演说者：Arthur Benjamin

演说题目：The magic of Fibonacci numbers

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So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration.

我们为什么要学数学？根本原因有三个：计算、 应用、最后一个，很不幸的从时间分配来看也是最少的，激发灵感。

Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics that we learn in school is not effectively motivated, and when our students ask, "Why are we learning this?" then they often hear that they'll need it in an upcoming math class or on a future test.

数学是研究规律的科学，我们通过学数学来训练逻辑思维能力、思辩能力以及创造力，但是我们在学校里面学到的数学，根本没有激起我们的兴趣。每当我们的学生问起 “我们为什么要学这个？” 他们得到的答案往往是考试要考，或者后续的数学课程中要用到。

But wouldn't it be great if every once in a while we did mathematics simply because it was fun or beautiful or because it excited the mind? Now, I know many people have not had the opportunity to see how this can happen, so let me give you a quick example with my favorite collection of numbers, the Fibonacci numbers.

有没有可能，哪怕只有那么一小会儿，我们研究数学仅仅是因为自己的兴趣，或是数学的优美，那岂不是很棒？现在我知道很多人一直没有机会来体验这一点，所以现在我们就来体验一下，以我最喜欢的数列——斐波纳契数列为例。

Yeah! I already have Fibonacci fans here. That's great.

太好了！看来在座的也有喜欢斐波纳契的。

Now these numbers can be appreciated in many different ways. From the standpoint of calculation, they're as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on.

非常好，我们可以从多种不同的角度来欣赏斐波纳契序列。从计算的角度，斐波纳契数列很容易被理解1加1，等于2 。1加2等于3，2加3等于5，3加5等于8以此类推。

Indeed, the person we call Fibonacci was actually named Leonardo of Pisa, and these numbers appear in his book "Liber Abaci," which taught the Western world the methods of arithmetic that we use today. In terms of applications, Fibonacci numbers appear in nature surprisingly often. The number of petals on a flower is typically a Fibonacci number, or the number of spirals on a sunflower or a pineapple tends to be a Fibonacci number as well.

事实上，那个我们称呼"斐波纳契"的人真实的名字叫列昂纳多，来自比萨，这个数列出自他的书《算盘宝典》（"Liber Abaci"）这本书奠定了西方世界的数学基础，其中的算术方法一直沿用至今。从应用的角度来看，斐波纳契数列在自然界中经常神奇的出现。一朵花的花瓣数量一般是一个斐波纳契数，向日葵的螺旋，菠萝表面的凸起也都对应着某个斐波纳契数。

In fact, there are many more applications of Fibonacci numbers, but what I find most inspirational about them are the beautiful number patterns they display. Let me show you one of my favorites. Suppose you like to square numbers, and frankly, who doesn't?

事实上还有很多斐波纳契数的应用实例，而我发现这其中最能给人启发的是这些数字呈现出来的漂亮模式。让我们看下我最喜欢的一个。假设你喜欢计算数的平方。坦白说，谁不喜欢？

Let's look at the squares of the first few Fibonacci numbers. So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on. Now, it's no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right?

让我们计算一下头几个斐波那契数的平方。1的平方是1，2的平方是4，3的平方是9，5的平方是25，以此类推。毫不意外的，当你加上两个连续的斐波那契数字时，你得到了下一个斐波那契数，没错吧？

That's how they're created. But you wouldn't expect anything special to happen when you add the squares together. But check this out. One plus one gives us two, and one plus four gives us five. And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues.

它就是这么定义的。但是你不知道把斐波那契数的平方加起来会得到什么有意思的结果。来尝试一下。1加1是2，1加4是5，4加9是13，9加25是34，没错，还是这个规律。

In fact, here's another one. Suppose you wanted to look at adding the squares of the first few Fibonacci numbers. Let's see what we get there. So one plus one plus four is six. Add nine to that, we get 15. Add 25, we get 40. Add 64, we get 104. Now look at those numbers. Those are not Fibonacci numbers, but if you look at them closely, you'll see the Fibonacci numbers buried inside of them.

