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So we've seen one example of this, this idea of walking through all the integers looking for the square root.
现在计算机速度很快了，我可以把这个数字设的更大点,计算机会去很快的做这个事情。
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And the fourth thing to notice is, I've used abstraction. I've said nothing about how I'm going to make square root.
第四件事情就是，我运用了抽象,我对怎么去求平方根没有提及。
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First of all, it's clear from the Pythagoras' theorem that a is the square root of ^2 + ^2.
首先,根据毕达哥拉斯定理,勾股定理在西方被称为"毕达哥拉斯定理"
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But he said the energy of an X-Y bond is going to be equal to the square root.
但他说ＸＹ键的键能,会等于ＸＸ键与ＹＹ键键能乘积的平方根。
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Because the square root of 10,000 is 100, whatever the standard deviation of the portfolio is, you would divide it by 100 and it would become really small.
因为一万的平方根是一百，无论这个投资组合的标准差是多大，当除以100后就都变得很小很小了。
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When we look at this angular part, we see that it's always the square root of 1 over 4 pi, it doesn't matter what the angle is, it's not dependent on the angle.
当我们看这角向部分，可以看到它总是等于１除以４pai开根号，这和是什么角度没有关系，它和角度无关。
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You know, if you're wandering through Harvard Square and you see an out-of-work Harvard grad, they're handing out examples of square roots, they'll give you an example and you can test it to see is the square root of 2, 1.41529 or whatever.
你知道，如果你从哈佛校园里穿过去,你看见了一个失业校友，正在派发平方根的示例，他们会给你一个例子,而你会检查2的平方根是1。41529或者别的什么。
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Square root of 16, and my plot basically shows it crosses at 4 and-- well, I think that's minus 4.
我的图清楚的显示这个函数,在4和-4两个点和x轴交合。
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I first tested it on the square root of 4 and in one iteration it found 2.
我首先测了下求4的平方根4,它只迭代了一次并返回了2。
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60 If we look at 435 times 160, take the square root of that, we will end up with 264 kilojoules per mole, which sensibly lies between these two values.
35乘以，取平方根，我们会得到264千焦每摩，很合理地落在两数值之间。
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Well I'm relying on the contract, if you like, that the manufacturer of square root put together, which is, if I know I'm giving it two floats, which I do because I make sure they're floats, the contract, if you like, of square root says I'll give you back a float.
好吧我能靠一些类似合约来保证，如果这么说你喜欢的话,其实就是当求平方根的方法,获得了两个我已经确认了是浮点数的数后,这个类似合约的东西就会保证，返回一个浮点数。
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It was over here. Because the square root of a quarter is not smaller than a quarter it's bigger than a quarter. Right?
答案超出了这个区间,因为0。25的平方根会比0。25大,是应该比0。25大对不对？
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Add the same thing to the y-values, squared, take the square root.
的差的平方，然后加起来开平方。
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Remember, I said when we do a bisection method, we're assuming the answer lies somewhere between Well, what is the square root of a quarter? It is a half.
记住，我提过当我们用二分法的时候,我们假设答案处于,上边界和下边界之间,0。25的平方根是多少？,是0。5。
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Or, it could act like a square root machine.
或者，它可以表现的像个开平方根的机器。
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The standard deviation is the square root of the variance.
标准差是方差的平方根
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In some sense, strictly speaking they shouldn't be necessary because the fact that my specification starts with an assumption, says, hey you, who might call square root make sure that the things you call me with obey the assumption.
在某种意义上严格的来讲,它们不是必要的,因为我已经对假设做了声明,也就是说，如果你要调用我的，求平方根的函数,请确保你传递给我的,参数满足假设条件。
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What we want to do now is compute the mean and variance of the portfolio-- or the mean and standard deviation, since standard deviation is the square root of the variance-- for different combinations of the portfolios.
我们现在要做的是,计算这个投资组合的均值和方差-,或者均值和标准差，因为标准差的平方就等于方差-,这对任何投资组合都是一样的。
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Let's call x the thing I'm trying to find the square root of. Let's start at 1. Square it.
让我们把求平方根的对象,这个数称为x，让我们从1开始。
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So we've looked at how to solve square root we've, looked at two problems, I've tried to instill in you this sense of paranoia which is so valuable, and now we're going to pull back Python and return to something much simpler than numbers, and that's Python. All right?
我们已经研究了如何去求平方根,研究了两种方法,我一直在尝试着给你们逐渐灌输,要偏执的理念，这样的理念非常有价值，现在我们回过头来去学习一些,比数字更简单的东西，也就是,对不对？数学问题很难？
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I like that answer. I then tested it on the square root of 9, and as I mentioned last time, I didn't find 3. I was not crushed.
我喜欢这个答案,然后我又试了试9的平方根，跟我上次说的一样，答案不是3。
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Then if I'm looking for the square root of something less than 1, I know it will be in my region, right?
如果我要求平方根的数小于,我就知道我们的答案应该,在0-1这个区间了，对不对？
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I want to then do, I need to find the square root b squared plus h squared, right?
我接下来要求b的平方和h的平方,的和的平方根对不对？
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In a high-level language, square root might simply be a primitive that you can use rather than you having to go over and code it.
一个高级的语言，平方根可能仅仅,只是你能使用的一个基本要素,而不是你需要去编译它。
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You have square root of minus one for something else.
你得到了某个负数的平方根。
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Computing square root. I'm going to capture it in a piece of code, I'm going to be able to refer to it by a name, and I'm going to suppress the details, meaning inside of that computation, you don't need to know what it does. You just need to know, if I give it the right kind of input, it'll give me back an input that satisfies the contract that I set up.
比如计算平方根,我会用一段代码来说明,我将给它命名,并且忽略细节,意思就是在这一段计算过程中,你不用知道它做了什么，你只需知道，如果我进行了正确的输入,它将会给我符合条件的输出。
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It's the square root of /100.
就是/100的开平方
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I'm given an integer that's a perfect square, and I want to write a little piece of code that's going to find the square root of it. All right so I'm cheating a little, I know it's a perfect square, somebody's given it to me, we'll come back in a second to generalizing it, so what would the steps be that I'd use to walk through it?
完美平方数的整数，我想写一段代码来求这个数的平方根,好，我这儿有点儿作弊了,我知道这是一个完美的平方数了,他们给我的，我们后面会讲怎么产生这个数的，那么我想解决这个问题,需要什么步骤呢？
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And the good news was we didn't have to worry about how math did square root or anything like that, we just got it and we used it.
也可能是其他数，好消息是,我们不必担心math函数是怎样开方的,我们只要使用它就好。
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