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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 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 Watch the promotional video:


  • 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, which gives an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. This formula can be used to calculate the nth Fibonacci number without having to sum the preceding terms in the sequence.
  • 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.
  • 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.

Taught by

Jeffrey R. Chasnov


4.9 rating, based on 242 Class Central reviews

4.8 rating at Coursera based on 1108 ratings

Start your review of Fibonacci Numbers and the Golden Ratio

  • Anonymous
    I recently had the pleasure of enrolling in the "Fibonacci and Golden Ratio" course, and I must say it was an enlightening and intellectually stimulating experience. This course offers a captivating exploration of two of nature's most intriguing mat…
  • 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 c…
  • 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
    While I am sure math enthusiasts liked the course more than I did, I was interested in the golden rule and benefited from the course. I wanted to know where it came from and would like to have seen more everyday applications/appearances of the golden ratio in nature or design. Thank you Professor!
  • Anonymous
    There was something to learn and enjoy in each lecture. The course expanded my knowledge of Fibonacci numbers as to how the sequence is not only applicable to describing natural phenomenon but is also fascinating mathematics.
  • 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
  • Profile image for Wandrers Nachtlied
    Wandrers Nachtlied
    I really liked this course, and I think the instructor is great. Everything is well explained, both in algebra as well as in theory and geometrically. Music and visual representations both fit, as well. I find incredible how these patterns are reproduced in nature and in ancient forms, and how they inner correlate numerically with each other, also. I've always wanted to explore better this issue, as I find it very interesting and I know these patterns are used in trading. It is still incredible to think that nature replicated this pattern into many of its designs.

    Best regards,

    Bruna Bonatto
  • Anonymous
    The course gives basic but complete research about Fibonacci numbers and how they can be used to create unique and beautiful mathematical expressions. Also, it helps the student practice some proof techniques (induction and direct methods) and remember some basics from linear algebra to understand the topics better.

    The course also allows the student to appreciate the relation between successions and ratios and how they can represent geometry in a practical form (spirals, rectangles, and squares).

    Great course and excellent material to learn.
  • Anonymous
    Its very usefull to learn.Really I had and spend time for learning this...Thanks for your great effect to teach us........I am really proud by learning in this.......thanks for making my time more usefull
  • 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
    A few of the most famous mathematical concepts are looked into and tied together in this course: Fibonacci numbers and the golder ratio, as the course title suggests. You'll also get to solve some exercise problems, including math proof. One thing in this course that surprised me is how continued fractions play an important role in nature, e.g., in sunflowers.
  • Anonymous
    For first time, i realized that maths is very interesting. All the course content is entertaining and glared. The real life examples of Fibonacci number and golden ratio especially in human being and teeth are enough to glared my mind. Overall course is just mind boggling.
  • 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 tha…
  • Anonymous
    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.
  • Anonymous
    Fantastic course! The details of the Fibonacci sequence and related topics were hardly taught in my undergraduate studies. Enlightening to say the least.
    Thank you!
  • Anonymous
    This course helped me to know more about Fibonacci and Golden Ratio which I was not aware. Thank you so much for the informative videos.
  • Anonymous
    Fue muy completo e interesante. Los temas estaban bien relacionados y la ayuda visual es impresionante y aclara muchas dudas.
  • Anonymous
    A great introduction to the sequence of Fibonacci and all the beautifull math thats is hidden in those concepts
  • Anonymous
    this course helped me to improve my skills and made me confident about my knowledge. it was beneficial
  • Anonymous
    Hi Coursera,
    Thank you for making this beautiful course material on Fibonacci Numbers available online especially us in the asian countries to learn with you and deepen our love for Mathematics. I love how you explained every concepts as well as for attaching pdf files that is a complete package. I have saved and downloaded lecture videos on my mobile so if I need to go back I can easily browse and watch it offline. Thank you professor. Wish you all the best.

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