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So why do we learn mathematics?

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Essentially, for three reasons:

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calculation,

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application,

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and last, and unfortunately least

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in terms of the time we give it,

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inspiration.

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Mathematics is the science of patterns,

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and we study it to learn how to think logically,

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critically and creatively,

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but too much of the mathematics
that we learn in school

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is not effectively motivated,

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and when our students ask,

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"Why are we learning this?"

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then they often hear that they'll need it

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in an upcoming math class or on a future test.

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But wouldn't it be great

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if every once in a while we did mathematics

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simply because it was fun or beautiful

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or because it excited the mind?

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Now, I know many people have not

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had the opportunity to see how this can happen,

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so let me give you a quick example

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with my favorite collection of numbers,

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the Fibonacci numbers. (Applause)

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Yeah! I already have Fibonacci fans here.

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That's great.

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Now these numbers can be appreciated

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in many different ways.

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From the standpoint of calculation,

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they're as easy to understand

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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

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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.

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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,

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and frankly, who doesn't? (Laughter)

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Let's look at the squares

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of the first few Fibonacci numbers.

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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

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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?

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(Laughter)

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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?

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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

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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?

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So the area is also eight times 13.

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Since we've correctly calculated the area

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two different ways,

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they have to be the same number,

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and that's why the squares of one,
one, two, three, five and eight

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add up to eight times 13.

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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.

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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,

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known to many people as the Golden Ratio,

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a number which has fascinated mathematicians,

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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

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in our schools.

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We spend lots of time learning about calculation,

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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.

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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)