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
от партнера
The Hong Kong University of Science and Technology
Программа курса: что вы изучите
Неделя
1Fibonacci: 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 видео ((всего 55 мин.)), 9 материалов для самостоятельного изучения, 4 тестов
7 видео
Course Overview6мин
The Fibonacci Sequence8мин
The Fibonacci Sequence Redux7мин
The Golden Ratio8мин
Fibonacci Numbers and the Golden Ratio6мин
Binet's Formula10мин
Mathematical Induction7мин
9 материала для самостоятельного изучения
Welcome and Course Information10мин
Get to Know Your Classmates10мин
Fibonacci Numbers with Negative Indices10мин
The Lucas Numbers10мин
Neighbour Swapping10мин
Some Algebra Practice10мин
Linearization of Powers of the Golden Ratio10мин
Another Derivation of Binet's formula10мин
Binet's Formula for the Lucas Numbers10мин
4 практического упражнения
Diagnostic Quiz10мин
The Fibonacci Numbers6мин
The Golden Ratio6мин
Week 120мин
Неделя
2Identities, 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 видео ((всего 65 мин.)), 10 материалов для самостоятельного изучения, 3 тестов
9 видео
Cassini's Identity8мин
The Fibonacci Bamboozlement6мин
Sum of Fibonacci Numbers8мин
Sum of Fibonacci Numbers Squared7мин
The Golden Rectangle5мин
Spiraling Squares3мин
Matrix Algebra: Addition and Multiplication5мин
Matrix Algebra: Determinants7мин
10 материала для самостоятельного изучения
Do You Know Matrices?10мин
The Fibonacci Addition Formula10мин
The Fibonacci Double Index Formula10мин
Do You Know Determinants?10мин
Proof of Cassini's Identity10мин
Catalan's Identity10мин
Sum of Lucas Numbers10мин
Sums of Even and Odd Fibonacci Numbers10мин
Sum of Lucas Numbers Squared10мин
Area of the Spiraling Squares10мин
3 практического упражнения
The Fibonacci Bamboozlement6мин
Fibonacci Sums6мин
Week 220мин
Неделя
3The 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 видео ((всего 61 мин.)), 8 материалов для самостоятельного изучения, 3 тестов
8 видео
An Inner Golden Rectangle5мин
The Fibonacci Spiral6мин
Fibonacci Numbers in Nature4мин
Continued Fractions15мин
The Golden Angle7мин
A Simple Model for the Growth of a Sunflower8мин
Concluding remarks4мин
8 материала для самостоятельного изучения
The Eye of God10мин
Area of the Inner Golden Rectangle10мин
Continued Fractions for Square Roots10мин
Continued Fraction for e10мин
The Golden Ratio and the Ratio of Fibonacci Numbers10мин
The Golden Angle and the Ratio of Fibonacci Numbers10мин
Please Rate this Course10мин
Acknowledgments10мин
3 практического упражнения
Spirals6мин
Fibonacci Numbers in Nature6мин
Week 320мин
4.7
Рецензии: 79Лучшие рецензии
автор: BS•Aug 30th 2017
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.
автор: HJ•Dec 4th 2016
Good course for introduction to Fibonacci Numbers. Should include more introduction lectures such as group theory, category theory, type theory, number theory, and algorithms.
Преподаватель
Jeffrey R. Chasnov
ProfessorDepartment of Mathematics
О 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.
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