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

阿瑟·本雅明: 神奇的斐波那契数列

Filmed:
7,057,274 views

数学不仅仅是一堆逻辑和函数,它还可以很酷。数学家阿瑟-本杰明向我们展示了斐波纳契数列的隐含魅力,以及种种看起来很神奇的巧合(同时提醒你,数学也可以是激动人心的!)。
- 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学习 mathematics数学?
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我们为什么要学习数学?
00:15
Essentially实质上, for three reasons原因:
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根本原因有三个:
00:18
calculation计算,
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计算,
00:19
application应用,
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应用,
00:21
and last, and unfortunately不幸 least最小
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最后一个,很不幸的,
00:24
in terms条款 of the time we give it,
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从时间分配来看也是最少的,
00:26
inspiration灵感.
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激发灵感.
00:28
Mathematics数学 is the science科学 of patterns模式,
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数学是研究规律的科学,
00:30
and we study研究 it to learn学习 how to think logically逻辑,
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我们通过学习数学来训练逻辑思维能力,
00:34
critically危重 and creatively创造性,
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思辩能力以及创造力,
00:36
but too much of the mathematics数学
that we learn学习 in school学校
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但是我们在学校里面学习到的数学,
00:39
is not effectively有效 motivated动机,
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根本没有激起我们的兴趣
00:41
and when our students学生们 ask,
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每当我们的学生问起
00:43
"Why are we learning学习 this?"
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"我们为什么要学这个?"
00:44
then they often经常 hear that they'll他们会 need it
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他们得到的答案往往是
00:46
in an upcoming即将到来 math数学 class or on a future未来 test测试.
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考试要考, 或者后续的数学课程中要用到.
00:50
But wouldn't不会 it be great
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有没有可能
00:51
if every一切 once一旦 in a while we did mathematics数学
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哪怕只有那么一小会儿, 我们研究数学
00:54
simply只是 because it was fun开玩笑 or beautiful美丽
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仅仅是因为自己的兴趣, 或是数学的优美
00:57
or because it excited兴奋 the mind心神?
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那岂不是很棒?
00:59
Now, I know many许多 people have not
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现在, 我知道很多人
01:01
had the opportunity机会 to see how this can happen发生,
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一直没有机会来体验这一点,
01:03
so let me give you a quick example
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所以现在我们就来体验一下
01:05
with my favorite喜爱 collection采集 of numbers数字,
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以我最喜欢的数列
01:07
the Fibonacci斐波那契 numbers数字. (Applause掌声)
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斐波纳契数列为例.(掌声)
01:10
Yeah! I already已经 have Fibonacci斐波那契 fans球迷 here.
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太好了! 看来在座的也有喜欢斐波纳契的.
01:12
That's great.
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非常好.
01:13
Now these numbers数字 can be appreciated赞赏
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我们可以从多种不同的角度
01:15
in many许多 different不同 ways方法.
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来欣赏斐波纳契序列.
01:17
From the standpoint立场 of calculation计算,
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从计算的角度
01:20
they're as easy简单 to understand理解
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斐波纳契数列很容易被理解
01:22
as one plus one, which哪一个 is two.
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1 加 1, 等于 2
01:24
Then one plus two is three,
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1 加 2 等于 3
01:26
two plus three is five, three plus five is eight,
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2 加 3 等于 5, 3 加 5 等于 8
01:29
and so on.
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以此类推.
01:31
Indeed确实, the person we call Fibonacci斐波那契
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事实上, 那个我们称呼"斐波纳契"的人
01:33
was actually其实 named命名 Leonardo莱昂纳多 of Pisa比萨,
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真实的名字叫列昂纳多, 来自比萨
01:36
and these numbers数字 appear出现 in his book "Liber莱博 Abaci算盘,"
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这个数列出自他的书《算盘宝典》("Liber Abaci")
01:39
which哪一个 taught the Western西 world世界
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这本书奠定了西方世界的数学基础
01:41
the methods方法 of arithmetic算术 that we use today今天.
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其中的算术方法一直沿用至今.
01:44
In terms条款 of applications应用,
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从应用的角度来看,
01:45
Fibonacci斐波那契 numbers数字 appear出现 in nature性质
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斐波纳契数列在自然界中经常
01:48
surprisingly出奇 often经常.
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神奇的出现.
01:49
The number of petals花瓣 on a flower
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一朵花的花瓣数量
01:51
is typically一般 a Fibonacci斐波那契 number,
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一般是一个斐波纳契数,
01:53
or the number of spirals螺旋 on a sunflower向日葵
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向日葵的螺旋,
01:56
or a pineapple菠萝
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菠萝表面的凸起,
01:57
tends趋向 to be a Fibonacci斐波那契 number as well.
