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The Hong Kong University of Science and Technology

Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology via Coursera


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.

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  • 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.
  • 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.
  • 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.

Taught by

Jeffrey R. Chasnov


4.9 rating, based on 221 Class Central reviews

4.8 rating at Coursera based on 1015 ratings

Start your review of Fibonacci Numbers and the Golden Ratio

  • Ryan Lam completed this course, spending 3 hours a week on it and found the course difficulty to be medium.

    A fun and engaging introductory mathematics course! Professor Chasnov did a really good job on introducing this topic! The exercises focused on the Fibonacci number and its counterpart, the Lucas Number, and it is aesthetically pleasing to see the connection...
  • I found this course most interesting as it relates mathematics to real-life biology. Fibonacci numbers, Lucas numbers, golden ratio, golden rectangle and what to say, just enjoy and get knowledge from this course. The positive part is that it is not at all lengthy and time spent in this course is worth it.
  • Anonymous

    Anonymous completed this course.

    If you like to dabble in mathematical proofs, quirks, and curiosities (okay, I'm a geek), this short course is for you! It requires nothing beyond algebra and geometry but opens up an entire world.

    With relatively simple tools and deep reasoning you'll see that some irrational numbers are more irrational than others and the Golden Ratio is the most irrational of all!

    I found some of the proofs to be a bit challenging but excellent course documentation and forums provided help where needed. The final lecture on the spiral pattern of sunflower seeds was truly memorable.

    Bottom line - - a short course but a joy for the mathematically inclined.
  • Profile image for Sreelakshmi K S
    Sreelakshmi K S
    Very interesting topic and an excellent class.Thank you sir.In the beginning of this course I only know the sequence of Fibonacci numbers.But at the end of the session I know the right information about Fibonacci sequence of numbers.And it is very interest to connect Fibonacci numbers and golden ratio with nature . Thank you so much sir for your valuable time.
  • Anonymous
    the negative point of this course was the huge amount of numerical mathematics and much less geometrical information which made it hard to link these two parts. especially the matrix lessons seemed some how not that much relevant. another thing that...
  • Anonymous
    Useful information . It is very useful for my study . Easy to understand .
    This is the first time I have attended a class in this format and wondered how effective it would be. It was very effective and therefore I would definitely be interested in attending other classes in the same format. The instructor was very knowlegeable and provided a wealth of information about the current version, especially since the last version I used was several releases ago."
  • Anonymous
    A really comprehensive explanation of the association between Fibonacci numbers and the Golden Ration. The course tells the whole story starting with elaborating on the basic concepts and concluding with real world example.
    I found optional assignment where you need to proof the mathematical statement particularly useful. It would be nice if students can find more help with these assignments in the course's discussion forums.
  • Profile image for Elmer Norvell
    Elmer Norvell
    This course has excellent graphics and detailed instruction. Prepare for proofs and discovery of unusual relationships found in the Fibonacci sequence. Have fun!
  • Anonymous
    The professor is one of the top instructors. It is a 3 week course which completely covers all the material plus the preliminaries. There is a 120 page syllabus with lecture notes, practice problems, and solutions. For fun or for a certificate, this is an excellent course! Highly Recommend!
  • Shristy Upadhyay
    This course is very good .I learnt many things from this course. It is really helpful to me. I am glad that I choose this course. It will help in future to get the good job. I very thankful . This course is very much important if we want to continue with mathematics in future.
  • Anonymous
    Its really great. keep making contents like this. I loved it through all the course. it is interactive, engaging. I hope I will find courses like this in the future. If you are reading this, I strongly recommend you take this course.
  • Anonymous
    This was really useful course
    It added my credit points
    The lectures , videos , try out questions, quiz and assignments were really helpful
    The way of approach is good
    The outcome is excellent
    Overall the best experience
  • Anonymous

    Thanks for the course, it was very interesting to study Fibonacci numbers. The teacher is excellent, everything is clear and explained in detail. I wish you success, and thank you for your cooperation!
  • Anonymous
    Foi um pouco difícil para mim,mas foi realmente muito interessante e divertido aprender como os números de Fibonacci se relacionam com a natureza e a proporção dourada. Muito obrigado.
  • Anonymous
    Although i did this course without intention of having a degree,but still this course pleased me lot.It is amazingly elegantly presented and thanks to professor for this initiative.
  • Anonymous
    I had so much fun doing this course. It helped me remember how much I loved doing math. It is easy to fallow, and makes you see the beauty in numbers
    best Course till the date...!!
    Fibonacci System are good examples that are present in Nature...
    ex.. 1,1,2,3,5,8,13,...... and so on...!!

  • Anonymous
    Nice explanation by instructor with simple language such that even a school student can understand the subject
  • Anonymous
    Extraordinary teacher and very amazing matter. Deep and clear introduction on Fibonacci numbers and main applications.
  • Anonymous
    The class was good and it was very easy to understand was very easy to understand comments it was good..

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