Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration.

主要有三個原因：計算、應用，以及最後且不幸地，就我們所給予它的時間來看也是最不重要的，靈感。

Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively.

數學是規律的科學，而我們學習數學是為了學習怎樣以有邏輯、 有批判性和創造性的方式思考。

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.

但是，有太多在學校進行的數學教育並沒有效地激勵學生思考，所以當學生問我們：「我們為什麼要學這個？」，他們通常只會聽到因為之後的數學課或者考試上要用到。

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?

可是，如果偶爾我們學數學僅僅是因為數學很有趣或迷人，或者因為它能激發思想的話，不是很好嗎？

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.

我知道很多人都還沒有機會了解到數學如何以有趣的方式呈現， 所以讓我用我最喜歡的一組數字： 費波那西數， 來為各位舉個小小的例子。

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

哇！這裡已經有費波那西數的愛好者了，真棒！

Now these numbers can be appreciated in many different ways.

我們可以從很多個方面來欣賞這組數字。

From the standpoint of calculation, they're as easy to understand as one plus one, which is two.

從計算上來看，它們非常易懂，比如，1 加 1 是 2。

Then one plus two is three, two plus three is five, three plus five is eight, and so on.

1 加 2 是 3，2 加 3 是 5，3 加 5 是 8 等等。

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.

事實上，我們稱做「費波那西」的這個人真正的名字其實是比薩的萊昂納多，而他在教授了西方世界今日所使用的算術方法的《計算之書》中描述了這些數字。

In terms of applications, Fibonacci numbers appear in nature surprisingly often.

從應用上來看，費波那西數意外地頗為頻繁地出現在自然界裡。

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.

花瓣的數目通常是一個費波那西數字， 而向日葵上、鳳梨上的螺旋數往往也是費波那西數字。

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.

事實上，費波那西數有更多的應用， 但我發現最振奮人心的是它們漂亮的數字規律。

Let me show you one of my favorites.

讓我給你看看我最愛的規律之一。

Suppose you like to square numbers, and frankly, who doesn't?

假設你喜歡平方數－老實說，誰不喜歡呢？

Let me show — let's look at the squares of the first few Fibonacci numbers, ok?

讓我來－讓我們看看頭幾個費波那西數的平方，好嗎？

So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on, alright?

1 的平方是 1， 2 的平方是 4，3 的平方是 9，5 的平方是 25，依此類推。

Now, it's no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right?

可想而知， 當你把相鄰的两個費波那西數加起來時，會得到下一個費波那西數。對吧？

That's how they're created.

這就是它們如何被定義的。

But you wouldn't expect anything special to happen when you add the squares together.

但你大概不會料到當你把這些數的平方加起來後，會有什麼特別的結果。

But check this out.

看看這個。

One plus one gives us two, and one plus four gives us five.

1 加 1 是 2，然後 1 加 4 是 5。

And four plus nine is 13, nine plus 25 is 34, and yes, the pattern continues.

4 加 9 是 13， 9 加 25 是 34，沒錯，這個規律會一直繼續下去。

In fact, here's another one.

事實上，還有另外一個規律。

Suppose you wanted to look at adding the squares of the first few Fibonacci numbers.

假設你想要看看把頭幾個費波那西數的平方值加起來會怎麼樣。

Let's see what we get there.

讓我們看看會有什麼結果。

So one plus one plus four is six.

1 加 1 加 4 等於 6。

Add nine to that, we get 15.

再加 9，我們得到 15。

Add 25, we get 40.

再加 25，我們得到 40。

Add 64, we get 104.

再加 64，我們得到 104。

Now look at those numbers.

現在來看看這些數字。

Those are not Fibonacci numbers, but if you look at them closely, you'll see the Fibonacci numbers buried inside of them.

它們不是費波那西數，但如果你再仔細審視這些數字，你會發現費波那西數就藏在它們裡面。

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

你找到了嗎？我指出來給各位看吧。

Six is two times three, 15 is three times five, 40 is five times eight. Two, three, five, eight, who do we appreciate?

6 是 2 乘 3、15 是 3 乘 5、40 是 5 乘 8。2、3、 5、 8，這是誰的功勞啊？

Fibonacci! Of course.

當然是費波那西囉！

Now, as much fun as it is to discover these patterns, it's even more satisfying to understand why they are true.

雖然找出這些規律確實有趣，但去瞭解為什麼會變成這樣的原理更能令人滿足。

Let's look at that last equation.

讓我們看看這最後的等式。

Why should the squares of one, one, two, three, five and eight add up to eight times 13?

為什麼 1、1、2、3、5 和 8 的平方加起來等於 8 乘以 13？

I'll show you by drawing a simple picture.

讓我畫一張簡單的圖來解釋給各位聽。

Alright? We'll start with a one-by-one square, and next to that put another one-by-one square.

好嗎？我們先由一個 1 x 1 的正方形開始，然後在旁邊再放一個 1 x 1 的正方形。

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

它們一起構成了一個 1 x 2 的矩形。

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?

接著，在下面再放一個 2 x 2 的正方形，旁邊再來一個 3 x 3 的正方形，在下方，放一個 5 x 5 的正方形， 然後在旁邊放一個 8 x 8 的正方形後，我們便能得到一個巨大的矩形，對吧？

Now let me ask you a simple question: what is the area of the rectangle?

現在讓我問你一個簡單的問題： 這個矩形的面積是多少？

Well, on the one hand, it's the sum of the areas of the squares inside it, right?

這個嘛，一方面， 它是所有這些所包含的正方形面積的總和，對吧？

Just as we created it.

正如我們如何創造了它。

It's one squared plus one squared, plus two squared, plus three squared, plus five squared, plus eight squared. Right?

它是 1 的平方加 1 的平方，加 2 的平方，再加 3 的平方，加 5 的平方，再加 8 的平方。對吧？

That's the area.

這就是總面積。

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?

另一方面，因為它是個矩形，面積等於高乘以底，而高顯然是 8， 而底是 5 加 8，又是一個費波那西數，13。對吧？

So the area is also eight times 13.

所以面積也是 8 乘以 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.

既然我們已經用兩種不同的方法正確地計算出了這個面積，它們必然是相同的數字， 而這就是為什麼 1、1、2、3、5 和 8 的平方加起來正好是 8 乘以 13。

Now, if we continue this process, we'll generate rectangles of the form 13 by 21, 21 by 34, and so on.

現在，如果我們繼續這一過程，我們會能創造出 13 x 21 的矩形、21 x 34 的矩形等等。

Now check this out.

再來看這個。

If you divide 13 by eight, you get 1.625.

如果用 13 除以 8，會得到 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.

如果你用較大的數除以較小的數，會發現這個比率越來越接近 1.618， 也就是數個世紀以來讓數學家，科學家和藝術家深深著迷，眾所周知的黃金比率。

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.

我會給各位看這些，是因為像很多數學原理一樣，它有著相當美麗的一面，而我認為這些美好並沒有在我們的學校中得到足夠的重視。

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.

我們花費大量的時間來學習如何計算，但我們不應忘了要學習應用數學的方式，而在其中包括或許是最重要的應用方式：學習如何去思考。

If I could summarize this in one sentence, it would be this:

如果要我用一句話來總結我想說的，那就是：

Mathematics is not just solving for X, it's also figuring out why.

數學不只是要解出解答，更要瞭解背後原因。

Thank you very much.

非常感謝各位聆聽。