4.8
stars
1,048 ratings

Jeffrey R. Chasnov

Top Instructor

33,568 already enrolled

Offered By

The Hong Kong University of Science and Technology

Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology

About this Course

12,795 recent views

Learn the mathematics behind the Fibonacci numbers, the golden ratio, and how they are related. These topics are not usually taught in a typical math curriculum, yet contain many fascinating results that are still accessible to an advanced high school student.  
The course culminates in an explanation of why the Fibonacci numbers appear unexpectedly in nature, such as the number of spirals in the head of a sunflower.

Download the lecture notes:
https://www.math.ust.hk/~machas/fibonacci.pdf

Watch the promotional video:
https://youtu.be/VWXeDFyB1hc

Flexible deadlines

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Shareable Certificate

Earn a Certificate upon completion

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Beginner Level

High school mathematics

Approx. 10 hours to complete

English

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What you will learn

Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions

Skills you will gain

Recreational Mathematics
Discrete Mathematics
Elementary Mathematics

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Beginner Level

High school mathematics

Approx. 10 hours to complete

English

Could your company benefit from training employees on in-demand skills?

Try Coursera for Business

Instructor

Instructor rating

4.79/5 (284 Ratings)

Jeffrey R. Chasnov
Top Instructor

Professor

Department of Mathematics

167,261 Learners

6 Courses

Offered by

The Hong Kong University of Science and Technology

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Syllabus - What you will learn from this course

Content Rating97%(3,030 ratings)

Week

Week 1

3 hours to complete

Fibonacci: It's as easy as 1, 1, 2, 3

3 hours to complete

7 videos (Total 55 min), 8 readings, 4 quizzes

7 videos

Promotional Video6m

The Fibonacci Sequence | Lecture 18m

The Fibonacci Sequence Redux | Lecture 27m

The Golden Ratio | Lecture 38m

Fibonacci Numbers and the Golden Ratio | Lecture 46m

Binet's Formula | Lecture 510m

Mathematical Induction7m

8 readings

Welcome and Course Information1m

Fibonacci Numbers with Negative Indices5m

The Lucas Numbers5m

Neighbour Swapping10m

Some Algebra Practice10m

Linearization of Powers of the Golden Ratio10m

Another Derivation of Binet's formula10m

Binet's Formula for the Lucas Numbers10m

4 practice exercises

Diagnostic Quiz5m

The Fibonacci Numbers15m

The Golden Ratio15m

Week 1 Assessment30m

Week

Week 2

3 hours to complete

Identities, sums and rectangles

We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for a famous dissection fallacy colourfully named the Fibonacci bamboozlement. A dissection fallacy is an apparent paradox arising from two arrangements of different area from one set of puzzle pieces. We also derive formulas for the sum of the first n Fibonacci numbers, and the sum of the first n Fibonacci numbers squared. Finally, we show how to construct a golden rectangle, and how this leads to the beautiful image of spiralling squares.

3 hours to complete

9 videos (Total 65 min), 10 readings, 3 quizzes

9 videos

The Fibonacci Q-matrix | Lecture 611m

Cassini's Identity | Lecture 78m

The Fibonacci Bamboozlement | Lecture 86m

Sum of Fibonacci Numbers | Lecture 98m

Sum of Fibonacci Numbers Squared | Lecture 107m

The Golden Rectangle | Lecture 115m

Spiraling Squares | Lecture 123m

Matrix Algebra: Addition and Multiplication5m

Matrix Algebra: Determinants7m

10 readings

Do You Know Matrices?

The Fibonacci Addition Formula10m

The Fibonacci Double Index Formula10m

Do You Know Determinants?

Proof of Cassini's Identity10m

Catalan's Identity10m

Sum of Lucas Numbers5m

Sums of Even and Odd Fibonacci Numbers5m

Sum of Lucas Numbers Squared5m

Area of the Spiraling Squares10m

3 practice exercises

The Fibonacci Bamboozlement15m

Fibonacci Sums15m

Week 2 Assessment30m

Week

Week 3

3 hours to complete

The most irrational number

We learn about the golden spiral and the Fibonacci spiral. Because of the relationship between the Fibonacci numbers and the golden ratio, the Fibonacci spiral eventually converges to the golden spiral. You will recognise the Fibonacci spiral because it is the icon of our course. We next learn about continued fractions. To construct a continued fraction is to construct a sequence of rational numbers that converges to a target irrational number. The golden ratio is the irrational number whose continued fraction converges the slowest. We say that the golden ratio is the irrational number that is the most difficult to approximate by a rational number, or that the golden ratio is the most irrational of the irrational numbers. We then define the golden angle, related to the golden ratio, and use it to model the growth of a sunflower head. Use of the golden angle in the model allows a fine packing of the florets, and results in the unexpected appearance of the Fibonacci numbers in the sunflower.

3 hours to complete

8 videos (Total 61 min), 8 readings, 3 quizzes

8 videos

The Golden Spiral | Lecture 138m

An Inner Golden Rectangle | Lecture 145m

The Fibonacci Spiral | Lecture 156m

Fibonacci Numbers in Nature | Lecture 164m

Continued Fractions | Lecture 1715m

The Golden Angle | Lecture 187m

A Simple Model for the Growth of a Sunflower | Lecture 198m

Concluding remarks4m

8 readings

The Eye of God30m

Area of the Inner Golden Rectangle10m

Continued Fractions for Square Roots10m

Continued Fraction for e10m

The Golden Ratio and the Ratio of Fibonacci Numbers10m

The Golden Angle and the Ratio of Fibonacci Numbers10m

Please Rate this Course1m

Acknowledgments

3 practice exercises

Spirals15m

Fibonacci Numbers in Nature15m

Week 3 Assessment30m

Reviews

4.8

359 reviews

5 stars
82.12%
4 stars
15.49%
3 stars
1.80%
2 stars
0.47%
1 star
0.09%

TOP REVIEWS FROM FIBONACCI NUMBERS AND THE GOLDEN RATIO

by UBJun 5, 2018

This course blends the rational thinking of mathematics and the aesthetics of art in nature. Enjoyed it. Thanks Coursera and Hong Kong University of Science and Technology

by CNFeb 13, 2021

Very neat and well organized, all material at hand. I liked the skipped math bits that the others mentioned, so that I could myself engage in figuring out.

by MHJul 12, 2020

It was a wonderful experience. I have learnt a lot more about Fibonacci numbers and the golden ratio during this course. I urge the professor to create more and more courses like this.

by HRNov 7, 2021

The course looks at perfection level, clear and fun. The patience and the generosity of Professor Chesnov throughout the whole course are commendable.

View all reviews

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Fibonacci Numbers and the Golden Ratio

Fibonacci Numbers and the Golden Ratio

About this Course

What you will learn

Skills you will gain