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
Université des sciences et technologies de Hong KongÀ propos de ce cours
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Niveau débutant
High school mathematics
Approx. 9 heures pour terminer
Anglais
Sous-titres : Arabe, Français, Portugais (européen), Chinois (simplifié), Italien, Vietnamien, Coréen, Allemand, Russe, Turc, Anglais, Espagnol
Ce que vous allez apprendre
Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions
Compétences que vous acquerrez
Recreational MathematicsDiscrete MathematicsElementary Mathematics
Résultats de carrière des étudiants
67%
ont commencé une nouvelle carrière après avoir terminé ce cours
22%
ont bénéficié d'un avantage concret dans leur carrières grâce à ce cours
12%
a obtenu une augmentation de salaire ou une promotion
Certificat partageable
Obtenez un Certificat lorsque vous terminez
100 % en ligne
Commencez dès maintenant et apprenez aux horaires qui vous conviennent.
Dates limites flexibles
Réinitialisez les dates limites selon votre disponibilité.
Niveau débutant
High school mathematics
Approx. 9 heures pour terminer
Anglais
Sous-titres : Arabe, Français, Portugais (européen), Chinois (simplifié), Italien, Vietnamien, Coréen, Allemand, Russe, Turc, Anglais, Espagnol
Offert par
Université des sciences et technologies de Hong Kong
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.
Programme du cours : ce que vous apprendrez dans ce cours
3 heures pour terminer
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 heures pour terminer
7 vidéos (Total 55 min), 8 lectures, 4 quiz
7 vidéos
Promotional Video6 min
The Fibonacci Sequence | Lecture 18 min
The Fibonacci Sequence Redux | Lecture 27 min
The Golden Ratio | Lecture 38 min
Fibonacci Numbers and the Golden Ratio | Lecture 46 min
Binet's Formula | Lecture 510 min
Mathematical Induction7 min
8 lectures
Welcome and Course Information1 min
Fibonacci Numbers with Negative Indices5 min
The Lucas Numbers5 min
Neighbour Swapping10 min
Some Algebra Practice10 min
Linearization of Powers of the Golden Ratio10 min
Another Derivation of Binet's formula10 min
Binet's Formula for the Lucas Numbers10 min
4 exercices pour s'entraîner
Diagnostic Quiz5 min
The Fibonacci Numbers15 min
The Golden Ratio15 min
Week 1 Assessment30 min
3 heures pour terminer
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 heures pour terminer
9 vidéos (Total 65 min), 10 lectures, 3 quiz
9 vidéos
Cassini's Identity | Lecture 78 min
The Fibonacci Bamboozlement | Lecture 86 min
Sum of Fibonacci Numbers | Lecture 98 min
Sum of Fibonacci Numbers Squared | Lecture 107 min
The Golden Rectangle | Lecture 115 min
Spiraling Squares | Lecture 123 min
Matrix Algebra: Addition and Multiplication5 min
Matrix Algebra: Determinants7 min
10 lectures
Do You Know Matrices?
The Fibonacci Addition Formula10 min
The Fibonacci Double Index Formula10 min
Do You Know Determinants?
Proof of Cassini's Identity10 min
Catalan's Identity10 min
Sum of Lucas Numbers5 min
Sums of Even and Odd Fibonacci Numbers5 min
Sum of Lucas Numbers Squared5 min
Area of the Spiraling Squares10 min
3 exercices pour s'entraîner
The Fibonacci Bamboozlement15 min
Fibonacci Sums15 min
Week 2 Assessment30 min
3 heures pour terminer
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 heures pour terminer
8 vidéos (Total 61 min), 8 lectures, 3 quiz
8 vidéos
An Inner Golden Rectangle | Lecture 145 min
The Fibonacci Spiral | Lecture 156 min
Fibonacci Numbers in Nature | Lecture 164 min
Continued Fractions | Lecture 1715 min
The Golden Angle | Lecture 187 min
A Simple Model for the Growth of a Sunflower | Lecture 198 min
Concluding remarks4 min
8 lectures
The Eye of God10 min
Area of the Inner Golden Rectangle10 min
Continued Fractions for Square Roots10 min
Continued Fraction for e10 min
The Golden Ratio and the Ratio of Fibonacci Numbers10 min
The Golden Angle and the Ratio of Fibonacci Numbers10 min
Please Rate this Course1 min
Acknowledgments
3 exercices pour s'entraîner
Spirals15 min
Fibonacci Numbers in Nature15 min
Week 3 Assessment30 min
Avis
Meilleurs avis pour FIBONACCI NUMBERS AND THE GOLDEN RATIO
par JR12 juil. 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.
par BH28 oct. 2020
Great course concept for about one of the most intriguing concepts in the mathematical world, however I found it on the difficult side especially for those who find math as a challenging topic.
par AK22 mars 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.
par MH12 juil. 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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