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.
Fibonacci Numbers and the Golden Ratio
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Fibonacci Numbers and the Golden Ratio
Universidad Científica y Tecnológica de Hong Kong
Acerca de este Curso
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Learner Career Outcomes
50%
comenzó una nueva carrera después de completar estos cursos
14%
consiguió un beneficio tangible en su carrera profesional gracias a este curso
Qué aprenderás
Fibonacci numbers
Golden ratio
Fibonacci identities and sums
Continued fractions
Habilidades que obtendrás
Recreational MathematicsDiscrete MathematicsElementary Mathematics
Learner Career Outcomes
50%
comenzó una nueva carrera después de completar estos cursos
14%
consiguió un beneficio tangible en su carrera profesional gracias a este curso
100 % en línea
Comienza de inmediato y aprende a tu propio ritmo.
Fechas límite flexibles
Restablece las fechas límite en función de tus horarios.
Nivel principiante
High school mathematics
Aprox. 11 horas para completar
Sugerido: 3 weeks of study, 3 hours/week
Inglés (English)
Subtítulos: Inglés (English), Griego, Árabe (Arabic)
Programa - Qué aprenderás en este curso
Semana
13 horas para completar
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.
7 videos (Total 55 minutos), 8 lecturas, 4 cuestionarios
7 videos
The Fibonacci Sequence | Lecture 18m
The Fibonacci Sequence Redux | Lecture 27m
The Golden Ratio | Lecture 38m
Fibonacci Numbers and the Golden Ratio | Lecture 46m
Binet's Formula | Lecture 510m
Mathematical Induction7m
8 lecturas
Welcome and Course Information1m
Fibonacci Numbers with Negative Indices5m
The Lucas Numbers5m
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 ejercicios de práctica
Diagnostic Quiz5m
The Fibonacci Numbers15m
The Golden Ratio15m
Week 1 Assessment30m
Semana
23 horas para completar
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.
9 videos (Total 65 minutos), 10 lecturas, 3 cuestionarios
9 videos
Cassini's Identity | Lecture 78m
The Fibonacci Bamboozlement | Lecture 86m
Sum of Fibonacci Numbers | Lecture 98m
Sum of Fibonacci Numbers Squared | Lecture 107m
The Golden Rectangle | Lecture 115m
Spiraling Squares | Lecture 123m
Matrix Algebra: Addition and Multiplication5m
Matrix Algebra: Determinants7m
10 lecturas
Do You Know Matrices?
The Fibonacci Addition Formula10m
The Fibonacci Double Index Formula10m
Do You Know Determinants?
Proof of Cassini's Identity10m
Catalan's Identity10m
Sum of Lucas Numbers5m
Sums of Even and Odd Fibonacci Numbers5m
Sum of Lucas Numbers Squared5m
Area of the Spiraling Squares10m
3 ejercicios de práctica
The Fibonacci Bamboozlement15m
Fibonacci Sums15m
Week 2 Assessment30m
Semana
33 horas para completar
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.
8 videos (Total 61 minutos), 8 lecturas, 3 cuestionarios
8 videos
An Inner Golden Rectangle | Lecture 145m
The Fibonacci Spiral | Lecture 156m
Fibonacci Numbers in Nature | Lecture 164m
Continued Fractions | Lecture 1715m
The Golden Angle | Lecture 187m
A Simple Model for the Growth of a Sunflower | Lecture 198m
Concluding remarks4m
8 lecturas
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 Course1m
Acknowledgments
3 ejercicios de práctica
Spirals15m
Fibonacci Numbers in Nature15m
Week 3 Assessment30m
Principales revisiones sobre Fibonacci Numbers and the Golden Ratio
Excellent exposition of a fascinating subject.\n\nVery well organized, including a handout with the course material.\n\nMany thanks to Prof. Chasnov and the HKUST for this worthwhile opportunity!
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.
Instructor
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
Acerca de Universidad Científica y Tecnológica 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.
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