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.
Fibonacci Numbers and the Golden Ratio
The Hong Kong University of Science and TechnologyAbout this Course
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Learner Career Outcomes
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started a new career after completing these courses
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Beginner Level
High school mathematics
Approx. 9 hours to complete
English
Subtitles: Arabic, French, Portuguese (Brazilian), Russian, English, Spanish
What you will learn
Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions
Skills you will gain
Recreational MathematicsDiscrete MathematicsElementary Mathematics
Learner Career Outcomes
50%
started a new career after completing these courses
14%
got a tangible career benefit from this course
Shareable Certificate
Earn a Certificate upon completion
100% online
Start instantly and learn at your own schedule.
Flexible deadlines
Reset deadlines in accordance to your schedule.
Beginner Level
High school mathematics
Approx. 9 hours to complete
English
Subtitles: Arabic, French, Portuguese (Brazilian), Russian, English, Spanish
Offered by
The Hong Kong University of Science and Technology
HKUST - A dynamic, international research university, in relentless pursuit of excellence, leading the advance of science and technology, and educating the new generation of front-runners for Asia and the world.
Syllabus - What you will learn from this course
3 hours to complete
Fibonacci: It's as easy as 1, 1, 2, 3
We learn about the Fibonacci numbers, the golden ratio, and their relationship. We derive the celebrated Binet's formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical.
3 hours to complete
7 videos (Total 55 min), 8 readings, 4 quizzes
7 videos
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
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
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
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
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 God10m
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
TOP REVIEWS FROM FIBONACCI NUMBERS AND THE GOLDEN RATIO
by NRAug 17, 2019
Excellent exposition of a fascinating subject. Very well organized, including a handout with the course material.\n\nMany thanks to Prof. Chasnov and the HKUST for this worthwhile opportunity!
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 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 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.
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