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
TEDGlobal 2013

Arthur Benjamin: The magic of Fibonacci numbers

Arthur Benjamin: 費波那西數列的魔力

Filmed:
7,057,274 views

數學是邏輯的、功能性的,並且簡直是......棒的。數學魔術師 Arthur Benjamin 探索了怪異又奇妙的費波那西數列的隱藏特性(並且提醒你,數學也是可以鼓舞人心的!)。
- Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio

Double-click the English transcript below to play the video.

00:12
So why do we learn學習 mathematics數學?
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我們為什麼要學數學?
00:15
Essentially實質上, for three reasons原因:
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主要有三個原因:
00:18
calculation計算,
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計算
00:19
application應用,
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應用
00:21
and last, and unfortunately不幸 least最小
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最後,不幸地,也是最不重要的,
00:24
in terms條款 of the time we give it,
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就我們所給予它的時間來看,
00:26
inspiration靈感.
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靈感。
00:28
Mathematics數學 is the science科學 of patterns模式,
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數學是规律的科學,
00:30
and we study研究 it to learn學習 how to think logically邏輯,
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而我們學習數學是為了學習怎樣邏輯地,
00:34
critically危重 and creatively創造性,
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批評地和有創造性地思考,
00:36
but too much of the mathematics數學
that we learn學習 in school學校
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但是,太多我們在學校學的數學
00:39
is not effectively有效 motivated動機,
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並沒有效地激勵學生思考
00:41
and when our students學生們 ask,
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所以當學生問我們,
00:43
"Why are we learning學習 this?"
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“我們為什麼要學這個?”
00:44
then they often經常 hear that they'll他們會 need it
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他們會聽到(我們說)因為下一節是數學課
00:46
in an upcoming即將到來 math數學 class or on a future未來 test測試.
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或者將來會有考試,他們需要這個。
00:50
But wouldn't不會 it be great
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可是,如果
00:51
if every一切 once一旦 in a while we did mathematics數學
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偶爾我們學數學
00:54
simply只是 because it was fun開玩笑 or beautiful美麗
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僅僅是因為數學很有趣或迷人,
00:57
or because it excited興奮 the mind心神?
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或者因為它激發思想,不是很好嗎?
00:59
Now, I know many許多 people have not
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我知道很多人都還沒有
01:01
had the opportunity機會 to see how this can happen發生,
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機會去看到數學如何可以有趣,
01:03
so let me give you a quick example
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所以讓我用我最喜歡的一組數字,
01:05
with my favorite喜愛 collection採集 of numbers數字,
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來給你舉個小小的例子,
01:07
the Fibonacci斐波那契 numbers數字. (Applause掌聲)
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費波那西數。(鼓掌)
01:10
Yeah! I already已經 have Fibonacci斐波那契 fans球迷 here.
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哇,這裡已經有費波那西數的愛好者了。
01:12
That's great.
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不錯。
01:13
Now these numbers數字 can be appreciated讚賞
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(我們可以)從很多個方面來
01:15
in many許多 different不同 ways方法.
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欣賞這組數字。
01:17
From the standpoint立場 of calculation計算,
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從計算上來看,
01:20
they're as easy簡單 to understand理解
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它們非常易懂
01:22
as one plus one, which哪一個 is two.
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比如,1加1,是2.
01:24
Then one plus two is three,
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1加2是3,
01:26
two plus three is five, three plus five is eight,
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2加3是5,3加5是8,
01:29
and so on.
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等等。
01:31
Indeed確實, the person we call Fibonacci斐波那契
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事實上,我們稱做“費波那西”的這個人
01:33
was actually其實 named命名 Leonardo萊昂納多 of Pisa比薩,
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是比薩的莱昂纳多,
01:36
and these numbers數字 appear出現 in his book "Liber萊博 Abaci算盤,"
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而這些數字是在他的“計算之書”中描述的,
01:39
which哪一個 taught the Western西 world世界
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這本書教授了西方世界
01:41
the methods方法 of arithmetic算術 that we use today今天.
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我們今天所使用的算術方法。
01:44
In terms條款 of applications應用,
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從應用上來看,
01:45
Fibonacci斐波那契 numbers數字 appear出現 in nature性質
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費波那西數讓人驚訝地
01:48
surprisingly出奇 often經常.
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頻繁出現在自然界裡。
01:49
The number of petals花瓣 on a flower
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花瓣的數目
01:51
is typically一般 a Fibonacci斐波那契 number,
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通常是一個費波那西數字,
01:53
or the number of spirals螺旋 on a sunflower向日葵
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或向日葵上、鳳梨上的螺旋數
01:56
or a pineapple菠蘿
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01:57
tends趨向 to be a Fibonacci斐波那契 number as well.
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往往也是費波那西數字。
02:00
In fact事實, there are many許多 more
applications應用 of Fibonacci斐波那契 numbers數字,
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事實上,費波那西數有更多的應用,
02:03
