TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Arthur Benjamin: 費波那西數列的魔力
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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！）。
Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
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 mathematics?
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我們為什麼要學數學？
00:15
Essentially, for three reasons:
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主要有三個原因：
00:18
calculation,
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計算
00:19
application,
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應用
00:21
and last, and unfortunately least
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最後，不幸地，也是最不重要的，
00:24
in terms 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.
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靈感。
00:28
Mathematics is the science of patterns,
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數學是规律的科學，
00:30
and we study it to learn how to think logically,
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而我們學習數學是為了學習怎樣邏輯地，
00:34
critically and creatively,
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批評地和有創造性地思考，
00:36
but too much of the mathematics
that we learn in school
that we learn in school
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但是，太多我們在學校學的數學
00:39
is not effectively motivated,
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並沒有效地激勵學生思考
00:41
and when our students ask,
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所以當學生問我們，
00:43
"Why are we learning this?"
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“我們為什麼要學這個？”
00:44
then they often hear that they'll 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 math class or on a future test.
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或者將來會有考試，他們需要這個。
00:50
But wouldn't it be great
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可是，如果
00:51
if every once in a while we did mathematics
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偶爾我們學數學
00:54
simply because it was fun or beautiful
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僅僅是因為數學很有趣或迷人，
00:57
or because it excited the mind?
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或者因為它激發思想，不是很好嗎？
00:59
Now, I know many people have not
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我知道很多人都還沒有
01:01
had the opportunity to see how this can happen,
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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 collection of numbers,
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來給你舉個小小的例子，
01:07
the Fibonacci numbers. (Applause)
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費波那西數。（鼓掌）
01:10
Yeah! I already have Fibonacci fans 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 can be appreciated
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（我們可以）從很多個方面來
01:15
in many different ways.
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欣賞這組數字。
01:17
From the standpoint of calculation,
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從計算上來看，
01:20
they're as easy to understand
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它們非常易懂
01:22
as one plus one, which 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, the person we call Fibonacci
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事實上，我們稱做“費波那西”的這個人
01:33
was actually named Leonardo of Pisa,
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是比薩的莱昂纳多，
01:36
and these numbers appear in his book "Liber Abaci,"
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而這些數字是在他的“計算之書”中描述的，
01:39
which taught the Western world
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這本書教授了西方世界
01:41
the methods of arithmetic that we use today.
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我們今天所使用的算術方法。
01:44
In terms of applications,
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從應用上來看，
01:45
Fibonacci numbers appear in nature
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費波那西數讓人驚訝地
01:48
surprisingly often.
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頻繁出現在自然界裡。
01:49
The number of petals on a flower
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花瓣的數目
01:51
is typically a Fibonacci number,
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通常是一個費波那西數字，
01:53
or the number of spirals on a sunflower
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或向日葵上、鳳梨上的螺旋數
01:56
or a pineapple
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01:57
tends to be a Fibonacci number as well.
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往往也是費波那西數字。
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
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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 about them
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但我發現最鼓舞人心的
02:06
are the beautiful number patterns they display.
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是它們所顯示的漂亮的數字规律。
02:08
Let me show you one of my favorites.
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讓我給你看看我的最愛之一。
02:11
Suppose you like to square numbers,
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假設你喜歡平方數，
02:13
and frankly, who doesn't? (Laughter)
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坦率地說，誰不喜歡？（笑聲）
02:16
Let's look at the squares
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讓我們看看頭幾個
02:18
of the first few Fibonacci numbers.
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費波那西數的平方。
02:20
So one squared is one,
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1的平方是1，
02:22
two squared is four, three squared is nine,
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2 的平方是4，3的平方是9，
02:24
five squared 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 Fibonacci numbers,
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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 Fibonacci 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.
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這就是它們如何被定義的。
02:35
But you wouldn't expect anything special
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但你大概不會料到
02:37
to happen when you add the squares together.
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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。
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 continues.
