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Fibonacci Numbers and the

Golden Ratio

Lecture Notes forJeffrey R. Chasnov

The Hong Kong University of Science and Technology

Department of Mathematics

Clear Water Bay, Kowloon

Hong KongCopyright

c○2016-2022 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

Preface

View the promotional video on YouTubeThese are my lecture notes for my online Coursera course,Fibonacci Numbers and the

Golden Ratio

. These lecture notes are divided into chapters called Lectures, and each Lecture corresponds to a video on Coursera. I have also uploaded the Coursera videos to YouTube, and links are placed at the top of each Lecture. Most of the Lectures also contain problems for students to solve. Less experienced students may find some of these problems difficult. Do not despair! The Lectures can be read and watched, and the material understood and enjoyed without actually solving any problems. But mathematicians do like to solve problems and I have selected those that I found to be interesting. Try some of them, but if you get stuck, full solutions can be read in the Appendix. My aim in writing these lecture notes was to place the mathematics at the level of an advanced high school student. Proof by mathematical induction and matrices, however, may be unfamiliar to a typical high school student and I have provided a short and hopefully readable discussion of these topics in the Appendix. Although all the material presented here can be considered elementary, I suspect that some, if not most, of the material may be unfamiliar to even professional mathematicians since Fibonacci numbers and the golden ratio are topics not usually covered in a University course. So I welcome both young and old, novice and experienced mathematicians to peruse these lecture notes, watch my lecture videos, solve some problems, and enjoy the wonders of the Fibonacci sequence and the golden ratio. For your interest, here are the links to my other online courses. If you are studying matrices and elementary linear algebra, have a look at

Matrix Algebra for Engineers

If your interests are differential equations, you may want to browse

Differential Equations for Engineers

For a course on multivariable calculus, try

Vector Calculus for Engineers

And for a course on numerical methods, enroll in

Numerical Methods for Engineers

Contents

I Fibonacci: It"s as Easy as1,1,2,311 The Fibonacci sequence2

2 The Fibonacci sequence redux

4

Practice quiz: The Fibonacci numbers

6

3 The golden ratio

7

4 Fibonacci numbers and the golden ratio

9

5 Binet"s formula

11

Practice quiz: The golden ratio

14 II Identities, Sums and Rectangles156 The Fibonacci Q-matrix16

7 Cassini"s identity

19

8 The Fibonacci bamboozlement

21

Practice quiz: The Fibonacci bamboozlement

24

9 Sum of Fibonacci numbers

25

10 Sum of Fibonacci numbers squared

27

Practice quiz: Fibonacci sums

29

11 The golden rectangle

30

12 Spiraling squares

32 III The Most Irrational Number3513 The golden spiral36

14 An inner golden rectangle

39

15 The Fibonacci spiral

42

Practice quiz: Spirals

44
iv

CONTENTSv

16 Fibonacci numbers in nature

45

17 Continued fractions

46

18 The golden angle

49

19 The growth of a sunflower

51

Practice quiz: Fibonacci numbers in nature

53 Appendices54A Mathematical induction55

B Matrix algebra

57

B.1 Addition and Multiplication

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

B.2 Determinants

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

C Problem and practice quiz solutions

60

Week I

Fibonacci: It"s as Easy as1,1,2,3

In this week"s lectures, we learn about the Fibonacci numbers, the golden ratio, and their relation- ship. We conclude the week by deriving the celebrated Binet"s formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. 1

Lecture 1| The Fibonacci

sequence View this lecture on YouTubeFibonacci published in the year 1202 his now famous rabbit puzzle: A man put a male-female pair of newly born rabbits in a field. Rabbits take a month to mature before mating. One month after mating, females give birth to one male-female pair and then mate again. No rabbits die. How many rabbit pairs are there after one year?

To solve, we construct Table

1.1 . At the start of each month, the number of juvenile pairs, adult pairs, and total number of pairs are shown. At the start of January, one pair of juvenile rabbits is introduced into the population. At the start of February, this pair of rabbits has matured. At the start of March, this pair has given birth to a new pair of juvenile rabbits. And so on.month J F M A M J J A S O N D J juvenile 1 0 1 1 2 3 5 8 13 21 34 55 89 adult 0 1 1 2 3 5 8 13 21 34 55 89 144 total 1 1 2 3 5 8 13 21 34 55 89 144 233

Table 1.1: Fibonacci"s rabbit population.