but what I find most inspirational勵志 about them
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但我發現最鼓舞人心的
02:06
are the beautiful美麗 number patterns模式 they display顯示.
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是它們所顯示的漂亮的數字规律。
02:08
Let me show顯示 you one of my favorites最愛.
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讓我給你看看我的最愛之一。
02:11
Suppose假設 you like to square廣場 numbers數字,
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假設你喜歡平方數,
02:13
and frankly坦率地說, who doesn't? (Laughter笑聲)
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坦率地說,誰不喜歡?(笑聲)
02:16
Let's look at the squares廣場
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讓我們看看頭幾個
02:18
of the first few少數 Fibonacci斐波那契 numbers數字.
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費波那西數的平方。
02:20
So one squared平方 is one,
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1的平方是1,
02:22
two squared平方 is four, three squared平方 is nine,
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2 的平方是4,3的平方是9,
02:24
five squared平方 is 25, and so on.
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5 的平方是 25,依此類推。
02:27
Now, it's no surprise
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可想而知,
02:29
that when you add consecutive連續 Fibonacci斐波那契 numbers數字,
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當你把相鄰的两個費波那西數加起來時,
02:32
you get the next下一個 Fibonacci斐波那契 number. Right?
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會得到下一個費波那西數。對吧?
02:34
That's how they're created創建.
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這就是它們如何被定義的。
02:35
But you wouldn't不會 expect期望 anything special特別
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但你大概不會料到
02:37
to happen發生 when you add the squares廣場 together一起.
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當你把這些數的平方加起來,
會有什麼特別的結果。
02:40
But check this out.
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看這個,
02:42
One plus one gives us two,
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1加1是2,
02:44
and one plus four gives us five.
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然後,1加4是5。
02:46
And four plus nine is 13,
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4加9是13,
02:48
nine plus 25 is 34,
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9 加 25 是 34,
02:52
and yes, the pattern模式 continues繼續.
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是的,這個規律一直繼續下去。
02:54
In fact事實, here's這裡的 another另一個 one.
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事實上,還有另外一個。
02:56
Suppose假設 you wanted to look at
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假設你想要看看
02:58
adding加入 the squares廣場 of
the first few少數 Fibonacci斐波那契 numbers數字.
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把頭幾個費波那西數的平方值加起來。
03:00
Let's see what we get there.
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讓我們看看會有什麼結果。
03:02
So one plus one plus four is six.
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1加1加4等於6。
03:04
Add nine to that, we get 15.
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再加9,我們得到15。
03:07
Add 25, we get 40.
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再加 25,我們得到 40。
03:09
Add 64, we get 104.
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再加 64,我們得到104。
03:12
Now look at those numbers數字.
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現在來看看這些數字。
03:14
Those are not Fibonacci斐波那契 numbers數字,
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那些不是費波那西數,
03:16
but if you look at them closely密切,
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但如果你仔細再看這些數字,
03:18
you'll你會 see the Fibonacci斐波那契 numbers數字
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你會看到費波那西數
03:20
buried隱藏 inside of them.
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藏在它們裡面。
03:22
Do you see it? I'll show顯示 it to you.
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你看到了嗎?讓我指出來給你。
03:24
Six is two times three, 15 is three times five,
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6是2乘3、 15 是3乘5、
03:28
40 is five times eight,
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40 是5乘8、
03:30
two, three, five, eight, who do we appreciate欣賞?
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2、3、 5、 8,我們在欣賞什麼?
03:33
(Laughter笑聲)
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(笑聲)
03:34
Fibonacci斐波那契! Of course課程.
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當然是費波那西數!
03:36
Now, as much fun開玩笑 as it is to discover發現 these patterns模式,
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正如找出這些規律是很好玩的,
03:40
it's even more satisfying滿意的 to understand理解
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更令人滿意的是瞭解
03:42
why they are true真正.
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為什麼它們是這樣的。
03:44
Let's look at that last equation方程.
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讓我們看看這最後的等式。
03:46
Why should the squares廣場 of one, one,
two, three, five and eight
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為什麼1,1,2,3,5和8的平方
03:50
add up to eight times 13?
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加起來等於8乘以13?
03:53
I'll show顯示 you by drawing畫畫 a simple簡單 picture圖片.
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我畫一張簡單的圖來解釋給你。
03:56
We'll start開始 with a one-by-one一個接一個 square廣場
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我們先由一個1x1的正方形開始
03:58
and next下一個 to that put another另一個 one-by-one一個接一個 square廣場.
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在旁邊再放一個1x1的正方形。
04:02
Together一起, they form形成 a one-by-two一對二 rectangle長方形.
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它們一起,構成一個1x2的矩形。
04:06
Beneath下面 that, I'll put a two-by-two兩兩 square廣場,
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接著,再放一個2x2的正方形,
04:08
and next下一個 to that, a three-by-three三乘三 square廣場,
