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Lecture 04 Floating Point.ass
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Lecture 04 Floating Point.ass
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Dialogue: 0,0:00:00.86,0:00:02.36,English,,0,0,0,,Good afternoon everybody
Dialogue: 0,0:00:00.86,0:00:02.36,Chinese,,0,0,0,,大家下午好
Dialogue: 0,0:00:03.42,0:00:04.36,English,,0,0,0,,Welcome
Dialogue: 0,0:00:03.42,0:00:04.36,Chinese,,0,0,0,,欢迎
Dialogue: 0,0:00:05.64,0:00:06.70,English,,0,0,0,,Good to see you
Dialogue: 0,0:00:05.64,0:00:06.70,Chinese,,0,0,0,,很高兴见到你们
Dialogue: 0,0:00:11.32,0:00:17.46,English,,0,0,0,,So today we're going to study an arcane area of computer systems known as floating point
Dialogue: 0,0:00:11.32,0:00:17.46,Chinese,,0,0,0,,今天要学习的是浮点数，它是计算机系统中比较难理解的内容
Dialogue: 0,0:00:17.98,0:00:22.34,English,,0,0,0,,Which is the way that computer systems represent the real numbers
Dialogue: 0,0:00:17.98,0:00:22.34,Chinese,,0,0,0,,它是计算机系统内部表示实数的方法
Dialogue: 0,0:00:24.54,0:00:31.06,English,,0,0,0,,So you're all familiar with decimal you know something like 3.14
Dialogue: 0,0:00:24.54,0:00:31.06,Chinese,,0,0,0,,你们都熟悉十进制，比方说 3.14
Dialogue: 0,0:00:31.52,0:00:36.36,English,,0,0,0,,And there's a similar idea with binary numbers
Dialogue: 0,0:00:31.52,0:00:36.36,Chinese,,0,0,0,,二进制数也有类似的东西
Dialogue: 0,0:00:36.82,0:00:50.04,English,,0,0,0,,Okay so what does a fractional binary number like 10111.01 represent and how does that work
Dialogue: 0,0:00:36.82,0:00:50.04,Chinese,,0,0,0,,那么像 10111.01 那样的二进制小数代表什么呢？它是如何使用的？
Dialogue: 0,0:00:51.96,0:00:58.34,English,,0,0,0,, Well, it works just like the fractional decimal numbers that you used to
Dialogue: 0,0:00:51.96,0:00:58.34,Chinese,,0,0,0,,它就像你熟悉的十进制小数一样使用
Dialogue: 0,0:01:04.92,0:01:08.76,English,,0,0,0,,So you have a... this case, it's a binary point
Dialogue: 0,0:01:04.92,0:01:08.76,Chinese,,0,0,0,,在二进制小数中有一个二进制小数点
Dialogue: 0,0:01:08.76,0:01:10.62,English,,0,0,0,,And to the left of the binary point
Dialogue: 0,0:01:08.76,0:01:10.62,Chinese,,0,0,0,,在二进制小数点的左边
Dialogue: 0,0:01:12.50,0:01:16.46,English,,0,0,0,,You have bit positions to representing powers-of-2
Dialogue: 0,0:01:12.50,0:01:16.46,Chinese,,0,0,0,,每一位代表 2 的正幂
Dialogue: 0,0:01:16.46,0:01:20.00,English,,0,0,0,,So 2^0, 2^1, 2^2
Dialogue: 0,0:01:16.46,0:01:20.00,Chinese,,0,0,0,,所以左边的每一位分别代表 2^0，2^1，2^2
Dialogue: 0,0:01:20.64,0:01:26.60,English,,0,0,0,,So 1/2... I'm sorry. 1, 2, 4 and so on
Dialogue: 0,0:01:20.64,0:01:26.60,Chinese,,0,0,0,,所以从小数点开始向左依次是 1，2，4 等等
Dialogue: 0,0:01:27.34,0:01:30.50,English,,0,0,0,,And to the right of the binary point
Dialogue: 0,0:01:27.34,0:01:30.50,Chinese,,0,0,0,,在二进制小数点的右侧
Dialogue: 0,0:01:31.04,0:01:40.46,English,,0,0,0,,You have 2^-1=1/2, 2^-2=1/4, 2^-3=1/8 and so on
