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初級

完成時間大約為11 小時

建議:7 hours/week...

英語(English)

字幕:英語(English)

100% 在線

立即開始,按照自己的計劃學習。

可靈活調整截止日期

根據您的日程表重置截止日期。

初級

完成時間大約為11 小時

建議:7 hours/week...

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 4 小時

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 個視頻 (總計 55 分鐘), 9 個閱讀材料, 4 個測驗
7 個視頻
The Golden Ratio8分鐘
Fibonacci Numbers and the Golden Ratio6分鐘
Binet's Formula10分鐘
Mathematical Induction7分鐘
9 個閱讀材料
Welcome and Course Information2分鐘
Get to Know Your Classmates3分鐘
Fibonacci Numbers with Negative Indices10分鐘
The Lucas Numbers10分鐘
Neighbour Swapping10分鐘
Some Algebra Practice10分鐘
Linearization of Powers of the Golden Ratio10分鐘
Another Derivation of Binet's formula10分鐘
Binet's Formula for the Lucas Numbers10分鐘
4 個練習
Diagnostic Quiz10分鐘
The Fibonacci Numbers15分鐘
The Golden Ratio15分鐘
Week 150分鐘
2
完成時間為 4 小時

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 個視頻 (總計 65 分鐘), 10 個閱讀材料, 3 個測驗
9 個視頻
Sum of Fibonacci Numbers8分鐘
Sum of Fibonacci Numbers Squared7分鐘
The Golden Rectangle5分鐘
Spiraling Squares3分鐘
Matrix Algebra: Addition and Multiplication5分鐘
Matrix Algebra: Determinants7分鐘
10 個閱讀材料
Do You Know Matrices?
The Fibonacci Addition Formula10分鐘
The Fibonacci Double Index Formula10分鐘
Do You Know Determinants?10分鐘
Proof of Cassini's Identity10分鐘
Catalan's Identity10分鐘
Sum of Lucas Numbers10分鐘
Sums of Even and Odd Fibonacci Numbers10分鐘
Sum of Lucas Numbers Squared10分鐘
Area of the Spiraling Squares10分鐘
3 個練習
The Fibonacci Bamboozlement15分鐘
Fibonacci Sums15分鐘
Week 250分鐘
3
完成時間為 4 小時

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 個視頻 (總計 61 分鐘), 8 個閱讀材料, 3 個測驗
8 個視頻
Fibonacci Numbers in Nature4分鐘
Continued Fractions15分鐘
The Golden Angle7分鐘
A Simple Model for the Growth of a Sunflower8分鐘
Concluding remarks4分鐘
8 個閱讀材料
The Eye of God10分鐘
Area of the Inner Golden Rectangle10分鐘
Continued Fractions for Square Roots10分鐘
Continued Fraction for e10分鐘
The Golden Ratio and the Ratio of Fibonacci Numbers10分鐘
The Golden Angle and the Ratio of Fibonacci Numbers10分鐘
Please Rate this Course10分鐘
Acknowledgments10分鐘
3 個練習
Spirals15分鐘
Fibonacci Numbers in Nature15分鐘
Week 350分鐘
4.7
92 個審閱Chevron Right

50%

完成這些課程後已開始新的職業生涯

14%

通過此課程獲得實實在在的工作福利

來自Fibonacci Numbers and the Golden Ratio的熱門評論

創建者 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.

創建者 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.

講師

Avatar

Jeffrey R. Chasnov

Professor
Department of Mathematics

關於 香港科技大学

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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