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.

Offered By

## Fibonacci Numbers and the Golden Ratio

The Hong Kong University of Science and Technology## About this Course

### Learner Career Outcomes

## 50%

## 14%

High school mathematics

### What you will learn

Fibonacci numbers

Golden ratio

Fibonacci identities and sums

Continued fractions

## Skills you will gain

### Learner Career Outcomes

## 50%

## 14%

High school mathematics

### Offered by

#### 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.

## Syllabus - What you will learn from this course

**3 hours to complete**

## 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.

**3 hours to complete**

**7 videos**

**8 readings**

**4 practice exercises**

**3 hours to complete**

## 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.

**3 hours to complete**

**9 videos**

**10 readings**

**3 practice exercises**

**3 hours to complete**

## 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.

**3 hours to complete**

**8 videos**

**8 readings**

**3 practice exercises**

## Reviews

### TOP REVIEWS FROM FIBONACCI NUMBERS AND THE GOLDEN RATIO

Excellent exposition of a fascinating subject. Very well organized, including a handout with the course material.\n\nMany thanks to Prof. Chasnov and the HKUST for this worthwhile opportunity!

Someone has said that God created the integers; all the rest is the work of man. After seeing how the Fibonacci numbers play out in nature, I am not so sure about that. A very enjoyable course.

Finally I studied the Fibonacci sequence and the golden spiral. I used to say: one day I will. Very interesting course and made simple by the teacher in spite of the challenging topics

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.

## Frequently Asked Questions

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