The magic of Fibonacci numbers | Arthur Benjamin

The magic of Fibonacci numbers | Arthur Benjamin


So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration. Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics
that we learn in school is not effectively motivated, and when our students ask, “Why are we learning this?” then they often hear that they’ll need it in an upcoming math class or on a future test. But wouldn’t it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind? Now, I know many people have not had the opportunity to see how this can happen, so let me give you a quick example with my favorite collection of numbers, the Fibonacci numbers. (Applause) Yeah! I already have Fibonacci fans here. That’s great. Now these numbers can be appreciated in many different ways. From the standpoint of calculation, they’re as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on. Indeed, the person we call Fibonacci was actually named Leonardo of Pisa, and these numbers appear in his book “Liber Abaci,” which taught the Western world the methods of arithmetic that we use today. In terms of applications, Fibonacci numbers appear in nature surprisingly often. The number of petals on a flower is typically a Fibonacci number, or the number of spirals on a sunflower or a pineapple tends to be a Fibonacci number as well. In fact, there are many more
applications of Fibonacci numbers, but what I find most inspirational about them are the beautiful number patterns they display. Let me show you one of my favorites. Suppose you like to square numbers, and frankly, who doesn’t? (Laughter) Let’s look at the squares of the first few Fibonacci numbers. So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on. Now, it’s no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right? That’s how they’re created. But you wouldn’t expect anything special to happen when you add the squares together. But check this out. One plus one gives us two, and one plus four gives us five. And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues. In fact, here’s another one. Suppose you wanted to look at adding the squares of
the first few Fibonacci numbers. Let’s see what we get there. So one plus one plus four is six. Add nine to that, we get 15. Add 25, we get 40. Add 64, we get 104. Now look at those numbers. Those are not Fibonacci numbers, but if you look at them closely, you’ll see the Fibonacci numbers buried inside of them. Do you see it? I’ll show it to you. Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate? (Laughter) Fibonacci! Of course. Now, as much fun as it is to discover these patterns, it’s even more satisfying to understand why they are true. Let’s look at that last equation. Why should the squares of one, one,
two, three, five and eight add up to eight times 13? I’ll show you by drawing a simple picture. We’ll start with a one-by-one square and next to that put another one-by-one square. Together, they form a one-by-two rectangle. Beneath that, I’ll put a two-by-two square, and next to that, a three-by-three square, beneath that, a five-by-five square, and then an eight-by-eight square, creating one giant rectangle, right? Now let me ask you a simple question: what is the area of the rectangle? Well, on the one hand, it’s the sum of the areas of the squares inside it, right? Just as we created it. It’s one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? That’s the area. On the other hand, because it’s a rectangle, the area is equal to its height times its base, and the height is clearly eight, and the base is five plus eight, which is the next Fibonacci number, 13. Right? So the area is also eight times 13. Since we’ve correctly calculated the area two different ways, they have to be the same number, and that’s why the squares of one,
one, two, three, five and eight add up to eight times 13. Now, if we continue this process, we’ll generate rectangles of the form 13 by 21, 21 by 34, and so on. Now check this out. If you divide 13 by eight, you get 1.625. And if you divide the larger number
by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians, scientists and artists for centuries. Now, I show all this to you because, like so much of mathematics, there’s a beautiful side to it that I fear does not get enough attention in our schools. We spend lots of time learning about calculation, but let’s not forget about application, including, perhaps, the most
important application of all, learning how to think. If I could summarize this in one sentence, it would be this: Mathematics is not just solving for x, it’s also figuring out why. Thank you very much. (Applause)

100 thoughts on “The magic of Fibonacci numbers | Arthur Benjamin

  1. Thank You!
    Whenever I told a teacher I didn't understand, they would show me exactly what they did before
    but I didn't understand it, not because what the teacher was showing was not understandable
    but because I could never wrap my head as to how this got to this.
    lol

  2. I would have done a lot better in math class as a kid if we would have considered Y creatively. Nice talk Arthur! I became fascinated in Fibonacci numbers in my 20s especially in music. It's been a life long passion ever since.

  3. Didn't know the comedian Martin short was a mathematician. Kept expecting Steve Martin to walk out dressed as a mariachi

  4. Granted, English isn't my native language, but I've spoken it for +30 years. Yet, it wasn't until today that I've given any thought as to what "squared" means in a mathematical sense, and then this one, all too short TED talk explains it in a matter of seconds.

    Just goes to show that the guy is right: It wouldn't take a whole lot of tweaks to make basic school maths a whole lot more interesting.

