Notice:  As of December 14, 2016, the MindBites Website and Service will cease its operations.  Further information can be found here.  

Hi! We show you're using Internet Explorer 6. Unfortunately, IE6 is an older browser and everything at MindBites may not work for you. We recommend upgrading (for free) to the latest version of Internet Explorer from Microsoft or Firefox from Mozilla.
Click here to read more about IE6 and why it makes sense to upgrade.

Calculus: Fibonacci Numbers


Like what you see? false to watch it online or download.

You Might Also Like

About this Lesson

  • Type: Video Tutorial
  • Length: 11:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 123 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus: Math Fun (6 lessons, $7.92)
Calculus: Fibonnaci Sequence & the Golden Triangle (2 lessons, $2.97)

Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

2174 lessons

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

This lesson has not been reviewed.
Please purchase the lesson to review.
This lesson has not been reviewed.
Please purchase the lesson to review.

Math Fun
Fibonaci Numbers Page [1 of 3]
Well, I thought I would tell you about my favorite collection of numbers. Everyone has their favorites, of course, and I thought I would show you mine. This is a great collection of numbers and, and you may have seen this before, but I thought I'd tell you about it. And, basically, it's like, it's like one of these guessing games, you know, we start off with a game. It's always fun to start with a game. It's like one of these game where I give you first two numbers and then you have to guess the next one. You know? It's sort of like, it's sort of like the SATs. Remember that? They'd give you a first few numbers and then they'd ask you what the next one would be. See, you know, here, if you've got, guess the wrong answer, not a big deal, no one will ever know. But the SATs, you get the wrong answer, you don't go to college, your life is ruined, you don't get a job. So, already this video is better than taking the SATs. I'm very, very proud of that.
So, here we go, let me give you the first few. Let's see if you can guess the next one. So, look for the pattern here. So, the first one, first one is 1. Okay, so maybe you need more help that that. The next one is 2. The second number is 1. And now, what's the next one. Well, if you wanted pattern recognition here, you would see the pattern and you'd see the pattern as 1, 1, 1, 1, 1, 1, 1, and so you'd guess 1. But you'd be wrong, because the next one turns out to be 2. Okay, well, if you were an experimental scientist, you would say, "Oh, okay, the next one is 3 and the first one must have been an error". And you'd be right, because the next one is 3. But the next number after that isn't 4, in fact, the next one is 5. And maybe you're beginning to see the pattern. The next one would be 8. You see the pattern here? The next one is 13. See what I'm doing? The next one would be 21. After that would be 34. And after that would be 55. And after that would be 89, and it goes on, and on, and on, and on forever. And maybe you're seeing the pattern. The pattern is, to figure out the next person, what you do is take the two previous people and add them together. So, for example, 3 and 5 make 8, and that's the next number. You take 5 and 8 and add them up, you get 13. That's the next number. 13 and 21 is 34, and so on. So, that's the pattern. And, in fact, this collection of numbers, and they get bigger, and bigger, and bigger, it's great fun, these are called Fibonacci numbers. And, in fact, they're called that because someone named Fibonacci actually first thought about them. So, I think it's great. Now, I think it's really fun, these great numbers, and they get bigger, and bigger and bigger, and bigger and they are really fun. But, you know, some guy just invented them, so who cares? Right? Not a big deal.
Okay, so let's put them over there. So, there they all are. You can look at the Fibonacci numbers and you can make them up. You can make them even bigger, if you want. Instead, I thought that I would actually change the subject a little bit here and talk about, talk about nature, because, you know, nature is much more interesting than, than numbers and stuff. So, I thought I'd show you some beautiful pictures. See, here's a beautiful picture of a sunflower. Isn't that pretty? Look at that. Beautiful picture of a sunflower. Isn't that gorgeous? Love it. In fact, what makes that sunflower so appealing? Well, it's like what makes this fresh flower sort of appealing. Look at this beautiful flower here. Isn't that pretty? Um, it smells so good. And, and you can sort, you know, she loves me, she loves me not. She love me, and you know this game, like she loves me not, and you hope that there's the right amount of flowers there. But it's really beautiful, really pretty flowers. What makes these things so appealing? Isn't that sort of an interesting question?
Well, let me show you some more photographs. In fact, let me show you some photographs that I, myself took a while back. Here's a picture of a daisy, a daisy, which I think is really pretty. Beautiful color inside there. Here's another flower, explosion of color. This is called the coneflower. I took this actually in one of my colleague's garden. Isn't that beautiful? Beautiful coneflower. Just an explosion of color. Now, you know what my favorite part about this flower is? It's not the white part, it's not the part you pick off, but it's that beautiful yellow part in there. Do you see that? That's really enticing and, in fact, if you look really, really closely, you'll see that, in fact, there are a lot of spirals there. It sort of makes this thing so pretty. And, in fact, if you look really, really closely, you'll see that there's actually two collections of spirals that are interlocking.
Let me show you one. See, one sort of comes out like this, clockwise. Whoooooo! Whooooo! See them? Whoooooo! Whoooooooo! See them all coming out there? But then there's another set of spirals that are sort of interlocking the other way. Watch this. Shshshshssh! Shshshshshsh! Shshshshshsh! Do you see those spirals coming out the other way? Two interlocking sets of spirals that come together to make this very pretty pattern. In fact, let me show you the two interlocking spirals here. You can see them very clearly. You come out here, you can see the spirals coming out this way, like this. But then the spirals come out the other way, like this. Explosion coming out clockwise and spirals coming out counterclockwise. Do you see them both? So really, it's pretty. And they come together and make this really attractive design. In fact, if you look back at the sunflower, you really can see this explosion of spiral, spirals coming out this way, which is really pretty, spirals coming out the other way, and together they make this beautiful pattern, beautiful pattern.
