregled rezultata iz teorije algebarskih krivulja koji obuhvaćaju opisane primjere kao specijalne slučajeve.

Abstract (english)

In this paper, the following interesting fact is presented and explained. When six ordered pairs are formed from the adjacent digits of the period 142857 in the decimal expansion of the rational number 1=7, in a cyclic order, and the ordered pairs are interpreted as coordinates of points in a Cartesian coordinate system, these six points belong to an ellipse. An analogous result is obtained by taking pairs of adjacent digits of the same period. A generalization of these remarks leads to a method of generating a sequence of further examples. To that purpose, it is necessary to observe a symmetry in the described sets of points and establish a connection with centrally symmetric curves of second order, that is, central conics. The final chapter consists of a brief overview of results from the theory of algebraic curves that contain the previously exhibited examples as their special cases.

• Keywords: Kartezijev sustav; centralnosimetrične krivulje 2. reda; centralne konike; teorija algebarskih krivulja
• Keywords (english): Cartesian coordinate system; centrally symmetric curves of second order; central conics; theory of algebraic curves
• Language: croatian
• URN:NBN: urn:nbn:hr:217:665053
• 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: 2021-11-08 10:04:00