TEDGlobal 2013

Arthur Benjamin: The magic of Fibonacci numbers

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Math is logical, functional and just ... awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

- Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio

So why do we learn mathematics?
00:12
Essentially, for three reasons:
00:15
calculation,
00:18
application,
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and last, and unfortunately least
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in terms of the time we give it,
00:24
inspiration.
00:26
Mathematics is the science of patterns,
00:28
and we study it to learn how to think logically,
00:30
critically and creatively,
00:33
but too much of the mathematics
that we learn in school
00:36
is not effectively motivated,
00:39
and when our students ask,
00:41
"Why are we learning this?"
00:43
then they often hear that they'll need it
00:44
in an upcoming math class or on a future test.
00:46
But wouldn't it be great
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if every once in a while we did mathematics
00:51
simply because it was fun or beautiful
00:54
or because it excited the mind?
00:57
Now, I know many people have not
00:59
had the opportunity to see how this can happen,
01:01
so let me give you a quick example
01:03
with my favorite collection of numbers,
01:05
the Fibonacci numbers. (Applause)
01:07
Yeah! I already have Fibonacci fans here.
01:10
That's great.
01:12
Now these numbers can be appreciated
01:13
in many different ways.
01:15
From the standpoint of calculation,
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they're as easy to understand
01:20
as one plus one, which is two.
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Then one plus two is three,
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two plus three is five, three plus five is eight,
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and so on.
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Indeed, the person we call Fibonacci
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was actually named Leonardo of Pisa,
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and these numbers appear in his book "Liber Abaci,"
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which taught the Western world
01:39
the methods of arithmetic that we use today.
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In terms of applications,
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Fibonacci numbers appear in nature
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surprisingly often.
01:47
The number of petals on a flower
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is typically a Fibonacci number,
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or the number of spirals on a sunflower
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or a pineapple
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tends to be a Fibonacci number as well.
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In fact, there are many more
applications of Fibonacci numbers,
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but what I find most inspirational about them
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are the beautiful number patterns they display.
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Let me show you one of my favorites.
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Suppose you like to square numbers,
02:10
and frankly, who doesn't? (Laughter)
02:13
Let's look at the squares
02:15
of the first few Fibonacci numbers.
02:18
So one squared is one,
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two squared is four, three squared is nine,
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five squared is 25, and so on.
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Now, it's no surprise
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that when you add consecutive Fibonacci numbers,
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you get the next Fibonacci number. Right?
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That's how they're created.
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But you wouldn't expect anything special
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to happen when you add the squares together.
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But check this out.
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One plus one gives us two,
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and one plus four gives us five.
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And four plus nine is 13,
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nine plus 25 is 34,
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and yes, the pattern continues.
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In fact, here's another one.
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Suppose you wanted to look at
02:56
adding the squares of
the first few Fibonacci numbers.
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Let's see what we get there.
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So one plus one plus four is six.
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Add nine to that, we get 15.
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Add 25, we get 40.
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Add 64, we get 104.
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Now look at those numbers.
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Those are not Fibonacci numbers,
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but if you look at them closely,
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you'll see the Fibonacci numbers
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buried inside of them.
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Do you see it? I'll show it to you.
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Six is two times three, 15 is three times five,
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40 is five times eight,
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two, three, five, eight, who do we appreciate?
03:30
(Laughter)
03:33
Fibonacci! Of course.
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Now, as much fun as it is to discover these patterns,
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it's even more satisfying to understand
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why they are true.
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Let's look at that last equation.
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Why should the squares of one, one,
two, three, five and eight
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add up to eight times 13?
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I'll show you by drawing a simple picture.
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We'll start with a one-by-one square
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and next to that put another one-by-one square.
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Together, they form a one-by-two rectangle.
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Beneath that, I'll put a two-by-two square,
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and next to that, a three-by-three square,
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beneath that, a five-by-five square,
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and then an eight-by-eight square,
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creating one giant rectangle, right?
04:15
Now let me ask you a simple question:
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what is the area of the rectangle?
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Well, on the one hand,
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it's the sum of the areas
04:25
of the squares inside it, right?
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Just as we created it.
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It's one squared plus one squared
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plus two squared plus three squared
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plus five squared plus eight squared. Right?
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That's the area.
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On the other hand, because it's a rectangle,
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the area is equal to its height times its base,
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and the height is clearly eight,
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and the base is five plus eight,
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which is the next Fibonacci number, 13. Right?
04:51
So the area is also eight times 13.
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Since we've correctly calculated the area
04:58
two different ways,
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they have to be the same number,
05:02
and that's why the squares of one,
one, two, three, five and eight
05:04
add up to eight times 13.
05:07
Now, if we continue this process,
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we'll generate rectangles of the form 13 by 21,
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21 by 34, and so on.
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Now check this out.
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If you divide 13 by eight,
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you get 1.625.
05:22
And if you divide the larger number
by the smaller number,
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then these ratios get closer and closer
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to about 1.618,
05:30
known to many people as the Golden Ratio,
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a number which has fascinated mathematicians,
05:36
scientists and artists for centuries.
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Now, I show all this to you because,
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like so much of mathematics,
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there's a beautiful side to it
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that I fear does not get enough attention
05:48
in our schools.
05:50
We spend lots of time learning about calculation,
05:52
but let's not forget about application,
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including, perhaps, the most
important application of all,
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learning how to think.
06:01
If I could summarize this in one sentence,
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it would be this:
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Mathematics is not just solving for x,
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it's also figuring out why.
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Thank you very much.
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(Applause)
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About the Speaker:

Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty.

Why you should listen

Arthur Benjamin makes numbers dance. In his day job, he's a professor of math at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage in his tuxedo to perform high-speed mental calculations, memorizations and other astounding math stunts. It's part of his drive to teach math and mental agility in interesting ways, following in the footsteps of such heroes as Martin Gardner.

Benjamin is the co-author, with Michael Shermer, of Secrets of Mental Math (which shares his secrets for rapid mental calculation), as well as the co-author of the MAA award-winning Proofs That Really Count: The Art of Combinatorial Proof. For a glimpse of his broad approach to math, see the list of research talks on his website, which seesaws between high-level math (such as his "Vandermonde's Determinant and Fibonacci SAWs," presented at MIT in 2004) and engaging math talks for the rest of us ("An Amazing Mathematical Card Trick").