Dialogue: 0,0:01:31.04,0:01:40.46,Chinese,,0,0,0,,存在 2^-1=1/2，2^-2=1/4，2^-3=1/8 等等
Dialogue: 0,0:01:41.20,0:01:46.80,English,,0,0,0,,So we can represent the... any fractional binary number
Dialogue: 0,0:01:41.20,0:01:46.80,Chinese,,0,0,0,,所以我们可以使用这种标记法表示任何二进制小数
Dialogue: 0,0:01:47.54,0:01:57.36,English,,0,0,0,,Using this notation where we sum each the bit at position k*2^k
Dialogue: 0,0:01:47.54,0:01:57.36,Chinese,,0,0,0,,我们对每个第 k 处的值乘 2^k 后求和
Dialogue: 0,0:01:58.76,0:02:05.46,English,,0,0,0,,Ok so it's really works exactly the same way as with base 10, just with base 2
Dialogue: 0,0:01:58.76,0:02:05.46,Chinese,,0,0,0,,这与 10 进制完全相同，只是将基数变成了 2
Dialogue: 0,0:02:08.94,0:02:13.94,English,,0,0,0,,So let's look at some of these values
Dialogue: 0,0:02:08.94,0:02:13.94,Chinese,,0,0,0,,那么让我们看看其中一些值
Dialogue: 0,0:02:13.94,0:02:20.24,English,,0,0,0,,So 5 3/4 we would represent as 101.11
Dialogue: 0,0:02:13.94,0:02:20.24,Chinese,,0,0,0,,所以 5 3/4 会被表示为 101.11
Dialogue: 0,0:02:21.96,0:02:27.96,English,,0,0,0,,So this is before, (1*2^1) plus 1 is 5
Dialogue: 0,0:02:21.96,0:02:27.96,Chinese,,0,0,0,,所以二进制小数点左边相加后是5
Dialogue: 0,0:02:29.30,0:02:32.58,English,,0,0,0,,+1/2, +1/4
Dialogue: 0,0:02:29.30,0:02:32.58,Chinese,,0,0,0,,（右边）+1/2, +1/4
Dialogue: 0,0:02:32.88,0:02:38.40,English,,0,0,0,,Ok so 2/4+1/4=3/4. so 5 3/4
Dialogue: 0,0:02:32.88,0:02:38.40,Chinese,,0,0,0,,2/4+1/4=3/4。最终结果 5 3/4
Dialogue: 0,0:02:38.92,0:02:40.50,English,,0,0,0,,Ok so that's pretty simple
Dialogue: 0,0:02:38.92,0:02:40.50,Chinese,,0,0,0,,好的，这很简单
Dialogue: 0,0:02:41.34,0:02:51.92,English,,0,0,0,,2 7/8 is 10.111 so that 10 is 2 of course
Dialogue: 0,0:02:41.34,0:02:51.92,Chinese,,0,0,0,,2 7/8 表示为 10.111。二进制中的 10 是十进制中的 2
Dialogue: 0,0:02:53.90,0:03:08.66,English,,0,0,0,,And then .111 is 1/2,so 4/8, 1/4=2/8+1/8, okay so 7/8
Dialogue: 0,0:02:53.90,0:03:08.66,Chinese,,0,0,0,,然后 .111 是 1/2（4/8）+ 1/4（2/8）+ 1/8 = 7/8
Dialogue: 0,0:03:11.58,0:03:21.04,English,,0,0,0,,And 1 7/16 would be 1.0111
Dialogue: 0,0:03:11.58,0:03:21.04,Chinese,,0,0,0,,1 7/16 将是 1.0111
Dialogue: 0,0:03:21.16,0:03:24.20,English,,0,0,0,,So that the 1 to the left of the binary point is 1
Dialogue: 0,0:03:21.16,0:03:24.20,Chinese,,0,0,0,,这样二进制小数点左边的 1 就是 1
Dialogue: 0,0:03:25.08,0:03:31.04,English,,0,0,0,,And then we have 1/4+1/8+1/16
Dialogue: 0,0:03:25.08,0:03:31.04,Chinese,,0,0,0,,然后有 1/4+1/8+1/16
Dialogue: 0,0:03:32.00,0:03:47.70,English,,0,0,0,,So 1/4 is 4/16, 1/8 is 2/16 and 1/16,so you add those up you get 7/16
Dialogue: 0,0:03:32.00,0:03:47.70,Chinese,,0,0,0,,所以 1/4 是 4/16，1/8 是 2/16，还有 1/16，所以总和为 7/16
Dialogue: 0,0:03:49.48,0:03:51.66,English,,0,0,0,,Okay now notice that for some of these numbers
