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TEDGlobal 2013

Arthur Benjamin: The magic of Fibonacci numbers

アーサー・ベンジャミン: フィボナッチ数の魅力

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数学は論理的かつ機能的そして・・・スゴいのです。数学マジシャンのアーサー・ベンジャミンが探るのは、不思議で奇妙な数の集合「フィボナッチ数列」の隠れた性質です。(それに数学は想像力を刺激することだってできるのです!)

- 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?
なぜ数学を学ぶのでしょうか?
00:12
Essentially, for three reasons:
本質的には3つの理由があります
00:15
calculation,
計算するため
00:18
application,
応用するため
00:19
and last, and unfortunately least
そして 発想するためです
00:21
in terms of the time we give it,
発想に時間をかけないのは
00:24
inspiration.
残念なことですが・・・
00:26
Mathematics is the science of patterns,
数学とはパターンの科学です
00:28
and we study it to learn how to think logically,
ここから論理的 批判的 創造的な
00:30
critically and creatively,
考え方を学べるのです
00:34
but too much of the mathematics
that we learn in school
一方 学校で習う数学は
00:36
is not effectively motivated,
効果的に意欲を
高めているとは言えません
00:39
and when our students ask,
数学を勉強する理由を
生徒がたずねても
00:41
"Why are we learning this?"
数学を勉強する理由を
生徒がたずねても
00:43
then they often hear that they'll need it
授業で いつか使うからとか
00:44
in an upcoming math class or on a future test.
テストに出るからと
言われることも多いのです
00:46
But wouldn't it be great
でも 時々でいいから
00:50
if every once in a while we did mathematics
面白くて美しくて
ワクワクするから
00:51
simply because it was fun or beautiful
数学を学ぶという
機会がもてたら
00:54
or because it excited the mind?
素敵だと思いませんか
00:57
Now, I know many people have not
でも そんな機会の作り方が
00:59
had the opportunity to see how this can happen,
わからないという
声も聞きます
01:01
so let me give you a quick example
そこで私のお気に入りの数から
01:03
with my favorite collection of numbers,
ちょっとした例を挙げましょう
01:05
the Fibonacci numbers. (Applause)
フィボナッチ数です (拍手)
01:07
Yeah! I already have Fibonacci fans here.
ここにもフィボナッチ・
ファンがいますね
01:10
That's great.
素晴らしい
01:12
Now these numbers can be appreciated
この数列はいろいろな角度から
01:13
in many different ways.
楽しむことができます
01:15
From the standpoint of calculation,
計算の面では
01:17
they're as easy to understand
わかりやすい数列です
01:20
as one plus one, which is two.
1足す 1は 2で
01:22
Then one plus two is three,
1足す 2で 3 —
01:24
two plus three is five, three plus five is eight,
2足す 3で 5
3足す 5で 8と
01:26
and so on.
続きます
01:29
Indeed, the person we call Fibonacci
「フィボナッチ」の本名は
01:31
was actually named Leonardo of Pisa,
ピサのレオナルドです
01:33
and these numbers appear in his book "Liber Abaci,"
彼の著書『算盤の書』で
この数列が紹介されました
01:36
which taught the Western world
現在使われる計算方法は
01:39
the methods of arithmetic that we use today.
この本を通して
西洋世界に伝わりました
01:41
In terms of applications,
応用の点から言うと
01:44
Fibonacci numbers appear in nature
フィボナッチ数は
01:45
surprisingly often.
自然界にあふれています
01:48
The number of petals on a flower
花びらの数は普通 —
01:49
is typically a Fibonacci number,
フィボナッチ数です
01:51
or the number of spirals on a sunflower
ひまわりの花や
パイナップルに見られる
01:53
or a pineapple
らせんの数も
01:56
tends to be a Fibonacci number as well.
フィボナッチ数が多いです
01:57
In fact, there are many more
applications of Fibonacci numbers,
この数は さらに
いろいろなものに見出せます
02:00
