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.
我們為什麼要學數學? 主要有三個原因: 計算 應用 最後,不幸地,也是最不重要的, 就我們所給予它的時間來看, 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. 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. (Applause)
數學是规律的科學, 而我們學習數學是為了學習怎樣邏輯地, 批評地和有創造性地思考, 但是,太多我們在學校學的數學 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?” 他們會聽到(我們說)因為下一節是數學課 或者將來會有考試,他們需要這個。 可是,如果 偶爾我們學數學 僅僅是因為數學很有趣或迷人, 或者因為它激發思想,不是很好嗎? 我知道很多人都還沒有 機會去看到數學如何可以有趣, 所以讓我用我最喜歡的一組數字, 來給你舉個小小的例子, 費波那西數。(鼓掌)
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。 不錯。
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. 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.
(我們可以)從很多個方面來 欣賞這組數字。 從計算上來看, 它們非常易懂 比如,1加1,是2. 1加2是3, 2加3是5,3加5是8, 等等。 事實上,我們稱做“費波那西”的這個人 是比薩的莱昂纳多, 而這些數字是在他的“計算之書”中描述的, 這本書教授了西方世界 我們今天所使用的算術方法。 從應用上來看, 費波那西數讓人驚訝地 頻繁出現在自然界裡。 花瓣的數目 通常是一個費波那西數字, 或向日葵上、鳳梨上的螺旋數 往往也是費波那西數字。 事實上,費波那西數有更多的應用,
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? (Laughter)
但我發現最鼓舞人心的 是它們所顯示的漂亮的數字规律。 讓我給你看看我的最愛之一。 假設你喜歡平方數, 坦率地說,誰不喜歡?(笑聲)
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? 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.
讓我們看看頭幾個 費波那西數的平方。 1的平方是1, 2 的平方是4,3的平方是9, 5 的平方是 25,依此類推。 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? 這就是它們如何被定義的。 但你大概不會料到 當你把這些數的平方加起來, 會有什麼特別的結果。 看這個, 1加1是2, 然後,1加4是5。 4加9是13, 9 加 25 是 34, 是的,這個規律一直繼續下去。
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,還有另外一個。 假設你想要看看 把頭幾個費波那西數的平方值加起來。 讓我們看看會有什麼結果。 1加1加4等於6。 再加9,我們得到15。 再加 25,我們得到 40。 再加 64,我們得到104。 現在來看看這些數字。 那些不是費波那西數, 但如果你仔細再看這些數字, 你會看到費波那西數 藏在它們裡面。
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?
你看到了嗎?讓我指出來給你。 6是2乘3、 15 是3乘5、 40 是5乘8、 2、3、 5、 8,我們在欣賞什麼?
(Laughter)
(笑聲)
Fibonacci! Of course.
當然是費波那西數!
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?
正如找出這些規律是很好玩的, 更令人滿意的是瞭解 為什麼它們是這樣的。 讓我們看看這最後的等式。 為什麼1,1,2,3,5和8的平方 加起來等於8乘以13? 我畫一張簡單的圖來解釋給你。 我們先由一個1x1的正方形開始 在旁邊再放一個1x1的正方形。 它們一起,構成一個1x2的矩形。 接著,再放一個2x2的正方形, 旁邊再來一個3x3的正方形, 在下方,放一個5x5的正方形, 然後旁邊一個8x8的正方形, 得到一個巨大的矩形,對嗎?
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.
現在讓我問你一個簡單的問題: 這個矩形的面積是多少? 好吧,一方面, 它是所有這些所包含的 正方形面積的總和,是吧? 正如我們如何創造了它, 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? 這就是總面積。 另一方面,因為它是個矩形 面積等於高乘以底, 高顯然是8, 而底是5加8, 這就是下一個費波那西數,13。對吧? 所以面積也是8乘以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, 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, 我們會生成13x21 的矩形, 21x34 的矩形等等。
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.
再來看這個。 如果你用 13除以8, 你得到 1.625。 如果你用較大的數除以較小的數, 會發現這些比率越來越接近 1.618, 眾所周知的黃金比率, 一個讓數學家,科學家和藝術家 著迷幾個世紀的數字。
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.
我給你看這些,是因為 像很多數學, 都有它美麗的一面 而我覺得(這些美麗)沒有在我們的學校 得到足夠的重視。 我們花費大量的時間來學習如何計算, 但別忘了要應用, 或許,包括,最重要的應用, 學習如何去思考。
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.
如果要用一句話來總結, 那就是: 數學不只是解出x, 也要知道為什麼。
Thank you very much.
謝謝。
(Applause)
(掌聲)