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旁邊再來一個3x3的正方形,
04:11
beneath下面 that, a five-by-five五乘以五 square廣場,
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在下方,放一個5x5的正方形,
04:13
and then an eight-by-eight八乘八 square廣場,
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然後旁邊一個8x8的正方形,
04:15
creating創建 one giant巨人 rectangle長方形, right?
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得到一個巨大的矩形,對嗎?
04:18
Now let me ask you a simple簡單 question:
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現在讓我問你一個簡單的問題:
04:20
what is the area of the rectangle長方形?
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這個矩形的面積是多少?
04:23
Well, on the one hand,
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好吧,一方面,
04:25
it's the sum of the areas
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它是所有這些所包含的
04:28
of the squares廣場 inside it, right?
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正方形面積的總和,是吧?
04:30
Just as we created創建 it.
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正如我們如何創造了它,
04:31
It's one squared平方 plus one squared平方
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它是1的平方加1的平方
04:33
plus two squared平方 plus three squared平方
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加2的平方再加3的平方
04:35
plus five squared平方 plus eight squared平方. Right?
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加 5 的平方再加8的平方。對吧?
04:38
That's the area.
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這就是總面積。
04:40
On the other hand, because it's a rectangle長方形,
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另一方面,因為它是個矩形
04:42
the area is equal等於 to its height高度 times its base基礎,
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面積等於高乘以底,
04:46
and the height高度 is clearly明確地 eight,
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高顯然是8,
04:48
and the base基礎 is five plus eight,
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而底是5加8,
04:51
which哪一個 is the next下一個 Fibonacci斐波那契 number, 13. Right?
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這就是下一個費波那西數,13。對吧?
04:55
So the area is also eight times 13.
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所以面積也是8乘以13。
04:58
Since以來 we've我們已經 correctly正確地 calculated計算 the area
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既然我們已經用兩種不同的方法,
05:00
two different不同 ways方法,
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正確地計算出了這個面積
05:02
they have to be the same相同 number,
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它們必然是相同的數字,
05:04
and that's why the squares廣場 of one,
one, two, three, five and eight
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這就是為什麼1,1,2,3,5和8的平方
05:08
add up to eight times 13.
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加起來正好是8乘以13。
05:10
Now, if we continue繼續 this process處理,
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現在,如果我們繼續這一過程,
05:12
we'll generate生成 rectangles矩形 of the form形成 13 by 21,
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我們會生成13x21 的矩形,
05:16
21 by 34, and so on.
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21x34 的矩形等等。
05:19
Now check this out.
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再來看這個。
05:20
If you divide劃分 13 by eight,
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如果你用 13除以8,
05:22
you get 1.625.
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你得到 1.625。
05:24
And if you divide劃分 the larger number
by the smaller number,
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如果你用較大的數除以較小的數,
05:28
then these ratios get closer接近 and closer接近
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會發現這些比率越來越接近
05:31
to about 1.618,
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1.618,
05:33
known已知 to many許多 people as the Golden金色 Ratio,
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眾所周知的黃金比率,
05:37
a number which哪一個 has fascinated入迷 mathematicians數學家,
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一個讓數學家,科學家和藝術家
05:39
scientists科學家們 and artists藝術家 for centuries百年.
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著迷幾個世紀的數字。
05:42
Now, I show顯示 all this to you because,
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我給你看這些,是因為
05:45
like so much of mathematics數學,
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像很多數學,
05:47
there's a beautiful美麗 side to it
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都有它美麗的一面
05:49
that I fear恐懼 does not get enough足夠 attention注意
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而我覺得(這些美麗)沒有在我們的學校
05:51
in our schools學校.
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得到足夠的重視。
05:52
We spend lots of time learning學習 about calculation計算,
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我們花費大量的時間來學習如何計算,
05:55
but let's not forget忘記 about application應用,
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但別忘了要應用,
05:58
including包含, perhaps也許, the most
important重要 application應用 of all,
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或許,包括,最重要的應用,
06:01
learning學習 how to think.
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學習如何去思考。
06:03
If I could summarize總結 this in one sentence句子,
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如果要用一句話來總結,
06:05
it would be this:
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那就是:
06:07
Mathematics數學 is not just solving for x,
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數學不只是解出x,
06:10
it's also figuring盤算 out why.
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也要知道為什麼。
06:13
Thank you very much.
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謝謝。
06:15
(Applause掌聲)
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(掌聲)
Translated by Yukun Chen
Reviewed by 宇凡 布

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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