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是的，這個規律一直繼續下去。
02:54
In fact, here's another one.
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事實上，還有另外一個。
02:56
Suppose you wanted to look at
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假設你想要看看
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
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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.
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現在來看看這些數字。
03:14
Those are not Fibonacci numbers,
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那些不是費波那西數，
03:16
but if you look at them closely,
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但如果你仔細再看這些數字，
03:18
you'll see the Fibonacci numbers
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你會看到費波那西數
03:20
buried inside of them.
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藏在它們裡面。
03:22
Do you see it? I'll show 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?
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2、3、 5、 8，我們在欣賞什麼？
03:33
(Laughter)
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（笑聲）
03:34
Fibonacci! Of course.
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當然是費波那西數！
03:36
Now, as much fun as it is to discover these patterns,
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正如找出這些規律是很好玩的，
03:40
it's even more satisfying to understand
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更令人滿意的是瞭解
03:42
why they are true.
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為什麼它們是這樣的。
03:44
Let's look at that last equation.
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讓我們看看這最後的等式。
03:46
Why should the squares of one, one,
two, three, five and eight
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 you by drawing a simple picture.
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我畫一張簡單的圖來解釋給你。
03:56
We'll start with a one-by-one square
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我們先由一個1x1的正方形開始
03:58
and next to that put another one-by-one square.
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在旁邊再放一個1x1的正方形。
04:02
Together, they form a one-by-two rectangle.
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它們一起，構成一個1x2的矩形。
04:06
Beneath that, I'll put a two-by-two square,
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接著，再放一個2x2的正方形，
04:08
and next to that, a three-by-three square,
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旁邊再來一個3x3的正方形，
04:11
beneath that, a five-by-five square,
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在下方，放一個5x5的正方形，
04:13
and then an eight-by-eight square,
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然後旁邊一個8x8的正方形，
04:15
creating one giant rectangle, 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 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?
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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 inside it, right?
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正方形面積的總和，是吧？
04:30
Just as we created it.
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正如我們如何創造了它，
04:31
It's one squared plus one squared
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它是1的平方加1的平方
04:33
plus two squared plus three squared
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加2的平方再加3的平方
04:35
plus five squared plus eight squared. 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,
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另一方面，因為它是個矩形
04:42
the area is equal to its height times its base,
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面積等於高乘以底，
04:46
and the height is clearly eight,
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高顯然是8，
04:48
and the base is five plus eight,
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而底是5加8，
04:51
which is the next Fibonacci 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 we've correctly calculated the area
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既然我們已經用兩種不同的方法，
05:00
two different ways,
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正確地計算出了這個面積
05:02
they have to be the same number,
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它們必然是相同的數字，
05:04
and that's why the squares of one,
one, two, three, five and eight
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 this process,
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現在，如果我們繼續這一過程，
05:12
we'll generate rectangles of the form 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 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 the larger number
by the smaller 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 and closer
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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 to many people as the Golden Ratio,
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眾所周知的黃金比率，
05:37
a number which has fascinated mathematicians,
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一個讓數學家，科學家和藝術家
05:39
scientists and artists for centuries.
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著迷幾個世紀的數字。
05:42
Now, I show 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,
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像很多數學，
05:47
there's a beautiful side to it
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都有它美麗的一面
05:49
that I fear does not get enough attention
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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.
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得到足夠的重視。
05:52
We spend lots of time learning about calculation,
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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 about application,
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但別忘了要應用，
05:58
including, perhaps, the most
important application of all,
important application of all,
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或許，包括，最重要的應用，
06:01
learning how to think.
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學習如何去思考。
06:03
If I could summarize this in one sentence,
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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 is not just solving for x,
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數學不只是解出x，
06:10
it's also figuring 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)
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（掌聲）
ABOUT THE SPEAKER
Arthur Benjamin - MathemagicianUsing 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").
Arthur Benjamin | Speaker | TED.com