Fibonacci Numbers and the Golden Ratio from Coursera

What's inside

Syllabus

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, which gives an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprocal. This formula can be used to calculate the nth Fibonacci number without having to sum the preceding terms in the sequence.

Identities, sums and rectangles

We learn about the Fibonacci Q-matrix and Cassini's identity. Cassini's identity is the basis for the famous dissection fallacy, 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 spiraling squares. This image is a drawing of a sequence of squares, each with side lengths equal to the golden ratio conjugate raised to an integer power, creating a visually appealing and mathematically intriguing pattern.

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 recognize 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, which is related to the golden ratio, and use it to model the growth of a sunflower head. The 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.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Explores sequences, numbers, and geometry, which are fundamental concepts in mathematics

Begins with the Fibonacci numbers, which are widely recognized in recreational mathematics

Taught by Jeffrey R. Chasnov, who is an experienced instructor and mathematics author

Examines the relationship between the Fibonacci numbers and the golden ratio, which are classical topics in number theory

Investigates the applications of the Fibonacci numbers and the golden ratio in nature and art, which provides practical examples of their significance

Reviews summary

Fibonacci and golden ratio: engaging math concepts

Learners say this course on Fibonacci numbers and the Golden Ratio is very informative, engaging, and interesting. It's a welcoming course for beginners in mathematics, but the mathematical proofs may pose a challenge for some. The curriculum is well-paced and includes lots of engaging exercises to help solidify concepts. Overall, students say this is a great course for anyone who loves math.

Mathematical proofs can be challenging, but rewarding.

"A really interesting course that fuels the interest of digging deeper into mathematics"

"The proofs by induction brought back fond memories of my university days many years ago."

"I have been interested in sacred geometry be years this course furthered my understanding of the symbols that are seen by everyone."

Concepts are explained clearly and engagingly.

"Clearly explained material most of which is accessible to everyone."

"The professor presented the course very well and made it interesting to follow."

"The relation of the golden ratio and fibonacci ratio is very interesting."

Exercises help understanding the material.

"This course is amazing! It was very interesting."

"Lot's of exercises which help understanding the material."

"Welly covered topics and excercises are enjoyable."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Fibonacci Numbers and the Golden Ratio with these activities:

Participate in peer study sessions to discuss course concepts

Show steps

Engage in discussions with peers to clarify concepts, share insights, and foster a deeper understanding of the course material.

Show steps

Identify a peer or group of peers to study with.
Schedule regular study sessions.
Prepare for sessions by reviewing course notes and materials.
Actively participate in discussions, asking questions and sharing your own understanding.

Review the Fibonacci sequence

Show steps

Brush up on the basics of the Fibonacci sequence to strengthen your understanding of the topics covered in this course.

Show steps

Recall the definition of the Fibonacci sequence.
Generate the first 10 terms of the sequence.

Attend a workshop on the golden ratio in design

Show steps

Attend a workshop to learn how the golden ratio is used in design, and how you can apply it to your own work.

Browse courses on Design

Show steps

Find a workshop on the golden ratio in design
Attend the workshop

Ten other activities

Expand to see all activities and additional details

Show all 13 activities

Read 'The Golden Ratio' by Mario Livio

Show steps

Review the book 'The Golden Ratio' to gain a comprehensive overview of its history, applications, and significance. This will deepen your understanding of the topic beyond the course material.

View The Golden Ratio: The Story of PHI, the World's... on Amazon

Show steps

Read the book, paying attention to the historical and cultural aspects of the golden ratio.
Take notes on the mathematical concepts and formulas related to the golden ratio.

Solve problems involving Fibonacci numbers and the Golden ratio

Show steps

Reinforce your problem-solving skills by tackling challenges that involve applying Fibonacci numbers and the Golden ratio, improving your proficiency.

Browse courses on Fibonacci Numbers

Show steps

Find practice problems online or in textbooks.
Set aside dedicated time for solving the problems.
Check your solutions against provided answers or work with a tutor to verify your understanding.

Explore interactive visualizations of the Golden ratio

Show steps

Delve deeper into the Golden ratio through interactive visualizations, enhancing your comprehension of its properties and applications.

Show steps

Locate interactive visualizations of the Golden ratio online.
Explore the visualizations, observing how the Golden ratio manifests in various geometric forms.

Fibonacci Formula Practice

Show steps

Apply your knowledge of the Fibonacci Formula by completing a series of practice problems.

Browse courses on Fibonacci Numbers

Show steps

Review the Fibonacci Formula: F(n) = F(n-1) + F(n-2)
Solve practice problems involving the Fibonacci Formula
Check your answers and identify areas for improvement

Fibonacci Problem-Solving Collaboration

Show steps

Engage in peer discussions and problem-solving sessions to deepen your understanding of Fibonacci-related concepts.