  5. relationship between fibnocassi series and cutting
    like if we cut one time we have two piece
    two cut give three piece
    three cut give four piece
    and so on
    so find out where the graph correlation

  6. What Fibonacci didn’t discover this sequence? Didn’t they exist? Why should I appreciate Fibonacci if the squad of Fibonacci sequences gives us another Fibonacci number!!! It is in nature and we should be thanks for the creator of nature not someone who finds it.

  7. I wonder: is the Golden Ratio the ratio for playing jazz (not quite 2 to 1, not quite 6/8 time). It would seem so, but I've never heard/read that theory put forth.

  8. i was just scrolling around on youtube then i found this video!! This was such a nice explanation!! <3 im greatly fascinated with the patterns, haha!

  9. If you want another example of how we see the Fibonacci number on nature just divide your total height by your height from your feet to the belly button. You should get in theory 1.6 or the "Fibonacci number"

  10. Google is listening, I was just learning fibonacci in school! Btw we also watched this video in school!

  11. It's the interpretation of mathematical results that often gets overlooked. A lot of connections may be found after interpretation of math results….For example the connection between Golden ratio and Fibonacci numbers was beautifully put…..Good talk👍

  12. This kind of Maths is really boring. Anything Calculus and beyond is something. If people fail to appreciate Maths at school, they can surely attempt to be lawyer.

  13. I haven’t done math in 5-6 years and now I can’t even see myself doing pre algebra. I stopped doing math when I was 16 when I passed algebra 2 in the 0 period computer make up class before 1st period.

  14. Columbus discovered America? Fibanocci discovered this equation? Please credit to what was there already. Indian scholars had this equation 1,000 years before Pizza man.

  15. I still don't get it, so I'm going to smoke another bong and go lay out in my backyard and look at the stars an figure out the observable universe

  16. Remember when our teachers said "You have to learn math because you're not always going to have a calculator with you."
    Wrong Ms. Davis 😀

  17. If you look for it hard enough… There is always a (co)relation in progression and order of numbers along with geometrical observations… nothing spectacular about it just common sense haha

  18. Funny how Fibonacci gets credit for something that was invented in India earlier than him(Pingala-Hemachandra series). Oh must be because he is Italian/western

  19. Totally wastage of time. He didn't tell even a single useful application of it. I mean HOW can we use this and where? What the heck in his examples: Petals in flower is fibonacci, 8×13 area of rectangle and blah blah… is fibonacci.

  20. Give your life to Jesus Christ. Believe and Pray: God, I know that I am a sinner and unless you save me I will be lost forever. I believe that Jesus Christ is the true Son of God, died on the cross for my sins and arose on the 3rd day. I accept Jesus as my Lord and Savior, in the name of Jesus, Amen.

  21. Remarkable maths dude, I just think you gotta find an additional skill, that'll help emphasis more, other than that, appreciations and best wishes, thank you

  22. It's a good video. But for the correction this was done in India by mathematician Pingla a thousand years before Fibonacci. Fibonacci just put his stamp on the stolen work.

  23. Numbers curl around orthogonal numbers. 3 and 4 and 5. Form an inward spiral. They are called Fibonacci.

  24. How many you guys wish if he is your math teacher when u were in elementary school? So you dont hate math, like you hate the teacher..

  25. That was lovely. Figuring out WHY! I always tell everyone, it's not good enough to just remember the equation, but why the equation exists, and how did we come up with it.

  26. This makes sense to me! I remember back in high school I'm always wondering the "why's" for the formula. I mean, math teachers force us to memorize the formula but doesn't explain to us why it was formed, what significance does it has, on what real life scenario we can apply it to.

  27. Holy crap !! I have known the Fibonacci sequence for 30 years, I even can calculate the general term (without the double recurrence formula) but I didn't know the square and sum stuff.
    If every math teacher in the world was as good and passionnate as Mr Benjamin, evreybody would love math !

  28. I had to stop watching this, I was afraid I might have too much fun and my brain would explode — I might take up stamp collecting or train spotting as they seem very exciting as well — colours, ha ha ha ha! ….different shapes, ha ha ha haA!!! jokes about stamps, haa haa haaa haaaa-Aaaaaaaaaaa!!!!

  29. I use Mathematics in my Art. Was never taught that it is in everything. Would have made it much more interesting. Also Sacred Geometry.

  30. Sometimes,
    somebody,
    tell us the Fibonacci numbers is responsible for everything.
    but I think this is wrong.
    the number is responsible for a lot of things .
    but not for everything.
    And so I couldn't understand, why the Fibonacci Schuld be responsible for everything .
    the fact is , you must find the relation between the numbers and the objects.
    and if you don't find the logic.
    you must find another Logik
    You must use another number.
    So I think the Fibonacci number is useful for metric measuring.
    Or a new number Base for Computer operation.
    Or for funk . Telecomunication.
    Or alghorytmic.
    Maybe approximation.

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