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Best of All-Time Online Course

Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology via Coursera

Overview

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.

Syllabus

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

Reviews

4.9 rating, based on 144 reviews

Start your review of Fibonacci Numbers and the Golden Ratio

  • 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.
  • Anonymous

    Anonymous 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

    Anonymous completed this course.

    It was a awesome video. I learned a lot from the video. I even learnt about some facts about mathematics. Actually the course fibonacci numbers and golden ratio was unknown to me. And I was curious one side to know about it and another side I was worried whether I could cope up with the course. But what happened was absolutely opposite of my thoughts . Hence I enjoyed and learnt it . Thank you for the course.
  • Anonymous

    Anonymous 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

    Anonymous 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
  • Profile image for Sreelakshmi K S
    Sreelakshmi 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

    Anonymous 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!
  • Anonymous

    Anonymous completed this course.

    The Fibonacci numbers and golden ratio Coursera was great learning experience it improves mathematical thinking, application of Fibonacci numbers are explained to the best...like Fibonacci rabbit problem and Fibonacci sequence in the nature... sunflower.
  • Profile image for Tarit Goswami
    Tarit G.

    Tarit 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

    Anonymous 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
  • Fernando M.

    Fernando 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

    Anonymous completed this course.

    This is the first course I have taken in coursera. The experience has been really very good. I have been able to learn about some of the very interesting characteristics of numbers.
  • Anonymous

    Anonymous completed this course.

    Really good course for beginners, especially interested in sunflowers ) Hope to read more about this in detail.

    Thanks a lot Dr Chasnov and Hong Kong University )
  • Anonymous

    Anonymous is taking this course right now.

    found it very balanced in depth, rigor and presentation . Would like to know if there are any physical applications of Fibonacci numbers
  • Anonymous

    Anonymous completed this course.

    If you are interested in learning deeper about Fibonacci then this course is for you. The course is very well planned and easy to understand.
  • Anonymous

    Anonymous completed this course.

    The course for fun.This course is great if you want to learn practical aplicability of maths.

    The course presentation is just amazing.
  • Profile image for Ryan Lam
    Ryan L.

    Ryan 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...
  • Anonymous

    Anonymous completed this course.

    Very interesting course and the topic is covered comprehensively. Though this course is run every week, the instructor replies promptly to any questions asked. Most of the exercises are proofs which was good practice, though I found them rather too challenging for my standard. However, this is inevitable, as these proofs directly relate to formulas which will be used and are not just random exercises and thus I am grateful for the solutions provided by the pdf. Any gaps are also filled in well by the lecture notes pdf provided, though some proofs will be more challenging to understand. Great course for anyone interested in a rather niche area of mathematics.
  • Anonymous

    Anonymous completed this course.

    Just incredible how much challenging mathematics could be derived from such a simple starting point! The lectures were short and informative and the quizzes and assessments were very do-able.

    Of course the coup de grace is at the end with the mind blowing occurrence of these seemingly abstract numbers in nature and in particular sunflowers; wow!

    Great to do a short course with accessible mathematics all relating back to the simple concept of the Fibonacci sequence.

    Really enjoyed it.
  • Anonymous

    Anonymous completed this course.

    The subject is appealing, the professor does a good job of selecting the material to emphasize, the quizzes and tests are relevant. Being just for fun, I skipped a couple of assigned problems because I understood the principles and didn't feel like wading through the necessary algebra. Aside from that, it was fun and rewarding to exercises and to see them produce the results. For such a short course, it managed to pack in everything I wanted to explore on this subject.

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