# Perspectives in Mathematics I

Lecturer Kentaro ITO, Associate Professor School of Science / Graduate School of Science, 2015 Spring 1st year students (2・週1回全15回)

### Course Overview

You will learn how to perceive rational numbers in a completely different way than in high school and to think in a more geometric manner. Specifically, we will discuss the order of continued fractions, Ford circles and hyperbolic geometry. Continued fractions are rational numbers presented in a nested fraction. They are not suitable for calculation but they have many interesting characteristics and provide a new perspective on rational and irrational numbers. Then you will learn about Ford circles corresponding to rational numbers. By using Ford circles, we can understand the geometric relationship between rational numbers.

Also, Ford circles connect well with continued fractions. The relationship between continued fractions and Ford circles mentioned above can be understood through hyperbolic geometry. We will discuss this towards the end of the lecture series. Hyperbolic geometry is different from the normal Euclidean geometry in that it works without the assumption of the "parallel postulate". We will do an introductory explanation on hyperbolic geometry using the inversion of a circle.

### Key features

I chose a fun topic which does not need any background knowledge and can be understood using knowledge from high school. Continued fractions and Ford circles are not taught in high school or in university, but can be very easily explained, while leading to a lot of themes present in modern mathematics. Therefore, I thought this topic was the perfect choice for this lecture. Although Ford circles are conceptually quite simple, I have not seen any Japanese books explaining them so I thought this would be a good time to introduce them. The class is designed to be easily visually understood by using extensive graphical representations. Also I will show a computer animation in which Ford circles converge into irrational numbers.

### Keywords

Rational numbers, irrational numbers, continued fractions, the golden ratio, Fibonacci series, Ford circles, heyperbolicgeometry,inversion of circles.

### Level of academic knowledge required for this course

Knowledge of scientific mathematics taught in high school

This course can be taken by students from different schools and departments.
The course is open to all, we welcome all students who are interested to come and take the course.

### Course Objectives

I have chosen to teach a mathematical topic which is not normally taught in either high school or university and does not require prior knowledge. The purpose of this course is to promote interest in mathematical phenomena. Specifically, we will be learning topics including continued fractions, Ford circles and hyperbolic geometry.

### Schedule

Session Contents]
Part 1 Continued fractions
1 What is a continued fraction?
2 The golden ratio as a continued fraction
3 Irrational numbers as continued fraction expansions
4 Convergence of fraction sequences
5 Two dimensions of irrational numbers
6 Summary of continued fractions
Part 2 Ford circles
7 Ford circles
8 Good convergence (part 1)
9 Good convergence (part 2)
10 Hurwitz's theorem
11 Summary of Ford circles
Part 3 Hyperbolic Geometry
12 Inversion in a circle
13 Hyperboloids
14 Hyperboloids and ford circles
15 Summary of hyperbolic geometry

### Course Evaluation

Grades will be evaluated through reports. There will be more than 30 problems, students may choose whichever problems they can solve and submit these as a report. Grades will be given based on how well the report is and a passing mark will be given for 10 correct answers.

#### Last updated

May 10, 2020

OCW website, and all course materials made public on this site,
may be used under the terms and conditions posted under "About OCW"
© 2020 Nagoya University Open Course Ware