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ArthurBenjamin_2013G-_神奇的斐波那契数列_
So why do we learn mathematics ? 我们为什么要学习数学?
mathematics:n.数学;数学运算;
Essentially , for three reasons: calculation, application , and last, and unfortunately least in terms of the time we give it, inspiration. 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.
Essentially:adv.本质上;本来; application:n.应用;申请;应用程序;敷用; unfortunately:adv.不幸地;
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?" 数学是研究规律的科学, 我们通过学习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习到的数学, 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?
logically:adv.逻辑上;合乎逻辑; critically:adv.精密地;危急地;批评性地;用钻研眼光地; creatively:adv.创造性地;有创造力地; motivated:adj.有动机的; v.使产生动机;
then they often hear that they'll need it in an upcoming math class or on a future test. 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.
upcoming:adj.即将来临的;
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? 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?
every once in a while:有时;偶然;偶尔;时而;
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) 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.(掌声)
Fibonacci:n.斐波纳契(一种整数数列);
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 in many different ways. 非常好. 来欣赏斐波纳契序列.
appreciated:v.欣赏;感激;理解;(appreciate的过去分词和过去式)
From the standpoint of calculation, they're as easy to understand as one plus one, which is two. 从计算的角度 斐波纳契数列很容易被理解 1 加 1, 等于 2
standpoint:n.立场;观点;
Then one plus two is three, two plus three is five, three plus five is eight, and so on. 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," 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 )
Pisa:n.比萨(意大利城市); Liber:n.书籍;契据登记簿;
which taught the Western world the methods of arithmetic that we use today. 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.
arithmetic:n.算术,算法;
In terms of applications, 从应用的角度来看,
Fibonacci numbers appear in nature surprisingly often. 斐波纳契数列在自然界中经常 神奇的出现.
surprisingly:adv.令人惊讶地;出乎意料地
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. 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.
petals:n.[植]花瓣;翼瓣(petal的复数形式);v.绽出花瓣(petal的第三人称单数形式); typically:adv.代表性地;作为特色地; spirals:螺旋;盘旋; sunflower:n.向日葵;
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.
inspirational:adj.鼓舞人心的;带有灵感的,给予灵感的; display:n.显示;炫耀;vt.显示;表现;陈列;vi.[动]作炫耀行为;adj.展览的;陈列用的;
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, and frankly, who doesn't? (Laughter) 假设你喜欢计算数的平方. 坦白说, 谁不喜欢?(笑声)
Suppose:v.推断:假定:假设:设想:
Let's look at the squares of the first few Fibonacci numbers. 让我们计算一下 头几个斐波纳契数的平方.
So one squared is one, two squared is four, three squared is nine, five squared is 25, and so on. 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.
Now, it's no surprise that when you add consecutive Fibonacci numbers, you get the next Fibonacci number. Right? 毫不意外的, 当你加上两个连续的斐波纳契数字时, 你得到了下一个斐波纳契数, 没错吧?
consecutive:adj.连贯的;连续不断的;
That's how they're created. 它们就是这么定义的.
But you wouldn't expect anything special to happen when you add the squares together. 但是你不知道把斐波纳契数的平方 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, and one plus four gives us five. 1 加 1 是 2, 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, 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 adding the squares of the first few Fibonacci numbers. 假如你想计算一下 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, but if you look at them closely, you'll see the Fibonacci numbers buried inside of them. 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. 看到了么? 让我写给你看.
Six is two times three, 15 is three times five, 40 is five times eight, two, three, five, eight, who do we appreciate? 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?
(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, it's even more satisfying to understand why they are true. 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.
Let's look at that last equation . 让我们看看最后这个等式.
equation:n.方程式,等式;相等;[化学]反应式;
Why should the squares of one, one, two, three, five and eight add up to eight times 13? 为什么 1, 1, 2, 3, 5 和 8 的平方 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 and next to that put another one-by-one square. 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.
Together, they form a one-by-two rectangle . 拼起来之后得到了一个 1 乘 2 的矩形.
rectangle:n.矩形;长方形;
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? 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?
Beneath:prep.在…之下;adv.在下方; five-by-five:adj.矮胖的;
Now let me ask you a simple question: what is the area of the rectangle? 现在问大家一个简单的问题: 这个矩形的面积是多少?
Well, on the one hand , it's the sum of the areas of the squares inside it, right? 一方面, 它的面积就是 组成它的小矩形的面积之和, 对吧?
on the one hand:一方面;
Just as we created it. 就是我们用到的矩形之和
It's one squared plus one squared plus two squared plus three squared plus five squared plus eight squared. Right? 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?
That's the area. 这就是面积.
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?
On the other hand:另一方面;
So the area is also eight times 13. 所以面积就是 8 乘 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. 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. 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.
process:v.处理;加工;列队行进;n.过程,进行;方法,adj.经过特殊加工(或处理)的; generate:v.产生;引起; rectangles:n.[数]矩形;长方形(rectangle的复数形式);
Now check this out. 再来看看这个.
If you divide 13 by eight, you get 1.625. 如果你用 8 去除 13, 结果是 1.625.
And if you divide the larger number by the smaller number, then these ratios get closer and closer to about 1.618, 1.618, known to many people as the Golden Ratio, a number which has fascinated mathematicians , scientists and artists for centuries. 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 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.
ratios:n.比率(ratio的复数); v.以比率表示; mathematicians:n.[数]数学家(mathematician的复数形式);
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.
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. 我们花了很多时间去学习算术, 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: 把我今天所说的浓缩成一句, 那就是:
summarize:v.总结;概述;概括;归纳;
Mathematics is not just solving for x, it's also figuring out why. 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. 感谢大家.
(Applause) (掌声)