but what I find most inspirational about them
ただ最も想像力を
かき立てられるのは
02:03
are the beautiful number patterns they display.
この数列の美しい規則性です
02:06
Let me show you one of my favorites.
お気に入りを一つ紹介します
02:08
Suppose you like to square numbers,
平方数は
02:11
and frankly, who doesn't? (Laughter)
皆さん お好きですよね(笑)
02:13
Let's look at the squares
フィボナッチ数の最初のいくつかを
02:16
of the first few Fibonacci numbers.
それぞれ 2乗してみましょう
02:18
So one squared is one,
1の 2乗は 1 —
02:20
two squared is four, three squared is nine,
2の 2乗は 4
3の 2乗は 9 —
02:22
five squared is 25, and so on.
5の 2乗は 25と続きます
02:24
Now, it's no surprise
さて 連続するフィボナッチ数を
02:27
that when you add consecutive Fibonacci numbers,
加えると次の数を得ることが
02:29
you get the next Fibonacci number. Right?
できますよね
02:32
That's how they're created.
そういう作り方ですから
02:34
But you wouldn't expect anything special
でも 2乗した数 同士を
02:35
to happen when you add the squares together.
加えても何も
起こらないと思うでしょう
02:37
But check this out.
でも ご覧ください
02:40
One plus one gives us two,
1 + 1 = 2 —
02:42
and one plus four gives us five.
1 + 4 = 5 —
02:44
And four plus nine is 13,
4 + 9 = 13 —
02:46
nine plus 25 is 34,
9 + 25 = 34 になり
02:48
and yes, the pattern continues.
このパターンが続くのです
02:52
In fact, here's another one.
実は もう一つあります
02:54
Suppose you wanted to look at
フィボナッチ数を2乗したものを
02:56
adding the squares of
the first few Fibonacci numbers.
最初から足していってみましょう
02:58
Let's see what we get there.
どうなるでしょうか
03:00
So one plus one plus four is six.
1 + 1 + 4 = 6 です
03:02
Add nine to that, we get 15.
これに 9を加えると 15になります
03:04
Add 25, we get 40.
25を加えると 40に
03:07
Add 64, we get 104.
64を加えると 104になります
03:09
Now look at those numbers.
出てきた数を調べましょう
03:12
Those are not Fibonacci numbers,
フィボナッチ数には
なっていませんが
03:14
but if you look at them closely,
よく見ると
03:16
you'll see the Fibonacci numbers
フィボナッチ数が
03:18
buried inside of them.
隠れていますよ
03:20
Do you see it? I'll show it to you.
わかりますか?
ご覧に入れましょう
03:22
Six is two times three, 15 is three times five,
6 = 2 x 3
15 = 3 x 5 —
03:24
40 is five times eight,
40 = 5 x 8 です
03:28
two, three, five, eight, who do we appreciate?
2 3 5 8 ・・・
わかりますか?
03:30
(Laughter)
(笑)
03:33
Fibonacci! Of course.
フィボナッチ数ですよね
03:34
Now, as much fun as it is to discover these patterns,
さて こんな規則性を
見つけるのは面白いですが
03:36
it's even more satisfying to understand
なぜそうなるかを理解すれば
03:40
why they are true.
さらに楽しくなります
03:42
Let's look at that last equation.
一番下の方程式を見てください
03:44
Why should the squares of one, one,
two, three, five and eight
なぜ 1 1 2 3 5 8 の平方数を足すと
03:46
add up to eight times 13?
8 x 13 になるのでしょうか
03:50
I'll show you by drawing a simple picture.
簡単な図で示します
03:53
We'll start with a one-by-one square
1 x 1 の正方形から始めて
03:56
and next to that put another one-by-one square.
隣に 1 x 1 の正方形を置きます
03:58
Together, they form a one-by-two rectangle.
合わせると 1 x 2 の
長方形ができます
04:02
Beneath that, I'll put a two-by-two square,
その下に 2 x 2 の正方形 —
04:06
and next to that, a three-by-three square,
隣に 3 x 3 の正方形を置き
04:08
beneath that, a five-by-five square,
また下に 5 x 5 の正方形 —
