Online Course
Fibonacci Numbers and the Golden Ratio
All-Time Top 100The Hong Kong University of Science and Technology via Coursera
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Mathematics Courses
- Provider Coursera
- Cost Free Online Course (Audit)
- Session Upcoming
- Language English
- Certificate Paid Certificate Available
- Effort 3 hours a week
- Duration 3 weeks long
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Overview
Class Central Tips
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.
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.
-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
Class Central Charts
- #1 in Subjects / Mathematics
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Reviews for Coursera's Fibonacci Numbers and the Golden Ratio Based on 53 reviews
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- 4 stars 6%
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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.
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.
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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!
In short, great course - highly recommended!
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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.
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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.
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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
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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!
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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. : )
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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
Recommended
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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.
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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 )
Thanks a lot Dr Chasnov and Hong Kong University )
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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
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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 behind mathematics and nature!
One thing I love about this course is that you don't have to know a lot of mathematics beforehand, just simple algebra will do. There is also a preliminary quiz to see if you are familiar with the basic algebra or not, which is a very good design!
A lot of the exercises in this course is about mathematical induction, which is an extremely important skill in university mathematics. So this course provides an extremely useful insight into it!
All in all, it is a wonderful journey through mathematics! Absolutely recommend to people who love mathematics and want to study it at university!
One thing I love about this course is that you don't have to know a lot of mathematics beforehand, just simple algebra will do. There is also a preliminary quiz to see if you are familiar with the basic algebra or not, which is a very good design!
A lot of the exercises in this course is about mathematical induction, which is an extremely important skill in university mathematics. So this course provides an extremely useful insight into it!
All in all, it is a wonderful journey through mathematics! Absolutely recommend to people who love mathematics and want to study it at university!
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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.
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Fayaz
completed this course.
First and foremost,I thank coursera to approving this course to me. In this course,I had learned more about Fibonacci numbers and golden ratio. I thank the professor who made this course easier to me. He explained with good example in Fibonacci numbers and also golden ratio.Finally I thank coursera and Hong Kong university.
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Anonymous
Anonymous
completed this course.
Found this course most interesting and informative. Prof Chasnov did an excellent job in conveying the pertinent information related to Fibonacci Numbers. Upon completing this course, I am encouraged to discover much more in detail regarding the Fibonacci Numbers and the Golden Ratio. Thank you Prof. Chasnov
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Duc T
by
Duc
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.
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Evelina
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.
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Chi D
by
Chi
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.
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Andreas
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.
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Luis
completed this course, spending 7 hours a week on it and found the course difficulty to be medium.
Very well explained with no overwhelming Maths
This course help me to understand Fibonacci numbers, Golden ratio and their relations
Very comprehensive and self contained, useful information in all videos
I strongly recommend this course
This course help me to understand Fibonacci numbers, Golden ratio and their relations
Very comprehensive and self contained, useful information in all videos
I strongly recommend this course
Was this review helpful to you?
Yes
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