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# Fibonacci Numbers and the Golden Ratio

30 Reviews 6591 students interested
Found in
• Provider Coursera
• Cost Free Online Course (Audit)
• Session In progress
• Language English
• Certificate Paid Certificate Available
• Start Date
• Duration 3 weeks long

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## Overview

This is a course about the Fibonacci numbers, the golden ratio, and their intimate relationship. In this course, we learn the origin of the Fibonacci numbers and the golden ratio, and derive a formula to compute any Fibonacci number from powers of the golden ratio. We learn how to add a series of Fibonacci numbers and their squares, and unveil the mathematics behind a famous paradox called the Fibonacci bamboozlement. We construct a beautiful golden spiral and an even more beautiful Fibonacci spiral, and we learn why the Fibonacci numbers may appear unexpectedly in nature.

http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook

Course Overview video: https://youtu.be/GRthNC0_mrU

## Syllabus

Fibonacci: It's as easy as 1, 1, 2, 3
-In this week's lectures, we learn about the Fibonacci numbers, the golden ratio, and their relationship. 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 reciprical.

Identities, sums and rectangles
-In this week's lectures, 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.

The most irrational number
-In this week's lectures, 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 head of a sunflower.

#### Taught by

Jeffrey R. Chasnov

## Reviews for Coursera's Fibonacci Numbers and the Golden Ratio 4.9 Based on 30 reviews

• 5 stars 93%
• 4 stars 7%
• 3 star 0%
• 2 star 0%
• 1 star 0%

Did you take this course? Share your experience with other students.

• 1
Anonymous
5.0 3 years ago
completed this course.
I really liked this short course and recommend it if you are interested in the Fibonacci numbers and related things. The proof questions in the discussion prompts can sometimes be quite challenging but they are very satisfying to prove and worthwhile to attempt.
Anonymous
5.0 2 years ago
is taking this course right now.
I took the course for personal enrichment and to fuel my (clueless, crazy and way too unreachable, for the most who claim to know me best) willing to learn more Maths and sciences! I started attending another MOOC about Mathematical thinking
Fernando M
5.0 9 months ago
completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
Excellent course. An extremelly rare opportunity for someone interested in the beauty of Mathematics. The course can be taken by anyone with elementary high school algebra skills.
Anonymous
5.0 10 months ago
completed this course.

Thanks a lot Dr Chasnov and Hong Kong University )
Anonymous
5.0 10 months ago
completed this course.
Excellent course. I enjoyed every second of it - finished all the course in less than 2 days. The prof. is great - his explanations are very clear. The course is built very systematically. Lot's of exercises which help understanding the material. I learnt new interesting facts about Fibonacci numbers and more and some interesting stuff about the related mathematicians.

In short, great course - highly recommended!
Anonymous
5.0 a year ago
completed this course.
This was a short, but well-done course about Fibonacci numbers and the golden ratio that greatly enhanced my understanding of the topic. A big thanks to the professor and team for creating the course material. The course length, 3 weeks, makes for a good "filler" course in between other courses or obligations. More people should discover this one!
Duc T
5.0 7 months ago
by completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
It is a great course where you can learn many cool things about Fibonacci numbers and the golden ratio. Fibonacci numbers are not just 1+1=2, 1+2=3, 2+3=5, etc. they have many other cool characteristics. This course also shows amazing connection between Fibonacci numbers and the golden ratio (1+sqrt(5))/2.
Evelina C
5.0 a year ago
completed this course.

I really enjoyed this short course and I recommend it if you are interested in the Fibonacci numbers and its link with the golden ratio. The exercises are well done and the lessons short but clear. There are several interesting things exiting in mathematical and in nature. I really enjoyed.
Chi D
5.0 a year ago
by is taking this course right now, spending 2 hours a week on it and found the course difficulty to be medium.
I think this is the fundamental course for Fibonacci sequence and golden ratio. After going through this course, there are lots of interesting things exiting in mathematical findings and in nature. So cool! This course would be a trigger to keep digging this field.
Andreas Z
5.0 4 months ago
completed this course, spending 7 hours a week on it and found the course difficulty to be medium.
This is a wonderful course that provides you with insights into a topic much neglected by the university mathematics curriculum but which is of great importance to art and nature. I recommend everyone with an interest in mathematics to follow this eye-opener.
Anonymous
5.0 5 months ago
completed this course.
I have been interested in sacred geometry be years this course furthered my understanding of the symbols that are seen by everyone. I wish that the course was a specialization to go further into this field of math, and geometry.
Anonymous
5.0 4 weeks ago
completed this course.
The notes are very useful for learning. I even printed out the entire set so I can keep it for future reference. Very interesting and not so difficult. Good for those with solid foundation in maths and want to explore more.
Mahendra S
5.0 a year ago
by completed this course, spending 2 hours a week on it and found the course difficulty to be medium.
A very well planned and interactive course prepared by the lecturer, surely it helped to gain basic cum intermediate level of knowledge of Fibonacci numbers, Lucas numbers and the golden ratio.

regards.

:)
Anonymous
5.0 a year ago
completed this course.
A must for anyone who loves mathematics. Rigorous derivations, and very rewarding results... because you keep working on the mathematics, and then suddenly you understand how this applies to natural models.
Anonymous
5.0 a year ago
completed this course.
Clearly explained material most of which is accessible to everyone. There is some algebra, but it isn't required, as the interesting properties of the Fibonacci numbers are understandable without it.

Recommended
Victoria D
5.0 a year ago
completed this course, spending 3 hours a week on it and found the course difficulty to be medium.
It was amazing. The mathematical fun)

The Fibonacci numbers is a very important instrument for understanding of the universe. And now i can complete my design formula for students. Thank you very much.
Vipul T
5.0 a year ago
is taking this course right now.
Very good introduction to Fibonacci series and explained in a simple manner and clearly . Also all the identities and formulas are properly explained

A must do for all mathematics students
Tarit G
4.0 a year ago
completed this course, spending 2 hours a week on it and found the course difficulty to be easy.
It was a good introduction to Fibonacci sequence and it's relation with golden ratio. Welly covered topics and excercises are enjoyable. I have enjoyed this course very much. : )
Anonymous
5.0 a year ago
completed this course.
Very informing and interesting. Teaches you step-by-step until it gets you to the interesting examples in nature of the Fibonacci sequence and golden ratio. Highly recommended!
Marianne V
5.0 a year ago
completed this course.
The course was great fun and provided some challenges, while not being too difficult overall. The proofs by induction brought back fond memories of my university days many years ago.