WEBVTT

00:00:00.613 --> 00:00:03.652
So why do we learn mathematics?

00:00:03.652 --> 00:00:06.002
Essentially, for three reasons:

00:00:06.002 --> 00:00:07.828
calculation,

00:00:07.828 --> 00:00:09.728
application,

00:00:09.728 --> 00:00:12.415
and last, and unfortunately least

00:00:12.415 --> 00:00:14.520
in terms of the time we give it,

00:00:14.520 --> 00:00:16.442
inspiration.

NOTE Paragraph

00:00:16.442 --> 00:00:18.714
Mathematics is the science of patterns,

00:00:18.714 --> 00:00:22.226
and we study it to learn how to think logically,

00:00:22.226 --> 00:00:24.599
critically, and creatively,

00:00:24.599 --> 00:00:27.525
but too much of the mathematics
that we learn in school

00:00:27.525 --> 00:00:29.844
is not effectively motivated,

00:00:29.844 --> 00:00:31.563
and when our students ask,

00:00:31.563 --> 00:00:33.329
"Why are we learning this?"

00:00:33.329 --> 00:00:35.152
then they often hear that they'll need it

00:00:35.152 --> 00:00:38.463
in an upcoming math class or on a future test.

00:00:38.463 --> 00:00:40.142
But wouldn't it be great

00:00:40.142 --> 00:00:42.490
if every once in a while we did mathematics

00:00:42.490 --> 00:00:45.439
simply because it was fun or beautiful

00:00:45.439 --> 00:00:47.837
or because it excited the mind?

00:00:47.837 --> 00:00:49.251
Now, I know many people have not

00:00:49.251 --> 00:00:51.570
had the opportunity to see how this can happen,

00:00:51.570 --> 00:00:53.399
so let me give you a quick example

00:00:53.399 --> 00:00:55.740
with my favorite collection of numbers,

00:00:55.740 --> 00:00:57.738
the Fibonacci numbers.

NOTE Paragraph

00:00:57.738 --> 00:00:58.823
(Applause)

NOTE Paragraph

00:00:58.823 --> 00:01:00.736
Yeah! I already have Fibonacci fans here.

00:01:00.736 --> 00:01:02.129
That's great.

NOTE Paragraph

00:01:02.129 --> 00:01:03.952
Now these numbers can be appreciated

00:01:03.952 --> 00:01:05.830
in many different ways.

00:01:05.830 --> 00:01:08.539
From the standpoint of calculation,

00:01:08.539 --> 00:01:10.216
they're as easy to understand

00:01:10.216 --> 00:01:12.770
as one plus one, which is two.

00:01:12.770 --> 00:01:14.773
Then one plus two is three,

00:01:14.773 --> 00:01:17.787
two plus three is five, three plus five is eight,

00:01:17.787 --> 00:01:19.543
and so on.

00:01:19.543 --> 00:01:21.489
Indeed, the person we call Fibonacci

00:01:21.489 --> 00:01:24.669
was actually named Leonardo of Pisa,

00:01:24.669 --> 00:01:27.722
and these numbers appear in his book "Liber Abaci,"

00:01:27.722 --> 00:01:29.372
which taught the Western world

00:01:29.372 --> 00:01:32.199
the methods of arithmetic that we use today.

00:01:32.199 --> 00:01:34.089
In terms of amplications,

00:01:34.089 --> 00:01:36.103
Fibonacci numbers appear in nature

00:01:36.103 --> 00:01:37.960
surprisingly often.

00:01:37.960 --> 00:01:39.700
The number of petals on a flower

00:01:39.700 --> 00:01:41.670
is typically a Fibonacci number,

00:01:41.670 --> 00:01:44.332
or the number of spirals on a sunflower

00:01:44.332 --> 00:01:45.943
or a pineapple

00:01:45.943 --> 00:01:48.137
tends to be a Fibonacci number as well.

NOTE Paragraph

00:01:48.137 --> 00:01:51.064
In fact, there are many more
applications of Fibonacci numbers,

00:01:51.064 --> 00:01:54.020
but what I find most inspirational about them

00:01:54.020 --> 00:01:56.934
are the beautiful number patterns they display.

00:01:56.934 --> 00:01:59.128
Let me show you one of my favorites.