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也都对应着某个斐波纳契数.
02:00
In fact事实, there are many许多 more
applications应用 of Fibonacci斐波那契 numbers数字,
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事实上还有很多斐波纳契数的应用实例,
02:03
but what I find most inspirational励志 about them
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而我发现这其中最能给人启发的
02:06
are the beautiful美丽 number patterns模式 they display显示.
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是这些数字呈现出来的漂亮模式.
02:08
Let me show显示 you one of my favorites最爱.
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让我们看下我最喜欢的一个.
02:11
Suppose假设 you like to square广场 numbers数字,
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假设你喜欢计算数的平方.
02:13
and frankly坦率地说, who doesn't? (Laughter笑声)
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坦白说, 谁不喜欢?(笑声)
02:16
Let's look at the squares广场
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让我们计算一下
02:18
of the first few少数 Fibonacci斐波那契 numbers数字.
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头几个斐波纳契数的平方.
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, 以此类推.
02:27
Now, it's no surprise
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毫不意外的,
02:29
that when you add consecutive连续 Fibonacci斐波那契 numbers数字,
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当你加上两个连续的斐波纳契数字时,
02:32
you get the next下一个 Fibonacci斐波那契 number. Right?
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你得到了下一个斐波纳契数, 没错吧?
02:34
That's how they're created创建.
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它们就是这么定义的.
02:35
But you wouldn't不会 expect期望 anything special特别
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但是你不知道把斐波纳契数的平方
02:37
to happen发生 when you add the squares广场 together一起.
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加起来会得到什么有意思的结果.
02:40
But check this out.
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来尝试一下.
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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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继续.
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没错, 还是这个规律.
02:54
In fact事实, here's这里的 another另一个 one.
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事实上, 还有一个规律.
02:56
Suppose假设 you wanted to look at
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假如你想计算一下
02:58
adding加入 the squares广场 of
the first few少数 Fibonacci斐波那契 numbers数字.
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头几个斐波纳契数的平方和,
03:00
Let's see what we get there.
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看看结果是什么.
03:02
So one plus one plus four is six.
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1 加 1 加 4 是 6,
03:04
Add nine to that, we get 15.
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再加上 9, 得到 15,
03:07
Add 25, we get 40.
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再加上 25, 得到 40,
03:09
Add 64, we get 104.
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再加上 64, 得到 104.
03:12
Now look at those numbers数字.
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回头来看看这些数字.
03:14
Those are not Fibonacci斐波那契 numbers数字,
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他们不是斐波纳契数,
03:16
but if you look at them closely密切,
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但是如果你看得够仔细,
03:18
you'll你会 see the Fibonacci斐波那契 numbers数字
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你能看到他们的背后
03:20
buried隐藏 inside of them.
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隐藏着的斐波纳契数.
03:22
Do you see it? I'll show显示 it to you.
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看到了么? 让我写给你看.
03:24
Six is two times three, 15 is three times five,
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6 等于 2 乘 3, 15 等于 3 乘 5,
03:28
40 is five times eight,
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40 等于 5 乘 8,
03:30
two, three, five, eight, who do we appreciate欣赏?
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2, 3, 5, 8 我们看到了什么?
03:33
(Laughter笑声)
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(笑声)
03:34
Fibonacci斐波那契! Of course课程.
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斐波纳契! 当然, 当然.
03:36
Now, as much fun开玩笑 as it is to discover发现 these patterns模式,
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现在我们已经发现了这些好玩的模式,
03:40
it's even more satisfying满意的 to understand理解
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更能满足你们好奇心的事情是
03:42
why they are true真正.
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弄清楚背后的原因.
03:44
Let's look at that last equation方程.
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让我们看看最后这个等式.
03:46
Why should the squares广场 of one, one,
two, three, five and eight
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为什么 1, 1, 2, 3, 5 和 8 的平方
03:50
add up to eight times 13?
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加起来等于 8 乘以 13?
03:53
I'll show显示 you by drawing画画 a simple简单 picture图片.
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我通过一个简单的图形来解释.
03:56
We'll start开始 with a one-by-one一个接一个 square广场
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首先我们画一个 1 乘 1 的方块,
03:58
and next下一个 to that put another另一个 one-by-one一个接一个 square广场.
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然后再在旁边放一个相同尺寸的方块.
04:02
Together一起, they form形成 a one-by-two一对二 rectangle长方形.
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拼起来之后得到了一个 1 乘 2 的矩形.