Dialogue: 0,0:03:49.48,0:03:51.66,Chinese,,0,0,0,,好的，现在注意到其中一些数字
Dialogue: 0,0:03:51.66,0:03:57.90,English,,0,0,0,,This bit pattern 101.11
Dialogue: 0,0:03:51.66,0:03:57.90,Chinese,,0,0,0,,这个位组合，101.11
Dialogue: 0,0:03:58.50,0:04:02.82,English,,0,0,0,,If we shift it to the right to get 10.111
Dialogue: 0,0:03:58.50,0:04:02.82,Chinese,,0,0,0,,如果我们将数字向右移动，获得 10.111
Dialogue: 0,0:04:04.36,0:04:08.58,English,,0,0,0,,We've divided by 2, right? So we've taken 5 3/4 and divided it by 2
Dialogue: 0,0:04:04.36,0:04:08.58,Chinese,,0,0,0,,就相当于用 5 3/4 除以2
Dialogue: 0,0:04:09.20,0:04:12.00,English,,0,0,0,,And it works the other way too
Dialogue: 0,0:04:09.20,0:04:12.00,Chinese,,0,0,0,,反方向也同样适用
Dialogue: 0,0:04:12.00,0:04:17.28,English,,0,0,0,,If you shift to the left you multiply. If you shift to the left by 1 you multiply by 1
Dialogue: 0,0:04:12.00,0:04:17.28,Chinese,,0,0,0,,如果将数字向左移动 1 位，则乘 2 一次
Dialogue: 0,0:04:19.02,0:04:26.12,English,,0,0,0,,And these numbers that are all ones to the right of the of the binary point
Dialogue: 0,0:04:19.02,0:04:26.12,Chinese,,0,0,0,,而二进制小数点右边的数字
Dialogue: 0,0:04:26.12,0:04:31.62,English,,0,0,0,,These are numbers that are just slightly less than 1
Dialogue: 0,0:04:26.12,0:04:31.62,Chinese,,0,0,0,,这些数字只比 1 小一点点
Dialogue: 0,0:04:33.50,0:04:36.14,English,,0,0,0,,So well when we want to represent those
Dialogue: 0,0:04:33.50,0:04:36.14,Chinese,,0,0,0,,因此，当我们想要表示这些值的时候
Dialogue: 0,0:04:36.14,0:04:43.36,English,,0,0,0,,We will sometimes refer to those numbers of that form is 1- ℇ
Dialogue: 0,0:04:36.14,0:04:43.36,Chinese,,0,0,0,,我们有时会以 1.0-ℇ 指代那些形式的数字
Dialogue: 0,0:04:43.36,0:04:50.16,English,,0,0,0,,And ℇ just depends on how many bits you have to the right of the binary point
Dialogue: 0,0:04:43.36,0:04:50.16,Chinese,,0,0,0,,而 ℇ 取决于你在二进制小数点右边有多少位
Dialogue: 0,0:04:50.38,0:04:56.40,English,,0,0,0,,So ℇ gets smaller the more,the more of those bits you have there, it gets closer to 1
Dialogue: 0,0:04:50.38,0:04:56.40,Chinese,,0,0,0,,因此，ℇ 变得越小，二进制数的值就越接近 1
Dialogue: 0,0:04:56.90,0:04:59.30,English,,0,0,0,,And ℇ gets smaller the more of those bits you have
Dialogue: 0,0:04:56.90,0:04:59.30,Chinese,,0,0,0,,并且 ℇ 越小，二进制小数点右边的位就越多
Dialogue: 0,0:05:00.92,0:05:08.08,English,,0,0,0,,Now there's limitations in this kind of representation
Dialogue: 0,0:05:00.92,0:05:08.08,Chinese,,0,0,0,,但是，这种表示法存在局限性
Dialogue: 0,0:05:08.08,0:05:14.18,English,,0,0,0,,So first of all we can only represent rational numbers of the form x/2^k
Dialogue: 0,0:05:08.08,0:05:14.18,Chinese,,0,0,0,,首先，我们只能表示形如 x/2^k 的有理数
Dialogue: 0,0:05:14.18,0:05:16.80,English,,0,0,0,,We can only represent numbers of that form exactly
Dialogue: 0,0:05:14.18,0:05:16.80,Chinese,,0,0,0,,可表示的数必须严格满足那种形式
