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100% online

Comece imediatamente e aprenda em seu próprio cronograma.

Prazos flexíveis

Redefinir os prazos de acordo com sua programação.

Nível iniciante

Aprox. 11 horas para completar

Sugerido: 7 hours/week...

Inglês

Legendas: Inglês

Programa - O que você aprenderá com este curso

Semana
1
4 horas para concluir

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.

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7 vídeos ((Total 55 mín.)), 9 leituras, 4 testes
7 videos
The Fibonacci Sequence8min
The Fibonacci Sequence Redux7min
The Golden Ratio8min
Fibonacci Numbers and the Golden Ratio6min
Binet's Formula10min
Mathematical Induction7min
9 leituras
Welcome and Course Information2min
Get to Know Your Classmates3min
Fibonacci Numbers with Negative Indices10min
The Lucas Numbers10min
Neighbour Swapping10min
Some Algebra Practice10min
Linearization of Powers of the Golden Ratio10min
Another Derivation of Binet's formula10min
Binet's Formula for the Lucas Numbers10min
4 exercícios práticos
Diagnostic Quiz10min
The Fibonacci Numbers15min
The Golden Ratio15min
Week 150min
Semana
2
4 horas para concluir

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.

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9 vídeos ((Total 65 mín.)), 10 leituras, 3 testes
9 videos
Cassini's Identity8min
The Fibonacci Bamboozlement6min
Sum of Fibonacci Numbers8min
Sum of Fibonacci Numbers Squared7min
The Golden Rectangle5min
Spiraling Squares3min
Matrix Algebra: Addition and Multiplication5min
Matrix Algebra: Determinants7min
10 leituras
Do You Know Matrices?
The Fibonacci Addition Formula10min
The Fibonacci Double Index Formula10min
Do You Know Determinants?10min
Proof of Cassini's Identity10min
Catalan's Identity10min
Sum of Lucas Numbers10min
Sums of Even and Odd Fibonacci Numbers10min
Sum of Lucas Numbers Squared10min
Area of the Spiraling Squares10min
3 exercícios práticos
The Fibonacci Bamboozlement15min
Fibonacci Sums15min
Week 250min
Semana
3
4 horas para concluir

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.

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8 vídeos ((Total 61 mín.)), 8 leituras, 3 testes
8 videos
An Inner Golden Rectangle5min
The Fibonacci Spiral6min
Fibonacci Numbers in Nature4min
Continued Fractions15min
The Golden Angle7min
A Simple Model for the Growth of a Sunflower8min
Concluding remarks4min
8 leituras
The Eye of God10min
Area of the Inner Golden Rectangle10min
Continued Fractions for Square Roots10min
Continued Fraction for e10min
The Golden Ratio and the Ratio of Fibonacci Numbers10min
The Golden Angle and the Ratio of Fibonacci Numbers10min
Please Rate this Course10min
Acknowledgments10min
3 exercícios práticos
Spirals15min
Fibonacci Numbers in Nature15min
Week 350min
4.7
88 avaliaçõesChevron Right

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Principais avaliações do Fibonacci Numbers and the Golden Ratio

por AKMar 23rd 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.

por BSAug 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.

Instrutores

Avatar

Jeffrey R. Chasnov

Professor
Department of Mathematics

Sobre Universidade de Ciência e Tecnologia 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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