04:11
and then an eight-by-eight square,
隣に 8 x 8 の正方形を置くと
04:13
creating one giant rectangle, right?
大きな長方形が出来ます
04:15
Now let me ask you a simple question:
さて 簡単な質問をしましょう
04:18
what is the area of the rectangle?
長方形の面積は?
04:20
Well, on the one hand,
一つのやり方は
04:23
it's the sum of the areas
面積は正方形の面積の
04:25
of the squares inside it, right?
合計ですね
04:28
Just as we created it.
そう作ったのですから
04:30
It's one squared plus one squared
1の2乗プラス 1の2乗プラス
04:31
plus two squared plus three squared
2の2乗プラス 3の2乗プラス —
04:33
plus five squared plus eight squared. Right?
5の2乗プラス 8の2乗ですよね
04:35
That's the area.
これが面積です
04:38
On the other hand, because it's a rectangle,
一方 これは長方形ですから
04:40
the area is equal to its height times its base,
面積は たて x よこ です
04:42
and the height is clearly eight,
たては 8ですね
04:46
and the base is five plus eight,
よこは 5 + 8 なので
04:48
which is the next Fibonacci number, 13. Right?
次のフィナボッチ数である
13です
04:51
So the area is also eight times 13.
だから面積は 8 x 13 です
04:55
Since we've correctly calculated the area
面積を2種類の方法で
04:58
two different ways,
計算できました
05:00
they have to be the same number,
結果はお互いに同じなので
05:02
and that's why the squares of one,
one, two, three, five and eight
1 1 2 3 5 8 の平方数を足すと
05:04
add up to eight times 13.
8 x 13 になると言えるのです
05:08
Now, if we continue this process,
さて このプロセスを続けると
05:10
we'll generate rectangles of the form 13 by 21,
13 x 21や 21 x 34といった長方形を
05:12
21 by 34, and so on.
作り続けることができます
05:16
Now check this out.
では今度は
05:19
If you divide 13 by eight,
13を 8で割ってみると
05:20
you get 1.625.
1.625になります
05:22
And if you divide the larger number
by the smaller number,
大きい方の数を
小さい方の数で割ると
05:24
then these ratios get closer and closer
その結果は次第に
05:28
to about 1.618,
およそ 1.618に近づいていきます
05:31
known to many people as the Golden Ratio,
この数こそ「黄金比」と
呼ばれる比率です
05:33
a number which has fascinated mathematicians,
多くの数学者 科学者 芸術家達を
05:37
scientists and artists for centuries.
何世紀もの間
魅了してきた数です
05:39
Now, I show all this to you because,
今回 この題材を取り上げた理由は
05:42
like so much of mathematics,
数学の大半がそうであるように
05:45
there's a beautiful side to it
美しい部分があるからです
05:47
that I fear does not get enough attention
ただ学校で このような美は
05:49
in our schools.
あまり注目されません
05:51
We spend lots of time learning about calculation,
計算の仕方は
長い期間をかけて学びますが
05:52
but let's not forget about application,
実際に応用することを
忘れてはいけません
05:55
including, perhaps, the most
important application of all,
とりわけ重要なのは
考え方を学ぶ時に
05:58
learning how to think.
数学を応用することです
06:01
If I could summarize this in one sentence,
一言でまとめるとすれば
06:03
it would be this:
こうなるでしょう
06:05
Mathematics is not just solving for x,
「数学とは xの解を
求めるだけでなく
06:07
it's also figuring out why.
理由 “why” を
解明する学問である」
06:10
Thank you very much.
どうもありがとうございました
06:13
(Applause)
(拍手)
06:15
Translated by Kazunori Akashi
Reviewed by Yuko Yoshida

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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").

More profile about the speaker
Arthur Benjamin | Speaker | TED.com