事实上，还有一个规律。假如你想计算一下头几个斐波纳契数的平方和，看看结果是什么。1加1加4是6，再加上9，得到15，再加上25，得到40，再加上64，得到104。回头来看看这些数字。他们不是斐波纳契数，但是如果你看得够仔细，你能看到他们的背后隐藏着的斐波纳契数。

Do you see it? I'll show it to you. Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate?Fibonacci! Of course.

看到了么？让我写给你看。6等于2乘3，15等于3乘5，40等于5乘8，2，3，5，8，我们看到了什么？（笑声）斐波纳契！当然，当然。

Now, as much fun as it is to discover these patterns, it's even more satisfying to understand why they are true. Let's look at that last equation. Why should the squares of one, one, two, three, five and eight add up to eight times 13? I'll show you by drawing a simple picture. We'll start with a one-by-one square and next to that put another one-by-one square. Together, they form a one-by-two rectangle. Beneath that, I'll put a two-by-two square, and next to that, a three-by-three square, beneath that, a five-by-five square, and then an eight-by-eight square, creating one giant rectangle, right?

现在我们已经发现了这些好玩的模式，更能满足你们好奇心的事情是弄清楚背后的原因。让我们看看最后这个等式。为什么1， 1， 2， 3， 5和8的平方加起来等于8乘以13？我通过一个简单的图形来解释。首先我们画一个1乘1的方块，然后再在旁边放一个相同尺寸的方块。拼起来之后得到了一个1乘2的矩形。在这个下面再放一个2乘2的方块，之后贴着再放一个3乘3的方块，然后再在下面放一个5乘5的矩形，之后是一个8乘8的方块。得到了一个大的矩形，对吧?

Now let me ask you a simple question: what is the area of the rectangle? Well, on the one hand, it's the sum of the areas of the squares inside it, right? Just as we created it. It's one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? That's the area. On the other hand, because it's a rectangle, the area is equal to its height times its base, and the height is clearly eight, and the base is five plus eight, which is the next Fibonacci number, 13. Right? So the area is also eight times 13.Since we've correctly calculated the area two different ways, they have to be the same number, and that's why the squares of one, one, two, three, five and eight add up to eight times 13.

现在问大家一个简单的问题：这个矩形的面积是多少？一方面，它的面积就是组成它的小矩形的面积之和，对吧？就是我们用到的矩形之和它的面积是 1的平方加上 1 的平方加上2的平方加上3的平方加上5的平方加上8的平方。对吧？这就是面积。另一方面，因为这是矩形，面积就等于长乘高，高等于 8，长是 5 加 8，也是一个斐波纳契数，13，是不是？所以面积就是8乘 13。因为我们用两种不同的方式计算面积，同样一个矩形的面积 一定是一样的，这样就是为什么1， 1， 2， 3， 5， 8 的平方和，等于8乘13。

Now, if we continue this process, we'll generate rectangles of the form 13 by 21, 21 by 34, and so on.Now check this out. If you divide 13 by eight, you get 1.625. And if you divide the larger number by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians, scientists and artists for centuries.

如果我们继续探索下去，我们会得到13乘21的矩形，21乘 34的矩形，以此类推。再来看看这个。如果你用8去除13，结果是1.625。如果用大的斐波纳契数除以前一个小的斐波纳契数他们的比例会越来越接近1.618，这就是很多人知道的黄金分割率，一个几个世纪以来，让无数数学家，科学家和艺术家都非常着迷的数字。

Now, I show all this to you because, like so much of mathematics, there's a beautiful side to it that I fear does not get enough attention in our schools. We spend lots of time learning about calculation, but let's not forget about application, including, perhaps, the most important application of all, learning how to think.

我之所以向你们展示这些是因为，很多这样的数学（知识），都有其秒不可言的一面而我担心这一面并没有在学校里得到展现。我们花了很多时间去学算术，但是请不要忘记数学在实际中的应用，包括可能是最重要的一种应用形式， 学会如何思考。

If I could summarize this in one sentence, it would be this: Mathematics is not just solving for x, it's also figuring out why.

把我今天所说的浓缩成一句，那就是： 数学，不仅仅是求出X等于多少，还要能指出为什么。

Thank you very much.(Applause)

感谢大家。（掌声）