我們?yōu)槭裁匆獙W數(shù)學？

我們?yōu)槭裁匆獙W數(shù)學？數(shù)學是研究規(guī)律的科學，我們通過學習它來訓練邏輯思維，思辯能力，創(chuàng)造力。但是學校里學的數(shù)學，卻激不起我們的興趣。每當學生問起“為什么要學?”得到的答案往往是“考試要考”。有沒有可能，哪怕只有一小會兒，我們研究數(shù)學僅僅因為興趣，或是數(shù)學的優(yōu)美？

演說者：Arthur Benjamin

演說題目：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.

我們?yōu)槭裁匆獙W數(shù)學？根本原因有三個：計算、 應用、最后一個，很不幸的從時間分配來看也是最少的，激發(fā)靈感。

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.?

數(shù)學是研究規(guī)律的科學，我們通過學數(shù)學來訓練邏輯思維能力、思辯能力以及創(chuàng)造力，但是我們在學校里面學到的數(shù)學，根本沒有激起我們的興趣。每當我們的學生問起?“我們?yōu)槭裁匆獙W這個？” 他們得到的答案往往是考試要考，或者后續(xù)的數(shù)學課程中要用到。

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.?

有沒有可能，哪怕只有那么一小會兒，我們研究數(shù)學僅僅是因為自己的興趣，或是數(shù)學的優(yōu)美，那豈不是很棒？現(xiàn)在我知道很多人一直沒有機會來體驗這一點，所以現(xiàn)在我們就來體驗一下，以我最喜歡的數(shù)列——斐波納契數(shù)列為例。

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

太好了！看來在座的也有喜歡斐波納契的。

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，我們可以從多種不同的角度來欣賞斐波納契序列。從計算的角度，斐波納契數(shù)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以此類推。

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.

事實上，那個我們稱呼"斐波納契"的人真實的名字叫列昂納多，來自比薩，這個數(shù)列出自他的書《算盤寶典》（"Liber Abaci"）這本書奠定了西方世界的數(shù)學基礎，其中的算術(shù)方法一直沿用至今。從應用的角度來看，斐波納契數(shù)列在自然界中經(jīng)常神奇的出現(xiàn)。一朵花的花瓣數(shù)量一般是一個斐波納契數(shù)，向日葵的螺旋，菠蘿表面的凸起也都對應著某個斐波納契數(shù)。

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?

事實上還有很多斐波納契數(shù)的應用實例，而我發(fā)現(xiàn)這其中最能給人啟發(fā)的是這些數(shù)字呈現(xiàn)出來的漂亮模式。讓我們看下我最喜歡的一個。假設你喜歡計算數(shù)的平方。坦白說，誰不喜歡？

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??

讓我們計算一下頭幾個斐波那契數(shù)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，以此類推。毫不意外的，當你加上兩個連續(xù)的斐波那契數(shù)字時，你得到了下一個斐波那契數(shù)，沒錯吧？

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.

它就是這么定義的。但是你不知道把斐波那契數(shù)的平方加起來會得到什么有意思的結(jié)果。來嘗試一下。1加1是2，1加4是5，4加9是13，9加25是34，沒錯，還是這個規(guī)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.

事實上，還有一個規(guī)律。假如你想計算一下頭幾個斐波納契數(shù)的平方和，看看結(jié)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。回頭來看看這些數(shù)字。他們不是斐波納契數(shù)，但是如果你看得夠仔細，你能看到他們的背后隱藏著的斐波納契數(shù)。?

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？讓我寫給你看。6等于2乘3，15等于3乘5，40等于5乘8，2，3，5，8，我們看到了什么？（笑聲）斐波納契！當然，當然。

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?

現(xiàn)在我們已經(jīng)發(fā)現(xiàn)了這些好玩的模式，更能滿足你們好奇心的事情是弄清楚背后的原因。讓我們看看最后這個等式。為什么1， 1， 2， 3， 5和8的平方加起來等于8乘以13？我通過一個簡單的圖形來解釋。首先我們畫一個1乘1的方塊，然后再在旁邊放一個相同尺寸的方塊。拼起來之后得到了一個1乘2的矩形。在這個下面再放一個2乘2的方塊，之后貼著再放一個3乘3的方塊，然后再在下面放一個5乘5的矩形，之后是一個8乘8的方塊。得到了一個大的矩形，對吧?

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.

現(xiàn)在問大家一個簡單的問題：這個矩形的面積是多少？一方面，它的面積就是組成它的小矩形的面積之和，對吧？就是我們用到的矩形之和它的面積是 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。對吧？這就是面積。另一方面，因為這是矩形，面積就等于長乘高，高等于 8，長是 5 加 8，也是一個斐波納契數(shù)，13，是不是？所以面積就是8乘 13。因為我們用兩種不同的方式計算面積，同樣一個矩形的面積 一定是一樣的，這樣就是為什么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.

如果我們繼續(xù)探索下去，我們會得到13乘21的矩形，21乘 34的矩形，以此類推。再來看看這個。如果你用8去除13，結(jié)果是1.625。如果用大的斐波納契數(shù)除以前一個小的斐波納契數(shù)他們的比例會越來越接近1.618，這就是很多人知道的黃金分割率，一個幾個世紀以來，讓無數(shù)數(shù)學家，科學家和藝術(shù)家都非常著迷的數(shù)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.

我之所以向你們展示這些是因為，很多這樣的數(shù)學（知識），都有其秒不可言的一面而我擔心這一面并沒有在學校里得到展現(xiàn)。我們花了很多時間去學算術(shù)，但是請不要忘記數(shù)學在實際中的應用，包括可能是最重要的一種應用形式， 學會如何思考。

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.

把我今天所說的濃縮成一句，那就是： 數(shù)學，不僅僅是求出X等于多少，還要能指出為什么。

Thank you very much.(Applause)

感謝大家。（掌聲）

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