Fibonacci Numbers and the Golden Ratio
- Methode
Fibonacci Numbers and the Golden Ratio
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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 fr…
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When you enroll for courses through Coursera you get to choose for a paid plan or for a free plan .
- Free plan: No certicification and/or audit only. You will have access to all course materials except graded items.
- Paid plan: Commit to earning a Certificate—it's a trusted, shareable way to showcase your new skills.
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
Each course is like an interactive textbook, featuring pre-recorded videos, quizzes and projects.
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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
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
- Video: Course Overview
- Reading: Assignments and Grading
- Practice Quiz: Pre-course Survey
- Discussion Prompt: Meet and greet
- Video: The Fibonacci sequence
- Video: The Fibonacci sequence redux
- Discussion Prompt: Fibonacci numbers with negative indices
- Discussion Prompt: The Lucas numbers
- Discussion Prompt: Neighbour swapping
- Video: The golden ratio
- Video: Fibonacci numbers and the golden ratio
- Video: Binet's formula
- Discussion Prompt: Some algebra practice
- Discussion Prompt: A Fibonacci-like relationship
- Discussion Prompt: The conjugate relationship
- Discussion Prompt: Ratio of separated Fibonacci numbers
- Discussion Prompt: Linearization of powers of the golden ratio
- Discussion Prompt: Proof of Binet's formula by induction
- Discussion Prompt: Prove the limit of the ratio of Fibonacci numbers
- Discussion Prompt: Binet's formula from the linearization formulas
- Discussion Prompt: Binet's formula for the Lucas numbers
- Discussion Prompt: Powers of the golden ratio
- Video: Mathematical induction
Graded: Week 1
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
- Video: The Fibonacci Q-matrix
- Video: Cassini's identity
- Video: The Fibonacci bamboozlement
- Discussion Prompt: Powers of the Q-matrix
- Discussion Prompt: Fibonacci addition formula
- Discussion Prompt: Fibonacci double angle formulas
- Discussion Prompt: Cassini's identity
- Discussion Prompt: Catalan's identity
- Discussion Prompt: Fibonacci bamboozlement
- Video: Sum of Fibonacci numbers
- Video: Sum of Fibonacci numbers squared
- Discussion Prompt: Sum of Fibonacci numbers
- Discussion Prompt: Sum of Lucas numbers
- Discussion Prompt: Sum of odd and even Fibonacci numbers
- Discussion Prompt: Sum of Fibonacci numbers squared
- Discussion Prompt: Sum of Lucas numbers squared
- Video: The golden rectangle
- Video: Spiraling squares
- Discussion Prompt: Construct a golden rectangle
- Discussion Prompt: Spiraling squares
- Video: Matrix algebra: addition and multiplication
- Video: Matrix algebra: determinants
Graded: Week 2
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
- Video: The golden spiral
- Video: An inner golden rectangle
- Video: The Fibonacci spiral
- Discussion Prompt: The eye of God
- Discussion Prompt: The inner golden rectangle
- Video: Fibonacci numbers in nature
- Video: Continued fractions
- Video: The golden angle
- Video: A simple model for the growth of a sunflower
- Discussion Prompt: Continued fractions for square roots
- Discussion Prompt: Continued fraction for e
- Discussion Prompt: The golden ratio and the ratio of Fibonacci numbers
- Discussion Prompt: The golden angle and the ratio of Fibonacci numbers
- Video: Concluding remarks
- Practice Quiz: Post-course survey
Graded: Week 3
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