Download presentation

Presentation is loading. Please wait.

Published byCoby Courtney Modified over 3 years ago

1
Farey Sequences and Applications Brandon Kriesten

2
Outline Definition of a Farey Sequence Properties of Farey Sequences Construction of the Farey Sequence Theorem of two successive elements Number of Terms in a Farey Sequence Applications of Farey Sequences Ford Circles

3
Definition of a Farey Sequence A Farey Sequence is defined as all of the reduced fractions from [0,1] with denominators no larger than n, for all n in the domain of counting numbers, arranged in the order of increasing size.

4
Some Examples of Farey Sequences F_1 = {0/1, 1/1} F_1 = {0/1, 1/1} F_2 = {0/1, 1/2, 1/1} F_2 = {0/1, 1/2, 1/1} F_3 = {0/1,1/3,1/2,2/3,1/1} F_3 = {0/1,1/3,1/2,2/3,1/1} F_7 = {0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1} F_7 = {0/1, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 3/7, 1/2, 4/7, 3/5, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 1/1}

5
How to Construct the Farey Sequence To construct the Farey Sequence we must understand the concept of the mediant property. The mediant property is stated as follows: Given a Farey Sequence F_(n) with the adjacent elements a/b n the number will appear in a later sequence. Now from this definition and given F_2, construct F_3 and F4.

6
Theorem Theorem: Given a Farey sequence F_(n) with two successive elements (p_1)/(q_1) and (p_2)/(q_2), where (p_2)/(q_2) > (p_1)/(q_1), show that (p_2)/(q_2) - (p_1)/(q_1) = 1/(q_1*q_2).

7
Euler’s Totient Function To construct an equation for the number of elements in a Farey Sequence, we must first understand the concept of Euler’s Totient Function. Euler’s Totient function represents the number of positive integers less than a number n that are coprime to n. It is represented as the Greek symbol “phi” φ. E.g. φ (24) = 8, φ (6) = 2.

8
The number of Terms in a Farey Sequence The number of terms in a Farey Sequence is given by 1 plus the summation from k=1 to “n” of Euler’s Totient function. This can also be written as the number of elements in F_(n-1) + φ (n).

9
The Ford Circle

10
Ford Circles cont… A Ford Circle is defined as: For every rational number p/q where gcd(p,q) = 1, there exists a Ford Circle C(p,q) with a center (p/q, 1/2q^2) and a radius of 1/2q^2, that is tangent to the x axis at the point (p/q, 0).

11
Theorem Theorem: The Largest Ford circle between tangent Ford circles. Suppose that C(a, b) and C(c, d) are tangent Ford circles. Then the largest Ford circle between them is C(a + c, b + d), the Ford circle associated with the mediant fraction.

12
For Further Research http://mathworld.wolfram.com/FareySequence.h tml http://mathworld.wolfram.com/FareySequence.h tml http://golem.ph.utexas.edu/category/2008/06/far ey_sequences_and_the_sternb.html http://golem.ph.utexas.edu/category/2008/06/far ey_sequences_and_the_sternb.html http://math.stanford.edu/circle/FareyFord.pdf

Similar presentations

OK

The Real Number System. The natural numbers are the counting numbers, without _______. Whole numbers include the natural numbers and ____________. Integers.

The Real Number System. The natural numbers are the counting numbers, without _______. Whole numbers include the natural numbers and ____________. Integers.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on grease lubrication for o-rings Upload and view ppt online ticket Ppt on grease lubrication lines Ppt on content development tools Ppt on history of australian Ppt on special types of chromosomes genes Download ppt on great indian mathematicians Ppt on principles of object-oriented programming interview questions Ppt on power system stability analysis Free ppt on self development