4.8
stars
1,057 ratings

Jeffrey R. Chasnov

Top Instructor

34,072 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

10,685 recent views

Learn the mathematics behind the Fibonacci numbers, the golden ratio, and their relationship to each other. These topics may not be taught as part of a typical math curriculum, but they contain many fascinating results that are still accessible to an advanced high school student.  
The course culminates in an exploration of the Fibonacci numbers appearing unexpectedly in nature, such as the number of spirals in the head of a sunflower.

Download the lecture notes from the link
https://www.math.hkust.edu.hk/~machas/fibonacci.pdf

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

Flexible deadlines

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Beginner Level

High school mathematics

Approx. 9 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.

Coursera Labs

Includes hands on learning projects.

Learn more about Coursera Labs

Beginner Level

High school mathematics

Approx. 9 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 (291 Ratings)

Jeffrey R. Chasnov
Top Instructor

Professor

Department of Mathematics

172,602 Learners

16 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%(3,036 ratings)

Week1

Week 1

3 hours to complete

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

3 hours to complete

6 videos (Total 48 min), 8 readings, 4 quizzes

6 videos

8 readings

Welcome and Course Information1mFibonacci Numbers with Negative Indices5mThe Lucas Numbers5mNeighbour Swapping10mSome Algebra Practice10mLinearization of Powers of the Golden Ratio10mAnother Derivation of Binet's formula10mBinet's Formula for the Lucas Numbers10m

4 practice exercises

Diagnostic Quiz5mThe Fibonacci Numbers15mThe Golden Ratio15mWeek 1 Assessment30m

Week2

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 the famous dissection fallacy, 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 spiraling squares. This image is a drawing of a sequence of squares, each with side lengths equal to the golden ratio conjugate raised to an integer power, creating a visually appealing and mathematically intriguing pattern.

3 hours to complete

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

9 videos

10 readings

Do You Know Matrices?The Fibonacci Addition Formula10mThe Fibonacci Double Index Formula10mDo You Know Determinants?Proof of Cassini's Identity10mCatalan's Identity10mSum of Lucas Numbers5mSums of Even and Odd Fibonacci Numbers5mSum of Lucas Numbers Squared5mArea of the Spiraling Squares10m

3 practice exercises

The Fibonacci Bamboozlement15mFibonacci Sums15mWeek 2 Assessment30m

Week3

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 recognize 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, which is related to the golden ratio, and use it to model the growth of a sunflower head. The 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

8 readings

The Eye of God30mArea of the Inner Golden Rectangle10mContinued Fractions for Square Roots10mContinued Fraction for e10mThe Golden Ratio and the Ratio of Fibonacci Numbers10mThe Golden Angle and the Ratio of Fibonacci Numbers10mPlease Rate this Course1mAcknowledgments

3 practice exercises

Spirals15mFibonacci Numbers in Nature15mWeek 3 Assessment30m

Reviews

4.8

361 reviews

5 stars
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3 stars
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2 stars
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1 star
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TOP REVIEWS FROM FIBONACCI NUMBERS AND THE GOLDEN RATIO

by AKMar 22, 2019

Absolutely loved the content discussed in this course! It was challenging but totally worth the effort. Seeing how numbers, patterns and functions pop up in nature was a real eye opener.

by JRJul 12, 2020

Someone has said that God created the integers; all the rest is the work of man. After seeing how the Fibonacci numbers play out in nature, I am not so sure about that. A very enjoyable course.

by BSAug 29, 2017

Very well designed. It was a lot of fun taking this course. It's the kind of course that can get you excited about higher mathematics. Sincere thanks to Prof. Chasnov and HKUST.

by GMMar 15, 2017

Finally I studied the Fibonacci sequence and the golden spiral. I used to say: one day I will.

Very interesting course and made simple by the teacher in spite of the challenging topics

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