00:01:59.128 --> 00:02:01.349
Suppose you like to square numbers,

00:02:01.349 --> 00:02:03.471
and frankly, who doesn't?

NOTE Paragraph

00:02:03.471 --> 00:02:05.077
(Laughter)

NOTE Paragraph

00:02:05.077 --> 00:02:06.635
Let's look at the squares

00:02:06.635 --> 00:02:08.316
of the first few Fibonacci numbers.

00:02:08.316 --> 00:02:10.161
Okay? So one squared is one,

00:02:10.161 --> 00:02:12.478
two squared is four, three squared is nine,

00:02:12.478 --> 00:02:15.651
five squared is 25, and so on.

00:02:15.651 --> 00:02:17.552
All right? Now, it's no surprise

00:02:17.552 --> 00:02:20.038
that when you add consecutive Fibonacci numbers,

00:02:20.038 --> 00:02:22.412
you get the next Fibonacci number. Right?

00:02:22.412 --> 00:02:24.067
That's how they're created.

00:02:24.067 --> 00:02:25.720
But you wouldn't expect anything special

00:02:25.720 --> 00:02:28.656
to happen when you add the squares together.

00:02:28.656 --> 00:02:30.085
But check this out.

00:02:30.085 --> 00:02:32.025
One plus one gives us two,

00:02:32.025 --> 00:02:34.765
and one plus four gives us five.

00:02:34.765 --> 00:02:36.960
And four plus nine is 13,

00:02:36.960 --> 00:02:40.042
nine plus 25 is 34,

00:02:40.042 --> 00:02:42.832
and yes, the pattern continues.

NOTE Paragraph

00:02:42.832 --> 00:02:44.453
In fact, here's another one.

00:02:44.453 --> 00:02:46.297
Suppose you wanted to look at

00:02:46.297 --> 00:02:48.795
adding the squares of
the first few Fibonacci numbers.

00:02:48.795 --> 00:02:50.403
Let's see what we get there.

00:02:50.403 --> 00:02:52.542
So one plus one plus four is six.

00:02:52.542 --> 00:02:55.547
Add nine to that, we get 15.

00:02:55.547 --> 00:02:57.760
Add 25, we get 40.

00:02:57.760 --> 00:03:00.998
Add 64, we get 104.

00:03:00.998 --> 00:03:02.696
Now look at those numbers.

00:03:02.696 --> 00:03:04.587
Those are not Fibonacci numbers,

00:03:04.587 --> 00:03:06.466
but if you look at them closely,

00:03:06.466 --> 00:03:08.349
you'll see the Fibonacci numbers

00:03:08.349 --> 00:03:10.527
buried inside of them.

NOTE Paragraph

00:03:10.527 --> 00:03:12.597
Do you see it? I'll show it to you.

00:03:12.597 --> 00:03:16.330
Six is two times three, 15 is three times five,

00:03:16.330 --> 00:03:18.389
40 is five times eight,

00:03:18.389 --> 00:03:21.317
two, three, five, eight, who do we appreciate?

NOTE Paragraph

00:03:21.317 --> 00:03:22.735
(Laughter)

NOTE Paragraph

00:03:22.735 --> 00:03:24.659
Fibonacci! Of course.

NOTE Paragraph

00:03:24.659 --> 00:03:28.442
Now, as much fun as it is to discover these patterns,

00:03:28.442 --> 00:03:30.924
it's even more satisfying to understand

00:03:30.924 --> 00:03:32.882
why they are true.

00:03:32.882 --> 00:03:34.771
Let's look at that last equation.

00:03:34.771 --> 00:03:38.639
Why should the squares of one, one,
two, three, five, and eight

00:03:38.639 --> 00:03:41.184
add up to eight times 13?

00:03:41.184 --> 00:03:44.145
I'll show you by drawing a simple picture.

00:03:44.145 --> 00:03:46.832
All right? We'll start with a one-by-one square

00:03:46.832 --> 00:03:50.997
and next to that put another one-by-one square.

00:03:50.997 --> 00:03:54.405
Together, they form a one-by-two rectangle.

