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The Fibonacci Sequence Redux

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Fibonacci Numbers and the Golden Ratio

홍콩과학기술대학

4.7(280개의 평가) | 16K명의 학생이 등록함

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

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4.7(280개의 평가)

5 stars
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BS

Aug 30, 2017

Very well designed. It was a lot of fun taking this course. It's the kind of course that can get you excited about higher mathematics. Sincere thanks to Prof. Chasnov and HKUST.

HJ

Dec 04, 2016

Good course for introduction to Fibonacci Numbers. Should include more introduction lectures such as group theory, category theory, type theory, number theory, and algorithms.

수업에서

Fibonacci: It's as easy as 1, 1, 2, 3

In this week's lectures, we learn about the Fibonacci numbers, the golden ratio, and their relationship. We conclude the week by deriving the celebrated Binet's formula, an explicit formula for the Fibonacci numbers in terms of powers of the golden ratio and its reciprical.

The Fibonacci Sequence8:14

The Fibonacci Sequence Redux7:20

강사:

Jeffrey R. Chasnov
Professor

스크립트

[MUSIC]

Welcome back.

In this video I want to introduce another problem

whose solution is the Fibonacci numbers.

I call this problem climbing the staircase.

So how do we pose the climbing the staircase problem.

The question to answer is how many ways can one climb a staircase with n steps.

Taking one or two steps at a time.

So we have a staircase, we can go climb the staircase as one step,

one step, two step, one step.

Or we can climb it as two step, two step, one step.

And if we have n stairs in the staircase,

then how many different ways can we climb the staircase?

To answer this problem we could make a table.

We can consider a small number of stairs, from one to five stairs,

and we can list the number of ways that we can climb this staircase.

So, if there's only one stair there's only one way to climb this staircase,

which is to go up by one stair.

So we call a sub n the number of ways you can climb a n stair staircase.

So a sub 1 is 1.

If we have two stairs in the staircase, then we can climb 11.

Or we can just climb 2, go up both stairs at the same time.

So, a sub 2 then is equal to 2.

If we have a 3 stair staircase,

we can go 111, 12, go two

stairs at a time at the end, or we can go two, one, two stairs at the beginning.

And there are three different ways we can climb a three stair staircase.

So we have one, two, three, looks like we're just counting, right?

But if we have a four step staircase, that in fact there are five different ways,

1111, 112, 121, 211 and finally 22.

We take two stairs first and then we take two more stairs.

So there are five different ways to climb a four step staircase.

So now we have the sequence 1, 2, 3, 5.

Five is equal to two plus three so it's part of the Fibonacci sequence.

Let's look at one more.

What if we have five stairs in the staircase.

Then if we list out all the possible ways to claim a five

stair staircase we see that there are eight different ways.

So now we're getting the Fibonacci sequence here.

1, 2, 3, 5, 8 just missing the first one.

So why are we getting the Fibonacci sequence, can we do some sort of

mathematical argument to show you that we should have the Fibonacci sequence.

So let me try and do that.

So, let's think about how to climb the staircase.

We can actually break climbing of the staircase into two different ways.

We can start by climbing the first stair.

We can climb by one stair, right?

And then so on climb the rest of the staircase, okay?

The other way is we can start with two stairs, so

we can start with two stairs and then climb the rest of the staircase.

So we have those two possibilities.

The first step we take can be one stair or

the first step we take that can be two stairs.

So how many ways are there then to climb n stairs,

staircase if the first step we take is one stair.

While there's n minus one stairs left, right?

We took one step already, there's n minus 1 steps left.

So the number of ways we can climb at n stairs staircase here

is n- 1 ways because we've already climbed the first step.

The number of ways we can climb and

n stair staircase if we take as our first step, two steps,

is the number of ways we can climb n minus 2 steps because we already took 2 steps.

So the number of ways here is, a sub n- 2, okay?

So the argument I'm making is that the number of ways to

climb an n stair staircase is equal to a number of ways to

climb that staircase where the first step is one step.

A sub n- 1, plus the number of ways to climb a staircase.

Where the first step is two steps, a sub n- 2.

A sub n = a sub n- 1 + a sub n- 2.

That's just the recursion relation for the Fibonacci sequence.

That's why we're getting Fibonacci numbers.

The difference though is we need the initial conditions, so the initial values,

so the number of ways to climb a one step staircase is one right?

The number of ways to climb a two steps staircase.

Remember it's 1, 1 or just 2, so that's 2.

So these are not the initial values for the Fibonacci sequence but

2 is a unique Fibonacci number, is the third Fibonacci number, and

one can be the second Fibonacci or the first Fibonacci number.

Here I take it to be the second Fibonacci number.

So a sub 1 is F sub 2, a sub 2 is f sub 3.

a satisfies the Fibonacci recursion relation so

we must have a sub n as a Fibonacci number but it's not F sub n,

it's one more than F sub n, it's F sub n + 1, okay?

So the number of ways to climb an n step staircase taking one step or

two step at a time is the n+1 Fibonacci numbers.

Fibonacci numbers showed up in this problem.

So we've seen now two problems with Fibonacci numbers.

The rabbit plot problem, the classic rabbit problem from Fibonacci and

the climbing the staircase problem.

In the next lecture, I want to talk about the golden ratio.

And then we'll find a connection between the golden ratio and

the Fibonacci numbers.

I'll see you next time.

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