The magic of Fibonacci numbers | Arthur Benjamin | TED

翻译人员: Wei Wu 校对人员: Tingting Zhao

So why do we learn mathematics?

我们为什么要学习数学?

Essentially, for three reasons:

根本原因有三个:

calculation,

计算,

application,

应用,

and last, and unfortunately least

最后一个,很不幸的,

in terms of the time we give it,

从时间分配来看也是最少的,

inspiration.

激发灵感.

Mathematics is the science of patterns,

数学是研究规律的科学,

and we study it to learn how to think logically,

我们通过学习数学来训练逻辑思维能力,

critically and creatively,

思辩能力以及创造力,

but too much of the mathematics that we learn in school

但是我们在学校里面学习到的数学,

is not effectively motivated,

根本没有激起我们的兴趣

and when our students ask,

每当我们的学生问起

"Why are we learning this?"

"我们为什么要学这个?"

then they often hear that they'll need it

他们得到的答案往往是

in an upcoming math class or on a future test.

考试要考, 或者后续的数学课程中要用到.

But wouldn't it be great

有没有可能

if every once in a while we did mathematics

哪怕只有那么一小会儿, 我们研究数学

simply because it was fun or beautiful

仅仅是因为自己的兴趣, 或是数学的优美

or because it excited the mind?

那岂不是很棒?

Now, I know many people have not

现在, 我知道很多人

had the opportunity to see how this can happen,

一直没有机会来体验这一点,

so let me give you a quick example

所以现在我们就来体验一下

with my favorite collection of numbers,

以我最喜欢的数列

the Fibonacci numbers. (Applause)

斐波纳契数列为例.(掌声)

Yeah! I already have Fibonacci fans here.

太好了! 看来在座的也有喜欢斐波纳契的.

That's great.

非常好.

Now these numbers can be appreciated

我们可以从多种不同的角度

in many different ways.

来欣赏斐波纳契序列.

From the standpoint of calculation,

从计算的角度

they're as easy to understand

斐波纳契数列很容易被理解

as one plus one, which is two.

1 加 1, 等于 2

Then one plus two is three,

1 加 2 等于 3

two plus three is five, three plus five is eight,

2 加 3 等于 5, 3 加 5 等于 8

and so on.

以此类推.

Indeed, the person we call Fibonacci

事实上, 那个我们称呼"斐波纳契"的人

was actually named Leonardo of Pisa,

真实的名字叫列昂纳多, 来自比萨

and these numbers appear in his book "Liber Abaci,"

这个数列出自他的书《算盘宝典》("Liber Abaci")

which taught the Western world

这本书奠定了西方世界的数学基础

the methods of arithmetic that we use today.

其中的算术方法一直沿用至今.

In terms of applications,

从应用的角度来看,

Fibonacci numbers appear in nature

斐波纳契数列在自然界中经常

surprisingly often.

神奇的出现.

The number of petals on a flower

一朵花的花瓣数量

is typically a Fibonacci number,

一般是一个斐波纳契数,

or the number of spirals on a sunflower

向日葵的螺旋,

or a pineapple

菠萝表面的凸起,

tends to be a Fibonacci number as well.

也都对应着某个斐波纳契数.

In fact, there are many more applications of Fibonacci numbers,

事实上还有很多斐波纳契数的应用实例,

but what I find most inspirational about them

而我发现这其中最能给人启发的

are the beautiful number patterns they display.

是这些数字呈现出来的漂亮模式.

Let me show you one of my favorites.

让我们看下我最喜欢的一个.

Suppose you like to square numbers,

假设你喜欢计算数的平方.

and frankly, who doesn't? (Laughter)

坦白说, 谁不喜欢?(笑声)

Let's look at the squares

让我们计算一下

of the first few Fibonacci numbers.

头几个斐波纳契数的平方.

So one squared is one,

1的平方是1,

two squared is four, three squared is nine,

2的平方是4, 3的平方是9,

five squared is 25, and so on.

5的平方是25, 以此类推.

Now, it's no surprise

毫不意外的,

that when you add consecutive Fibonacci numbers,

当你加上两个连续的斐波纳契数字时,

you get the next Fibonacci number. Right?

你得到了下一个斐波纳契数, 没错吧?

That's how they're created.

它们就是这么定义的.

But you wouldn't expect anything special

但是你不知道把斐波纳契数的平方

to happen when you add the squares together.

加起来会得到什么有意思的结果.

But check this out.

来尝试一下.

One plus one gives us two,

1 加 1 是 2,

and one plus four gives us five.

1 加 4 是 5,

And four plus nine is 13,

4 加 9 是 13,

nine plus 25 is 34,

9 加 25 是 34,

and yes, the pattern continues.

没错, 还是这个规律.

In fact, here's another one.

事实上, 还有一个规律.

Suppose you wanted to look at

假如你想计算一下

adding the squares of the first few Fibonacci numbers.

头几个斐波纳契数的平方和,

Let's see what we get there.

看看结果是什么.

So one plus one plus four is six.

1 加 1 加 4 是 6,

Add nine to that, we get 15.

再加上 9, 得到 15,

Add 25, we get 40.