00:03:54.405 --> 00:03:56.954
Beneath that, I'll put a two-by-two square,

00:03:56.954 --> 00:03:59.749
and next to that, a three-by-three square,

00:03:59.749 --> 00:04:01.750
beneath that, a five-by-five square,

00:04:01.750 --> 00:04:03.662
and then an eight-by-eight square,

00:04:03.662 --> 00:04:06.234
creating one giant rectangle, right?

NOTE Paragraph

00:04:06.234 --> 00:04:08.150
Now let me ask you a simple question:

00:04:08.150 --> 00:04:11.806
what is the area of the rectangle?

00:04:11.806 --> 00:04:13.977
Well, on the one hand,

00:04:13.977 --> 00:04:16.006
it's the sum of the areas

00:04:16.006 --> 00:04:18.173
of the squares inside it, right?

00:04:18.173 --> 00:04:19.917
Just as we created it.

00:04:19.917 --> 00:04:21.889
It's one squared plus one squared

00:04:21.889 --> 00:04:23.937
plus two squared plus three squared

00:04:23.937 --> 00:04:26.536
plus five squared plus eight squared. Right?

00:04:26.536 --> 00:04:28.393
That's the area.

00:04:28.393 --> 00:04:30.719
On the other hand, because it's a rectangle,

00:04:30.719 --> 00:04:34.367
the area is equal to its height times its base,

00:04:34.367 --> 00:04:36.414
and the height is clearly eight,

00:04:36.414 --> 00:04:39.317
and the base is five plus eight,

00:04:39.317 --> 00:04:43.255
which is the next Fibonacci number, 13. Right?

00:04:43.255 --> 00:04:46.618
So the area is also eight times 13.

00:04:46.618 --> 00:04:48.880
Since we've correctly calculated the area

00:04:48.880 --> 00:04:50.691
two different ways,

00:04:50.691 --> 00:04:52.739
they have to be the same number,

00:04:52.739 --> 00:04:56.013
and that's why the squares of one,
one, two, three, five, and eight

00:04:56.013 --> 00:04:58.421
add up to eight times 13.

NOTE Paragraph

00:04:58.421 --> 00:05:00.795
Now, if we continue this process,

00:05:00.795 --> 00:05:04.773
we'll generate rectangles of the form 13 by 21,

00:05:04.773 --> 00:05:07.521
21 by 34, and so on.

NOTE Paragraph

00:05:07.521 --> 00:05:09.115
Now check this out.

00:05:09.115 --> 00:05:10.908
If you divide 13 by eight,

00:05:10.908 --> 00:05:13.012
you get 1.625.

00:05:13.012 --> 00:05:16.084
And if you divide the larger number
by the smaller number,

00:05:16.084 --> 00:05:19.112
then these ratios get closer and closer

00:05:19.112 --> 00:05:21.765
to about 1.618,

00:05:21.765 --> 00:05:25.039
known to many people as the Golden Ratio,

00:05:25.039 --> 00:05:27.662
a number which has fascinated mathematicians,

00:05:27.662 --> 00:05:31.017
scientists, and artists for centuries.

NOTE Paragraph

00:05:31.017 --> 00:05:33.139
Now, I show all this to you because,

00:05:33.139 --> 00:05:35.164
like so much of mathematics,

00:05:35.164 --> 00:05:37.131
there's a beautiful side to it

00:05:37.131 --> 00:05:39.316
that I fear does not get enough attention

00:05:39.316 --> 00:05:40.929
in our schools.

00:05:40.929 --> 00:05:43.546
We spend lots of time learning about calculation,

00:05:43.546 --> 00:05:46.302
but let's not forget about application,

00:05:46.302 --> 00:05:49.756
including, perhaps, the most
important application of all,

00:05:49.756 --> 00:05:51.832
learning how to think.

NOTE Paragraph

00:05:51.832 --> 00:05:53.943
If I could summarize this in one sentence,

00:05:53.943 --> 00:05:55.820
it would be this:

00:05:55.820 --> 00:05:58.610
mathematics is not just solving for x,

00:05:58.610 --> 00:06:01.535
it's also figuring out why.

NOTE Paragraph

00:06:01.535 --> 00:06:03.920
Thank you very much.

NOTE Paragraph

00:06:03.920 --> 00:06:07.757
(Applause)