# Fibonacci Numbers and the Golden Ratio

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## Fibonacci Numbers and the Golden Ratio Coursera (CC) Opleiderscore: starstarstarstar_halfstar_border 7,2

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### Beschrijving

About this course: 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 fr…

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About this course: 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

Who is this class for: This course is suitable for anyone who loves mathematics, and should be accessible to those that remember their high school algebra.

Created by:  The Hong Kong University of Science and Technology
• Taught by:  Jeffrey R. Chasnov, Professor

Department of Mathematics
Level Beginner Language English Hardware Req Pen, paper and the power of your brain. How To Pass Pass all graded assignments to complete the course. User Ratings 4.7 stars Average User Rating 4.7See what learners said Coursework

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Syllabus

WEEK 1

Dip your toes in the water

By the end of this week, you will be able to: 1) describe the origin of the Fibonacci sequence; 2) describe the origin of the golden ratio; 3) find the relationship between the Fibonacci sequence and the golden ratio, including derive Binet’s formula. Download the lecture notes, problems, and the professor's suggested solutions from http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook

7 videos, 1 reading, 1 practice quiz expand

1. Video: Course Overview
3. Practice Quiz: Pre-course Survey
4. Discussion Prompt: Meet and greet
5. Video: The Fibonacci sequence
6. Video: The Fibonacci sequence redux
7. Discussion Prompt: Fibonacci numbers with negative indices
8. Discussion Prompt: The Lucas numbers
9. Discussion Prompt: Neighbour swapping
10. Video: The golden ratio
11. Video: Fibonacci numbers and the golden ratio
12. Video: Binet's formula
13. Discussion Prompt: Some algebra practice
14. Discussion Prompt: A Fibonacci-like relationship
15. Discussion Prompt: The conjugate relationship
16. Discussion Prompt: Ratio of separated Fibonacci numbers
17. Discussion Prompt: Linearization of powers of the golden ratio
18. Discussion Prompt: Proof of Binet's formula by induction
19. Discussion Prompt: Prove the limit of the ratio of Fibonacci numbers
20. Discussion Prompt: Binet's formula from the linearization formulas
21. Discussion Prompt: Binet's formula for the Lucas numbers
22. Discussion Prompt: Powers of the golden ratio
23. Video: Mathematical induction

WEEK 2

Dive deeper

By the end of this week, you will be able to: 1) identify the Fibonacci Q-matrix and derive Cassini’s identity; 2) explain the Fibonacci bamboozlement; 3) derive and prove the sum of the first n Fibonacci numbers, and the sum of the squares of the first n Fibonacci numbers; 4) construct a golden rectangle and 5) draw a figure with spiraling squares. Download the lecture notes, problems, and the professor's suggested solutions from http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook

9 videos expand

1. Video: The Fibonacci Q-matrix
2. Video: Cassini's identity
3. Video: The Fibonacci bamboozlement
4. Discussion Prompt: Powers of the Q-matrix
5. Discussion Prompt: Fibonacci addition formula
6. Discussion Prompt: Fibonacci double angle formulas
7. Discussion Prompt: Cassini's identity
8. Discussion Prompt: Catalan's identity
9. Discussion Prompt: Fibonacci bamboozlement
10. Video: Sum of Fibonacci numbers
11. Video: Sum of Fibonacci numbers squared
12. Discussion Prompt: Sum of Fibonacci numbers
13. Discussion Prompt: Sum of Lucas numbers
14. Discussion Prompt: Sum of odd and even Fibonacci numbers
15. Discussion Prompt: Sum of Fibonacci numbers squared
16. Discussion Prompt: Sum of Lucas numbers squared
17. Video: The golden rectangle
18. Video: Spiraling squares
19. Discussion Prompt: Construct a golden rectangle
20. Discussion Prompt: Spiraling squares
21. Video: Matrix algebra: addition and multiplication
22. Video: Matrix algebra: determinants

WEEK 3

Swim with the big fish

By the end of this week, you will be able to: 1) describe the golden spiral and its relationship to the spiraling squares; 2) construct an inner golden rectangle; 3) explain the continued fraction and be able to compute them; 4) explain why the golden ratio is called the most irrational of the irrational numbers; 5) understand why the golden ratio and the Fibonacci numbers may show up unexpectedly in nature. Download the lecture notes, problems, and the professor's suggested solutions from http://bookboon.com/en/fibonacci-numbers-and-the-golden-ratio-ebook

8 videos, 1 practice quiz expand

1. Video: The golden spiral
2. Video: An inner golden rectangle
3. Video: The Fibonacci spiral
4. Discussion Prompt: The eye of God
5. Discussion Prompt: The inner golden rectangle
6. Video: Fibonacci numbers in nature
7. Video: Continued fractions
8. Video: The golden angle
9. Video: A simple model for the growth of a sunflower
10. Discussion Prompt: Continued fractions for square roots
11. Discussion Prompt: Continued fraction for e
12. Discussion Prompt: The golden ratio and the ratio of Fibonacci numbers
13. Discussion Prompt: The golden angle and the ratio of Fibonacci numbers
14. Video: Concluding remarks
15. Practice Quiz: Post-course survey

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