Dialogue: 0,0:05:17.20,0:05:23.70,English,,0,0,0,,Other numbers have to get have repeating bit representations that can't be represented
Dialogue: 0,0:05:17.20,0:05:23.70,Chinese,,0,0,0,,不满足上述形式的数需要循环重复位才能被准确表示
Dialogue: 0,0:05:24.06,0:05:27.82,English,,0,0,0,,with a finite number of bits that we have in a computer system
Dialogue: 0,0:05:24.06,0:05:27.82,Chinese,,0,0,0,,由于现在计算机中的位数是有限的，所以需要循环重复位的数不能被准确表示
Dialogue: 0,0:05:29.14,0:05:38.54,English,,0,0,0,,So for example things like a 1/3 is this repeating representation 1/5 and 1/10 or other examples of that
Dialogue: 0,0:05:29.14,0:05:38.54,Chinese,,0,0,0,,因此，例如 1/3 就是需要循环重复位才能被准确表示的数，1/5 和 1/10 也是
Dialogue: 0,0:05:39.28,0:05:42.06,English,,0,0,0,,Now the other problem we have is that
Dialogue: 0,0:05:39.28,0:05:42.06,Chinese,,0,0,0,,现在我们遇到的另一个问题是
Dialogue: 0,0:05:42.68,0:05:45.18,English,,0,0,0,,We can only...
Dialogue: 0,0:05:42.68,0:05:45.18,Chinese,,0,0,0,,我们只能……
Dialogue: 0,0:05:45.18,0:05:49.02,English,,0,0,0,,There's only so many bits to the left and the right of the the binary point
Dialogue: 0,0:05:45.18,0:05:49.02,Chinese,,0,0,0,,二进制小数点的左侧和右侧只有那么多位
Dialogue: 0,0:05:49.88,0:05:52.06,English,,0,0,0,,So we're kind of...
Dialogue: 0,0:05:49.88,0:05:52.06,Chinese,,0,0,0,,所以
Dialogue: 0,0:05:52.12,0:05:55.48,English,,0,0,0,,If we move the binary point to the left
Dialogue: 0,0:05:52.12,0:05:55.48,Chinese,,0,0,0,,如果我们将二进制小数点左移
Dialogue: 0,0:05:56.98,0:06:02.90,English,,0,0,0,,Then we don't have as many... We can't represent as many large numbers
Dialogue: 0,0:05:56.98,0:06:02.90,Chinese,,0,0,0,,那么，很多原本能表示的数值较大的数现在就不能被表示了
Dialogue: 0,0:06:02.90,0:06:04.62,English,,0,0,0,,We can only represent small numbers
Dialogue: 0,0:06:02.90,0:06:04.62,Chinese,,0,0,0,,我们只能表示小一点的数字
Dialogue: 0,0:06:04.62,0:06:07.54,English,,0,0,0,,But we have more precision to the right of the binary point
Dialogue: 0,0:06:04.62,0:06:07.54,Chinese,,0,0,0,,但是在二进制小数部分有更高的精度
Dialogue: 0,0:06:07.76,0:06:10.18,English,,0,0,0,,So we can represent more fractional values
Dialogue: 0,0:06:07.76,0:06:10.18,Chinese,,0,0,0,,所以我们可以表示更多的小数值
Dialogue: 0,0:06:10.60,0:06:13.70,English,,0,0,0,,Just the range of those values will be much smaller
Dialogue: 0,0:06:10.60,0:06:13.70,Chinese,,0,0,0,,只是总的取值范围会小得多
Dialogue: 0,0:06:14.56,0:06:16.98,English,,0,0,0,,Similarly if we move the binary point to the right
Dialogue: 0,0:06:14.56,0:06:16.98,Chinese,,0,0,0,,同样，如果我们将二进制小数点向右移动
Dialogue: 0,0:06:17.78,0:06:20.32,English,,0,0,0,,We'll have a larger range of values
Dialogue: 0,0:06:17.78,0:06:20.32,Chinese,,0,0,0,,我们将拥有更大的取值范围
Dialogue: 0,0:06:20.54,0:06:23.18,English,,0,0,0,,But we won't build represent as many fractions
Dialogue: 0,0:06:20.54,0:06:23.18,Chinese,,0,0,0,,但我们不能表示同样多的小数