Browse courses on Fibonacci Numbers

Show steps

Join or form a study group with other students
Discuss complex Fibonacci problems and share different approaches
Collaborate on solving challenging problems and provide constructive feedback

Golden Ratio in Nature

Show steps

Explore the fascinating applications of the Golden Ratio in nature, from the spiral patterns of seashells to the arrangement of leaves on a plant stem.

Show steps

Research the Golden Ratio and its significance in nature
Identify examples of the Golden Ratio in various natural phenomena
Analyze how the Golden Ratio contributes to the aesthetics and functionality of natural structures

Write a blog post about the applications of Fibonacci numbers

Show steps

Write a blog post to demonstrate your understanding of the applications of Fibonacci numbers in various fields, such as nature, art, and architecture.

Browse courses on Applications

Show steps

Research the applications of Fibonacci numbers
Write a blog post outlining your findings

Create a visual representation of the Fibonacci spiral

Show steps

Solidify your understanding of the Fibonacci spiral by creating your own visual representation, fostering a deeper connection with the concept.

Show steps

Choose a method for creating your visual representation (e.g., drawing, digital art, physical model).
Plan the layout and incorporate the Fibonacci spiral into your design.
Execute your design, accurately representing the proportions and patterns of the Fibonacci spiral.

Fibonacci Sequence in Music

Show steps

Demonstrate your understanding of the Fibonacci Sequence by creating a piece of music that incorporates its principles.

Browse courses on Fibonacci Numbers

Show steps

Study the Fibonacci Sequence and its mathematical properties
Compose a musical piece using the Fibonacci Sequence as a guide for note durations, rhythms, or melodic patterns
Share your musical composition with others for feedback and appreciation

Fibonacci Challenge

Show steps

Test your skills and knowledge by participating in a Fibonacci-themed competition or challenge.

Browse courses on Fibonacci Numbers

Show steps

Identify and register for a suitable Fibonacci competition or challenge
Prepare thoroughly by studying Fibonacci-related concepts and practicing problem-solving
Compete in the challenge and strive for excellence
Reflect on your performance and identify areas for improvement

Career center

Learners who complete Fibonacci Numbers and the Golden Ratio will develop knowledge and skills that may be useful to these careers:

Mathematician

Mathematicians conduct research in mathematics and develop new mathematical theories. In order to do this, mathematicians need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Mathematician

Physics Teacher

Physics teachers teach physics to high school students. In order to do this, physics teachers need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Physics Teacher

Math Teacher

Math teachers teach mathematics to high school students. In order to do this, math teachers need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Math Teacher

Computer Science Teacher

Computer science teachers teach computer science to high school students. In order to do this, computer science teachers need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Computer Science Teacher

Quantitative Analyst

Quantitative analysts use mathematical and statistical techniques to develop trading strategies. In order to do this, quantitative analysts need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. In order to do this, actuaries need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Actuary

Software Developer

Software developers design, develop, and maintain software systems. In order to do this, software developers need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Software Developer

Operations Research Analyst

Operations research analysts use mathematical and statistical techniques to solve problems in business and industry. In order to do this, operations research analysts need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Operations Research Analyst

Statistician

Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. In order to do this, statisticians need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Statistician

Financial Analyst

Financial analysts use mathematical and statistical techniques to evaluate financial performance. In order to do this, financial analysts need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Financial Analyst

Risk Analyst

Risk analysts use mathematical and statistical techniques to assess risk. In order to do this, risk analysts need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Risk Analyst

Investment Analyst

Investment analysts use mathematical and statistical techniques to evaluate investments. In order to do this, investment analysts need to have a strong foundation in mathematics, including number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of Fibonacci numbers, the golden ratio, and their relationship to one another.

See salaries and explore the career path for Investment Analyst

Computer Scientist

Computer scientists work with large datasets to discover new trends as well as model and solve complex problems. In order to do this, computer scientists need to have a theoretical foundation in mathematics with a focus in related areas such as number theory and algebra. This course can help build a foundation in these areas, by providing an exploration of topics including Fibonacci numbers, the golden ratio, and their relationship to one another. Familiarity with these theoretical concepts is thus important for computer scientists since they form the basis of many modern algorithms.

See salaries and explore the career path for Computer Scientist

Data Analyst

Data analysts collect, clean, analyze, and interpret data. This course may be useful for a data analyst because it involves the study of sequences and patterns, which are also important in the field of data analysis.

See salaries and explore the career path for Data Analyst

Market Researcher

Market researchers collect and analyze data about consumer behavior in order to identify patterns and trends. This course may be useful for a market researcher because it involves the study of sequences and patterns, which are also important in the field of market research.

See salaries and explore the career path for Market Researcher