Fibonacci Numbers (and the Euler-Binet Formula) [Video]

In this video, I explain what the Fibonacci numbers are and how they arise. Because the Fibonacci sequence have many exciting properties, I demonstrate several of them in this video, many of them using mathematical induction. Then I explain the golden ratio and its’ connection with the Fibonacci sequence. After that, I state and prove the Euler-Binet Formula. Finally, I explore the Fibonacci Prime Conjecture and the growth of the Fibonacci sequence.

Hi everyone, welcome back to my channel, I’m Dave. 

In today’s episode, I define the Fibonacci numbers as a recursive sequence starting with zero and one and then adding the previous two integers together. Because of the beauty, many people have long noticed that the Fibonacci sequence arises in many places throughout the natural world. In other words, the Fibonacci sequence has many unique mathematical properties.

Before you continue, you may want to download the video notes here: Fibonacci Numbers pdf notes. You may also be interested to know that this video is part of the Number Theory Series playlist.

Fibonacci Numbers (and the Euler-Binet Formula)

This video is for anyone looking for an introduction to the Fibonacci numbers and wanting to practice mathematical induction.

How-to Learn Fibonacci Numbers

By studying this video, you learn about the Fibonacci numbers and many of its properties.

Total Time Needed :

49

Minutes

Total Cost:

0

USD

Required Tools:

– A computing device with display screen.

Things Needed?

– An internet connection.

Steps to Learn Fibonacci Numbers

Fibonacci Numbers Learn what the Fibonacci Numbers are

Step 1: Learn what the Fibonacci Numbers are.

Their definition is recursive, so make sure and work out some basic examples.

Fibonacci Numbers Practice proving some Fibonacci identities using mathematical induction

Step 2: Practice proving some Fibonacci identities using mathematical induction.

So, here are some examples of this.

Fibonacci Numbers Explore the the Euler Binet formula

Step 3: Explore the the Euler-Binet formula.

After that, learn that the golden ratio is the positive root of a quadratic equation and that the Euler-Binet formula gives a relationship between the golden ratio and Fibonacci numbers.

Fibonacci Numbers Understand that there are many open questions about the Fibonacci sequence

Step 4: Understand that there are many open questions about the Fibonacci sequence.

After that, you should realize that there are many open questions involving the Fibonacci sequence. In particular, you will learn about the Fibonacci sequence’s growth, particularly about exponential growth and a fundamental inequality between the n-th Fibonacci number and the n-th power of the golden ratio.

FAQ Fibonacci Numbers

What are the Fibonacci numbers?

The Fibonacci numbers are recursively defined. Starting at 0 and 1, and then from here, add the previous two Fibonacci numbers to get the next Fibonacci number. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. 

Why is the Fibonacci sequence so important?

Many natural things follow the Fibonacci sequence, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, and an artichoke flowering. In some hurricanes and galaxies, the body rotation spawns spiral shapes (that follow the Fibonacci sequence) when the center turns faster than the periphery.

How many Fibonacci prime numbers are there?

The conjecture is that there is an infinite number of Fibonacci numbers that are prime numbers, such as 19, 134, 702, 400, 093, 278, 081, 449, 423, 917. But no one knows for sure. This question is still an open conjecture. 

What is the Euler-Binet formula?

The Euler-Binet formula is a closed formula for finding the n-th Fibonacci number (defined recursively). The formula states that the n-th Fibonacci number is equal to the difference of the n-th powers of the golden ratio and its conjugate divided by the square root of 5.

In conclusion, I want to turn it over to you. 

Do you think there are infinitely many Fibonacci primes? 

So, either way, let us know what you think in the comments for this video: Fibonacci Numbers right now.

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