This is a course about the Fibonacci numbers, the golden ratio, and their intimate relationship. In this course, we learn the origin of the Fibonacci numbers and the golden ratio, and derive a formula to compute any Fibonacci number from powers of the golden ratio. We learn how to add a series of Fibonacci numbers and their squares, and unveil the mathematics behind a famous paradox called the Fibonacci bamboozlement. We construct a beautiful golden spiral and an even more beautiful Fibonacci spiral, and we learn why the Fibonacci numbers may appear unexpectedly in nature.
The course lecture notes, problems, and professor's suggested solutions can be downloaded for free from
http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook
Course Overview video: https://youtu.be/GRthNC0_mrU
Fibonacci Numbers and the Golden Ratio
von
The Hong Kong University of Science and Technology
Lehrplan - Was Sie in diesem Kurs lernen werden
Woche
14 Stunden zum Abschließen
Fibonacci: It's as easy as 1, 1, 2, 3
In this week's lectures, we learn about the Fibonacci numbers, the golden ratio, and their relationship. We conclude the week by deriving the celebrated Binet's formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical.
7 Videos (Gesamt 55 min), 9 Lektüren, 4 Quiz
7 Videos
The Fibonacci Sequence8m
The Fibonacci Sequence Redux7m
The Golden Ratio8m
Fibonacci Numbers and the Golden Ratio6m
Binet's Formula10m
Mathematical Induction7m
9 Lektüren
Welcome and Course Information2m
Get to Know Your Classmates3m
Fibonacci Numbers with Negative Indices10m
The Lucas Numbers10m
Neighbour Swapping10m
Some Algebra Practice10m
Linearization of Powers of the Golden Ratio10m
Another Derivation of Binet's formula10m
Binet's Formula for the Lucas Numbers10m
4 praktische Übungen
Diagnostic Quiz10m
The Fibonacci Numbers15m
The Golden Ratio15m
Week 150m
Woche
24 Stunden zum Abschließen
Identities, sums and rectangles
In this week's lectures, 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.
9 Videos (Gesamt 65 min), 10 Lektüren, 3 Quiz
9 Videos
Cassini's Identity8m
The Fibonacci Bamboozlement6m
Sum of Fibonacci Numbers8m
Sum of Fibonacci Numbers Squared7m
The Golden Rectangle5m
Spiraling Squares3m
Matrix Algebra: Addition and Multiplication5m
Matrix Algebra: Determinants7m
10 Lektüren
Do You Know Matrices?
The Fibonacci Addition Formula10m
The Fibonacci Double Index Formula10m
Do You Know Determinants?10m
Proof of Cassini's Identity10m
Catalan's Identity10m
Sum of Lucas Numbers10m
Sums of Even and Odd Fibonacci Numbers10m
Sum of Lucas Numbers Squared10m
Area of the Spiraling Squares10m
3 praktische Übungen
The Fibonacci Bamboozlement15m
Fibonacci Sums15m
Week 250m
Woche
34 Stunden zum Abschließen
The most irrational number
In this week's lectures, 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 head of a sunflower.
8 Videos (Gesamt 61 min), 8 Lektüren, 3 Quiz
8 Videos
An Inner Golden Rectangle5m
The Fibonacci Spiral6m
Fibonacci Numbers in Nature4m
Continued Fractions15m
The Golden Angle7m
A Simple Model for the Growth of a Sunflower8m
Concluding remarks4m
8 Lektüren
The Eye of God10m
Area of the Inner Golden Rectangle10m
Continued Fractions for Square Roots10m
Continued Fraction for e10m
The Golden Ratio and the Ratio of Fibonacci Numbers10m
The Golden Angle and the Ratio of Fibonacci Numbers10m
Please Rate this Course10m
Acknowledgments10m
3 praktische Übungen
Spirals15m
Fibonacci Numbers in Nature15m
Week 350m
50%
nahm einen neuen Beruf nach Abschluss dieser Kurse auf
17%
ziehen Sie für Ihren Beruf greifbaren Nutzen aus diesem Kurs
Top-Bewertungen
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.
Very well designed. It was a lot of fun taking this course. It's the kind of course that can get you excited about higher mathematics. Sincere thanks to Prof. Chasnov and HKUST.
Dozent
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
Über The Hong Kong University of Science and Technology
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.
Häufig gestellte Fragen
Wann erhalte ich Zugang zu den Vorträgen und Aufgaben?
Sobald Sie sich für ein Zertifikat angemeldet haben, haben Sie Zugriff auf alle Videos, Quizspiele und Programmieraufgaben (falls zutreffend). Aufgaben, die von anderen Kursteilnehmern bewertet werden, können erst dann eingereicht und überprüft werden, wenn Ihr Unterricht begonnen hat. Wenn Sie sich den Kurs anschauen möchten, ohne ihn zu kaufen, können Sie womöglich auf bestimmte Aufgaben nicht zugreifen.
Was bekomme ich, wenn ich das Zertifikat erwerbe?
Wenn Sie ein Zertifikat erwerben, erhalten Sie Zugriff auf alle Kursmaterialien, einschließlich bewerteter Aufgaben. Nach Abschluss des Kurses wird Ihr elektronisches Zertifikat zu Ihrer Seite „Errungenschaften“ hinzugefügt – von dort können Sie Ihr Zertifikat ausdrucken oder es zu Ihrem LinkedIn Profil hinzufügen. Wenn Sie nur lesen und den Inhalt des Kurses anzeigen möchten, können Sie kostenlos als Gast an dem Kurs teilnehmen.
Wie erfolgen Rückerstattungen?
Ist finanzielle Unterstützung möglich?
Haben Sie weitere Fragen? Besuchen Sie das Hilfe-Center für Teiln..
Coursera arbeitet mit erstklassigen Universitäten und Organisationen zusammen, um Online-Kurse anzubieten und dadurch universellen Zugriff auf die weltweit beste Ausbildung zu ermöglichen.
© 2019 Coursera Inc. Alle Rechte vorbehalten.
