4.8
stars
1,055 ratings

Jeffrey R. Chasnov

Top Instructor

33,917 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

8,608 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

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. 9 hours to complete

English

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

Try Coursera for Business

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. 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 (290 Ratings)

Jeffrey R. Chasnov
Top Instructor

Professor

Department of Mathematics

170,998 Learners

16 Courses

Offered by

The Hong Kong University of Science and Technology

See how employees at top companies are mastering in-demand skills

Learn more about Coursera for Business

Syllabus - What you will learn from this course

Content Rating97%(3,035 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
82.18%
4 stars
15.36%
3 stars
1.88%
2 stars
0.47%
1 star
0.09%

TOP REVIEWS FROM FIBONACCI NUMBERS AND THE GOLDEN RATIO

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

by BSJun 24, 2020

The course is concise, to the point and very interesting. Though it looks easy at first sight, without practice, you cannot complete it.

Thank you, it was great fun.

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 CKMar 14, 2017

Thoroughly enjoyed this course. Interested in the Fibonacci Numbers and Golden Ratio and was looking for more information, this was clearly provided.

View all reviews

Frequently Asked Questions

When will I have access to the lectures and assignments?
Access to lectures and assignments depends on your type of enrollment. If you take a course in audit mode, you will be able to see most course materials for free. To access graded assignments and to earn a Certificate, you will need to purchase the Certificate experience, during or after your audit. If you don't see the audit option:
The course may not offer an audit option. You can try a Free Trial instead, or apply for Financial Aid.
The course may offer 'Full Course, No Certificate' instead. This option lets you see all course materials, submit required assessments, and get a final grade. This also means that you will not be able to purchase a Certificate experience.
What will I get if I purchase the Certificate?
When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
What is the refund policy?
You will be eligible for a full refund until two weeks after your payment date, or (for courses that have just launched) until two weeks after the first session of the course begins, whichever is later. You cannot receive a refund once you’ve earned a Course Certificate, even if you complete the course within the two-week refund period. See our full refund policy.
Is financial aid available?
Yes. In select learning programs, you can apply for financial aid or a scholarship if you can’t afford the enrollment fee. If fin aid or scholarship is available for your learning program selection, you’ll find a link to apply on the description page.

More questions? Visit the Learner Help Center.

Build employee skills, drive business results

Discover Coursera for Business

Fibonacci Numbers and the Golden Ratio

Fibonacci Numbers and the Golden Ratio

About this Course

What you will learn

Skills you will gain