04:06
Beneath下面 that, I'll put a two-by-two两两 square广场,
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在这个下面再放一个 2 乘 2 的方块,
04:08
and next下一个 to that, a three-by-three三乘三 square广场,
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之后贴着再放一个 3 乘 3 的方块,
04:11
beneath下面 that, a five-by-five五乘以五 square广场,
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然后再在下面放一个 5 乘 5 的矩形,
04:13
and then an eight-by-eight八乘八 square广场,
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之后是一个 8 乘 8 的方块.
04:15
creating创建 one giant巨人 rectangle长方形, right?
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得到了一个大的矩形, 对吧?
04:18
Now let me ask you a simple简单 question:
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现在问大家一个简单的问题:
04:20
what is the area of the rectangle长方形?
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这个矩形的面积是多少?
04:23
Well, on the one hand,
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一方面,
04:25
it's the sum of the areas
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它的面积就是
04:28
of the squares广场 inside it, right?
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组成它的小矩形的面积之和, 对吧?
04:30
Just as we created创建 it.
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就是我们用到的矩形之和
04:31
It's one squared平方 plus one squared平方
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它的面积是 1 的平方加上 1 的平方
04:33
plus two squared平方 plus three squared平方
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加上 2 的平方加上 3 的平方
04:35
plus five squared平方 plus eight squared平方. Right?
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加上 5 的平方加上 8 的平方. 对吧?
04:38
That's the area.
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这就是面积.
04:40
On the other hand, because it's a rectangle长方形,
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另一方面, 因为这是矩形,
04:42
the area is equal等于 to its height高度 times its base基础,
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面积就等于长乘高,
04:46
and the height高度 is clearly明确地 eight,
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高等于 8,
04:48
and the base基础 is five plus eight,
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长是 5 加 8,
04:51
which哪一个 is the next下一个 Fibonacci斐波那契 number, 13. Right?
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也是一个斐波纳契数, 13, 是不是?
04:55
So the area is also eight times 13.
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所以面积就是 8 乘 13.
04:58
Since以来 we've我们已经 correctly正确地 calculated计算 the area
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因为我们用两种不同的方式计算面积,
05:00
two different不同 ways方法,
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同样一个矩形的面积
05:02
they have to be the same相同 number,
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一定是一样的,
05:04
and that's why the squares广场 of one,
one, two, three, five and eight
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这样就是为什么 1, 1, 2, 3, 5, 8 的平方和,
05:08
add up to eight times 13.
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等于 8 乘 13.
05:10
Now, if we continue继续 this process处理,
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如果我们继续探索下去,
05:12
we'll generate生成 rectangles矩形 of the form形成 13 by 21,
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我们会得到 13 乘 21 的矩形,
05:16
21 by 34, and so on.
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21 乘 34 的矩形, 以此类推.
05:19
Now check this out.
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再来看看这个.
05:20
If you divide划分 13 by eight,
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如果你用 8 去除 13,
05:22
you get 1.625.
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结果是 1.625.
05:24
And if you divide划分 the larger number
by the smaller number,
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如果用大的斐波纳契数除以前一个小的斐波纳契数
05:28
then these ratios get closer接近 and closer接近
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他们的比例会越来越接近
05:31
to about 1.618,
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1.618,
05:33
known已知 to many许多 people as the Golden金色 Ratio,
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这就是很多人知道的黄金分割率,
05:37
a number which哪一个 has fascinated入迷 mathematicians数学家,
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一个几个世纪以来, 让无数数学家, 科学家和艺术家
05:39
scientists科学家们 and artists艺术家 for centuries百年.
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都非常着迷的数字.
05:42
Now, I show显示 all this to you because,
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我之所以向你们展示这些是因为,
05:45
like so much of mathematics数学,
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很多这样的数学(知识),
05:47
there's a beautiful美丽 side to it
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都有其秒不可言的一面
05:49
that I fear恐惧 does not get enough足够 attention注意
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而我担心这一面并没有在学校里
05:51
in our schools学校.
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得到展现.
05:52
We spend lots of time learning学习 about calculation计算,
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我们花了很多时间去学习算术,
05:55
but let's not forget忘记 about application应用,
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但是请不要忘记数学在实际中的应用,
05:58
including包含, perhaps也许, the most
important重要 application应用 of all,
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包括可能是最重要的一种应用形式,
06:01
learning学习 how to think.
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学会如何思考.
06:03
If I could summarize总结 this in one sentence句子,
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把我今天所说的浓缩成一句,
06:05
it would be this:
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那就是:
06:07
Mathematics数学 is not just solving for x,
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数学, 不仅仅是求出X等于多少,
06:10
it's also figuring盘算 out why.
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还要能指出为什么.
06:13
Thank you very much.
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感谢大家.
06:15
(Applause掌声)
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(掌声)
Translated by Psycho Decoder
Reviewed by Tingting Zhao

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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