再加上 25, 得到 40,

Add 64, we get 104.

再加上 64, 得到 104.

Now look at those numbers.

回头来看看这些数字.

Those are not Fibonacci numbers,

他们不是斐波纳契数,

but if you look at them closely,

但是如果你看得够仔细,

you'll see the Fibonacci numbers

你能看到他们的背后

buried inside of them.

隐藏着的斐波纳契数.

Do you see it? I'll show it to you.

看到了么? 让我写给你看.

Six is two times three, 15 is three times five,

6 等于 2 乘 3, 15 等于 3 乘 5,

40 is five times eight,

40 等于 5 乘 8,

two, three, five, eight, who do we appreciate?

2, 3, 5, 8 我们看到了什么?

(Laughter)

(笑声)

Fibonacci! Of course.

斐波纳契! 当然, 当然.

Now, as much fun as it is to discover these patterns,

现在我们已经发现了这些好玩的模式,

it's even more satisfying to understand

更能满足你们好奇心的事情是

why they are true.

弄清楚背后的原因.

Let's look at that last equation.

让我们看看最后这个等式.

Why should the squares of one, one, two, three, five and eight

为什么 1, 1, 2, 3, 5 和 8 的平方

add up to eight times 13?

加起来等于 8 乘以 13?

I'll show you by drawing a simple picture.

我通过一个简单的图形来解释.

We'll start with a one-by-one square

首先我们画一个 1 乘 1 的方块,

and next to that put another one-by-one square.

然后再在旁边放一个相同尺寸的方块.

Together, they form a one-by-two rectangle.

拼起来之后得到了一个 1 乘 2 的矩形.

Beneath that, I'll put a two-by-two square,

在这个下面再放一个 2 乘 2 的方块,

and next to that, a three-by-three square,

之后贴着再放一个 3 乘 3 的方块,

beneath that, a five-by-five square,

然后再在下面放一个 5 乘 5 的矩形,

and then an eight-by-eight square,

之后是一个 8 乘 8 的方块.

creating one giant rectangle, right?

得到了一个大的矩形, 对吧?

Now let me ask you a simple question:

现在问大家一个简单的问题:

what is the area of the rectangle?

这个矩形的面积是多少?

Well, on the one hand,

一方面,

it's the sum of the areas

它的面积就是

of the squares inside it, right?

组成它的小矩形的面积之和, 对吧?

Just as we created it.

就是我们用到的矩形之和

It's one squared plus one squared

它的面积是 1 的平方加上 1 的平方

plus two squared plus three squared

加上 2 的平方加上 3 的平方

plus five squared plus eight squared. Right?

加上 5 的平方加上 8 的平方. 对吧?

That's the area.

这就是面积.

On the other hand, because it's a rectangle,

另一方面, 因为这是矩形,

the area is equal to its height times its base,

面积就等于长乘高,

and the height is clearly eight,

高等于 8,

and the base is five plus eight,

长是 5 加 8,

which is the next Fibonacci number, 13. Right?

也是一个斐波纳契数, 13, 是不是?

So the area is also eight times 13.

所以面积就是 8 乘 13.

Since we've correctly calculated the area

因为我们用两种不同的方式计算面积,

two different ways,

同样一个矩形的面积

they have to be the same number,

一定是一样的,

and that's why the squares of one, one, two, three, five and eight

这样就是为什么 1, 1, 2, 3, 5, 8 的平方和,

add up to eight times 13.

等于 8 乘 13.

Now, if we continue this process,

如果我们继续探索下去,

we'll generate rectangles of the form 13 by 21,

我们会得到 13 乘 21 的矩形,

21 by 34, and so on.

21 乘 34 的矩形, 以此类推.

Now check this out.

再来看看这个.

If you divide 13 by eight,

如果你用 8 去除 13,

you get 1.625.

结果是 1.625.

And if you divide the larger number by the smaller number,

如果用大的斐波纳契数除以前一个小的斐波纳契数

then these ratios get closer and closer

他们的比例会越来越接近

to about 1.618,

1.618,

known to many people as the Golden Ratio,

这就是很多人知道的黄金分割率,

a number which has fascinated mathematicians,

一个几个世纪以来, 让无数数学家, 科学家和艺术家

scientists and artists for centuries.

都非常着迷的数字.

Now, I show all this to you because,

我之所以向你们展示这些是因为,

like so much of mathematics,

很多这样的数学(知识),

there's a beautiful side to it

都有其秒不可言的一面

that I fear does not get enough attention

而我担心这一面并没有在学校里

in our schools.

得到展现.

We spend lots of time learning about calculation,

我们花了很多时间去学习算术,

but let's not forget about application,

但是请不要忘记数学在实际中的应用,

including, perhaps, the most important application of all,

包括可能是最重要的一种应用形式,

learning how to think.

学会如何思考.

If I could summarize this in one sentence,

把我今天所说的浓缩成一句,

it would be this:

那就是:

Mathematics is not just solving for x,

数学, 不仅仅是求出X等于多少,

it's also figuring out why.

还要能指出为什么.

Thank you very much.

感谢大家.

(Applause)

(掌声)