Gee, I wonder - sort of a fun question - coming back to this flower, this daisy, I wonder how many spirals there are there. Boy, there are a lot of spirals! Whew! A lot of spirals, lot of spirals. Well, in fact, let's give us an attempt to actually count the number of spirals. Shall we try that right now? Let's try to count the number of spirals. I have a picture here of the same thing as before, but just of the spiral part. I don't know if you can see that too well, but, hopefully, you can see the spirals going both ways. And I thought it would be fun just to try to count to see how many spirals there are. Now, it's sort of a hard thing to do, so I helped you out a little bit here by making a little schematic, where I actually marked the spirals. So, now I'm going to start counting, and I'll start counting, let's say, right here, at that spiral, and I'll count around. You can count with me, if you want. Of course, I won't know because I can't hear you, but I'll count out loud here. So, we have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two. Let's see, did I count that correctly? Let's try that again, so, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one. Okay. Twenty-one spirals going that way.
Okay, now let's see how many spirals there are going this way. Let's see the other direction now. Actually, it looks like there are a lot more, doesn't it? Looks like there's a lot more, compared to here. That one doesn't look as many has here. You might have thought there was the same number of spirals. Turns out, looks like there's more here. Let's count these ones. So, we'll start here at the mark, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, twenty-two, twenty-three, twenty-four, twenty-five, twenty-six, twenty-seven, twenty-eight, twenty-nine, thirty, thirty-one, thirty-two, thirty-three, thirty-four.
So, it looks like there's thirty-four going this way. So, that's interesting. Twenty-one going this way and thirty-four going that way. Hmm, sort of interesting. Twenty-one and thirty-four. Twenty-one and thirty... Well! Well, look over there! Twenty-one and thirty-four! Those are two consecutive Fibonacci numbers! Holy moly! How is it possible that, in fact, we see two consecutive Fibonacci numbers in the count of the two different spirals here? That's utterly amazing! Well, is that a coincidence or freak of nature? Answer is no, it's not. That's the wonderful thing. If you look at spirals in nature, like the beautiful cone flower here, and count out the number of spirals here, this direction, you'll see twenty-one. Count them this way, you'll see thirty-four, if you actually count it. I actually tried this one year. And spirals in nature always capture the property that they're adjacent... When you count the two spirals, you get two adjacent Fibonacci numbers. With the sunflower, you have this explosion of spirals and, if you count, you usually get fifty-five and eighty-nine, which you notice are two Fibonacci numbers.
Why does nature want to capture this intrinsic beauty? Well, it actually has to do with growth and how these small, little pieces of flower inside there actually grow out and want to pack up space. And they pack space efficiently. It turns out you get these beautiful Fibonacci patterns. But, you see this all over the place. In fact, if you go to the produce section of the grocery store and look at a pineapple, well, it turns out, you'll see Fibonacci numbers there. In fact, in a pineapple, you'll notice there are spirals along the side of it. See, there are spirals sort of this way and another spiral going this way. Do you see that spiral going this way? And if you count them, let's see what we get. Boy, this is a hard thing to count, actually, but, but, happily, I actually got a spiral here, an actual live - and it smells so good, by the way. Boy, this pineapple, I wish, I wish we had smell surround or something and you could just smell it. You know, just smell that pineapple. It really smells oomph! Anyway, you can see there are two sets of spirals here. There's one set of spirals right here, going this way. See that? And there's another set of spirals going this way and sort of interlocking. And, in fact, if we marked them - let me see if I can mark them really, really fast here - if you mark them - so, you can mark them just with the `s' of a sort of sticky white paper. You can actually mark them pretty easily. So, let's mark this spiral right here. Let's mark this spiral. So, I'm putting this white tape - I hope you can see this pretty easily, for a little bit anyway - I'm putting this white tape - it's hard to get white tape to stick on a pineapple, apparently - but I'm putting this along one of the spirals, so you can really get a chance to see one of them. Let's count how many spirals this has. I'm going to start the white tape and count these spirals. So, the white tape is one, then two, three, four, five, six, seven, eight, and look where I am. Eight, I'm back to where I started. So there are eight spirals going that way. And how many spirals going the other way? Well, spirals going the other way, I've got to put the tape on the other direction. And if we put the tape here, what do you see? Let's start counting here. Here I see one, two, three, four, five, six, seven, eight - can't lose my place - nine, ten, eleven, twelve. Uh, no, let's see, I don't think that's right. Let's try it again. One, two, three, four, five, six, seven - I think I did lose my place when I changed over seven - eight, nine, ten, eleven, twelve, thirteen. There we go. Thirteen. So, look, we see eight and thirteen. Look! Two adjacent Fibonacci numbers.
So, spirals in nature turn out to really capture adjacent Fibonacci numbers. So, these Fibonacci numbers are actually quite natural, even though you may think that someone just sort of invented them. It turns out that nature really wants to capture these things and, in fact, in esthetics, Fibonacci numbers actually come into play. And, if you don't believe me, take a look at the golden rectangle math breakup ahead, and you'll see that, in fact, the Fibonacci numbers, oomph, will lead to beauty and esthetics. I'll see you back at math. `Bye.

Embed this video on your site

Copy and paste the following snippet: