Continued fractions and their usage

Republic of Kazakhstan Ministry of Education and Science

Karagandy Regional Specialized Boarding School for Gifted Children “Daryn”

Direction: Research of the properties and

the area of usage of continued fraction

Section: Mathematics

Continued fractions and their usage

Made by:

Muhammedzhanova Dinargul

Myrzagul Aigerim 10 - class

Scientific leader: Akishev Gabdolla, Karaganda State University professer

Leader: Mekeeva Arailym

teacher of Mathematics

Kazakhstan 2014

Abstract

Purpose: To study t he theories of the continued fractions, their use and application, research properties.

The actuality. Continued fractions are used in Mathematical theories, as well and other natural areas of science such as physics, astronomy, botany, and in such arts as music and architecture.

Problems. 1) to study the properties of continued fractions in mathematical theories (algebra, geometry).

2) Conversion of continued fractions in simple and easy-to-chain; consideration using of continued fractions in the derivation of the square root, as well as in solving equations.

3) Determination of the differences and the relationship of simple and continued fractions in astronomy: Julian calendar, the Persian calendar (calendar Omar Khayyam) Gregorian, Midler and other calendars.

4) Using continued fractions in physics when creating circuits.

5) Using continued fractions in architecture, music and other arts.

Object. Continued fractions

News. The methods of application of continued fractions in the natural science field were researched.

Аннотация

Цель. Исследование теорий цепных дробей, их использование и применение, исследование свойств.

Актуальность темы. Цепные дроби используются в математических теориях, в естественно-научных областях науки, как физика, астрономия, ботаника, а также в таких видах искусства, как музыка, архитектура.

Задачи. 1) Исследование свойств цепных дробей в математике, в естественно-научных областях науки.

2) Превращение цепных дробей в простые и простых в цепные; рассмотрение (наблюдения) использования цепных дробей при выведении квадратного корня, а также при решении уравнений.

3) Определение различий и взаимосвязи простых и цепных дробей в астрономии: в календаре Юлиана, в персидском календаре (календарь Омар Хайяма), Гриагорианском и Мидлерском календарях.

4) Использование цепных дробей в физике при создании электрических цепей.

5) Использование цепных дробей в архитектуре, музыке и других видах искусства.

Объект. Цепные дроби.

Новизна. Рассмотрены методы применения цепных дробей в естественно-научных областях физики, кроме математики.

Contents

Introduction….……………………………………………...………………..……1

Chapter I……………………………………………………………………………2

Continued fractions and their usage ……………..……………………...…………2

§1. Scheme of the continued fractions…………………………………..…………..2

§2. Convergent fractions……………………………………………..……..……….6

§3. Some ways to use the continued fractions ……………………………….……11

§4. Finding a square root…………………………………………………..……….13

§5.Finding one pair roots of the equation performed through the continued

fractions………………………………………………………………/……….……15

Chapter II …………………………………………………………………….……17

§1. Continued fraction and calendar …….…………………………………….…..17

§2. Usage in physics …………………………………………………………….…19

§3. Usage in music …………………………………………………………....……21

§4. Usage of the continued fractions in other spheres ………….……………...….22

Conclusion ……...………………………………………………...……………..….23

Introduction

Chapter 1 is devoted to the definition of the term continued fraction and proving of statements about the its connection to the common fraction. Also there are proved statements about the features of the continued fractions. Examples on finding roots of continued fractions equations are given.

Chapter 2 considers the usage of continued fractions in other natural sciences. They were used to find the differences among Julian, Gregorian, Omar Khayyam’s and Midler’s calendars.

It is shown that it is possible to use continued fractions in physics to find the method of connection electric conductors to each other. Also the are given examples on usage of continued fractions in botany, mechanics and architecture.

Chapter І

Continued fractions and their usage

Дата добавления: 2015-11-14