We define the Fibonacci numbersFnto be the total number of rabbit pairs at the start of thenth month. The number of rabbits pairs at the start of the 13th month,F13=233, can be taken as the solution to Fibonacci"s puzzle. Further examination of the Fibonacci numbers listed in Table 1.1 , reveals that these numbers satisfy the recursion relation F n+1=Fn+Fn1. (1.1) This recursion relation gives the next Fibonacci number as the sum of the preceeding two numbers. To start the recursion, we need to specifyF1andF2. In Fibonacci"s rabbit problem, the initial month starts with only one rabbit pair so thatF1=1. And this initial rabbit pair is newborn and takes one month to mature before mating soF2=1. The first few Fibonacci numbers, read from the table, are given by

1,1,2,3,5,8,13,21,34,55,89,144,233,...

and has become one of the most famous sequences in mathematics. 2

WEEK I. FIBONACCI: IT"S AS EASY AS1,1,2,33

Problems for Lecture 1

1.The Fibonacci numbers can be extended to zero and negative indices using the relation

F n=Fn+2Fn+1. DetermineF0and find a general formula forFnin terms ofFn. Prove your result using mathematical induction.

2.The Lucas numbers are closely related to the Fibonacci numbers and satisfy the same

recursion relationLn+1=Ln+Ln1, but with starting valuesL1=1 andL2=3. Deter- mine the first 12 Lucas numbers.

3.The generalized Fibonacci sequence satisfiesfn+1=fn+fn1with starting values

f

1=pandf2=q. Using mathematical induction, prove that

f n+2=Fnp+Fn+1q. (1.2)

4.Prove that

L n=Fn1+Fn+1. (1.3)

5.Prove that

F n=15 (Ln1+Ln+1).

6.The generating function for the Fibonacci sequence is given by the power series

f(x) =¥å n=1F nxn.

Assuming the power series converges, prove that

f(x) =x1xx2.

Solutions to the Problems

Lecture 2| The Fibonacci

sequence redux View this lecture on YouTubeWe can solve another puzzle that also leads to the Fibonacci sequence: How many ways can one climb a staircase withnsteps, taking one or two steps at a time? Any single climb can be represented by a string of ones and twos which sum ton. We defineanas the number of different strings that sum ton. In Table 1, we list the possible strings for the first five values ofn. It appears that thean"s form the beginning of the

Fibonacci sequence.

To derive a relationship betweenanand the Fibonacci numbers, consider the set of strings that sum ton. This set may be divided into two nonoverlapping subsets: those strings that start with one and those strings that start with two. For the subset of strings that start with one, the remaining part of the string must sum ton1; for the subset of strings that start with two, the remaining part of the string must sum ton2. Therefore, the number of strings that sum tonis equal to the number of strings that sum ton1 plus the number of strings that sum ton2. The number of strings that sum ton1 is given byan1and the number of strings that sum ton2 is given byan2, so that a n=an1+an2. And from the table we havea1=1=F2anda2=2=F3, so thatan=Fn+1for all positive integersn.nstringsan1 1 1

2 11, 2 2

3 111, 12, 21 3

4 1111, 112, 121, 211, 22 5

5 11111, 1112, 1121, 1211, 2111, 122, 212, 221 8

Table 2.1: Strings of ones and twos that add up ton. 4

WEEK I. FIBONACCI: IT"S AS EASY AS1,1,2,35

Problems for Lecture 2

1.Consider a string consisting of the firstnnatural numbers, 123...n. For each number

in the string, allow it to either stay fixed or change places with one of its neighbors. Define a nto be the number of different strings that can be formed. Examples for the first four values ofnare shown in Table2.2 . Prove thatan=Fn+1.nstringsan1 1 1

2 12, 21 2

3 123, 132, 213 3

4 1234, 1243, 1324, 2134, 2143 5

Table 2.2: Strings of natural numbers obtained by allowing a number to stay fixed or change places with its neighbor.

2.Consider a problem similar to that above, but now allow the first 1 to change places

with the lastn, as if the string lies on a circle. Supposen3, and definebnas the number of different strings that can be formed. Show thatbn=Ln, whereLnis thenth Lucas number.

Solutions to the Problems

Practice Quiz| The Fibonacci

numbers

1.In Fibonacci"s rabbit problem, the number of adult rabbit pairs in the fifth month is

a)1 b)2 c)3 d)5quotesdbs_dbs16.pdfusesText_22

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