Dialogue: 0,0:06:24.82,0:06:32.06,English,,0,0,0,,So there's this kind of trade-off and floating point is a representation
Dialogue: 0,0:06:24.82,0:06:32.06,Chinese,,0,0,0,,因此，选择浮点作为折中的表示方式
Dialogue: 0,0:06:32.80,0:06:42.76,English,,0,0,0,,to try that enables us to move that basically to move that binary point to represent sort of as wide a range as possible
Dialogue: 0,0:06:32.80,0:06:42.76,Chinese,,0,0,0,,浮点主要通过移动二进制小数点来表示尽可能大的取值范围
Dialogue: 0,0:06:43.14,0:06:46.26,English,,0,0,0,,with as much precision given the number of bits
Dialogue: 0,0:06:43.14,0:06:46.26,Chinese,,0,0,0,,兼顾尽可能高的精度
Dialogue: 0,0:06:46.78,0:06:49.14,English,,0,0,0,,the finite number of bits that we have to work with
Dialogue: 0,0:06:46.78,0:06:49.14,Chinese,,0,0,0,,同时还要受到位数有限的制约
Dialogue: 0,0:06:49.50,0:06:54.56,English,,0,0,0,,So that the float and floating point is this sort of shifting binary point. Right
Dialogue: 0,0:06:49.50,0:06:54.56,Chinese,,0,0,0,,因此，浮点就是这种移动的二进制小数点
Dialogue: 0,0:06:54.56,0:06:57.16,English,,0,0,0,,This floating, the floating binary point
Dialogue: 0,0:06:54.56,0:06:57.16,Chinese,,0,0,0,,能够不断变换位置的二进制小数点
Dialogue: 0,0:06:59.24,0:07:02.24,English,,0,0,0,,Now up until 1985
Dialogue: 0,0:06:59.24,0:07:02.24,Chinese,,0,0,0,,在 1985 年之前
Dialogue: 0,0:07:02.64,0:07:05.94,English,,0,0,0,,A floating point was just a disaster
Dialogue: 0,0:07:02.64,0:07:05.94,Chinese,,0,0,0,,浮点数一直是一场灾难
Dialogue: 0,0:07:06.48,0:07:11.18,English,,0,0,0,,And every computer manufacturer had their own representation
Dialogue: 0,0:07:06.48,0:07:11.18,Chinese,,0,0,0,,每个计算机制造商都有自己的实现方式
Dialogue: 0,0:07:12.40,0:07:16.22,English,,0,0,0,,You couldn't...and it was just terrible,because you couldn't predict
Dialogue: 0,0:07:12.40,0:07:16.22,Chinese,,0,0,0,,它太可怕了，因为你无法预测会发什么
Dialogue: 0,0:07:16.90,0:07:21.46,English,,0,0,0,,What would happen if you moved your program from one machine to another
Dialogue: 0,0:07:16.90,0:07:21.46,Chinese,,0,0,0,,如果你将程序换一台机器运行
Dialogue: 0,0:07:21.84,0:07:24.66,English,,0,0,0,,The behavior would be different the rounding would be different
Dialogue: 0,0:07:21.84,0:07:24.66,Chinese,,0,0,0,,（程序的）行为会变得不同，舍入也会变得不同
Dialogue: 0,0:07:25.86,0:07:35.13,English,,0,0,0,,It was a very difficult situation and people really, computer scientists in particular, really disliked floating point
Dialogue: 0,0:07:25.86,0:07:35.13,Chinese,,0,0,0,,这是一个非常糟糕的状况，人们特别是计算机科学家真的不喜欢浮点
Dialogue: 0,0:07:35.62,0:07:45.02,English,,0,0,0,,In 1985 IEEE which is the professional association for electrical engineers
Dialogue: 0,0:07:35.62,0:07:45.02,Chinese,,0,0,0,,1985 年，电气电子工程师协会（IEEE）
Dialogue: 0,0:07:45.44,0:07:51.02,English,,0,0,0,,Introduced a...and it's also a very important standards body
Dialogue: 0,0:07:45.44,0:07:51.02,Chinese,,0,0,0,,引入了一个非常重要的标准化表示
Dialogue: 0,0:07:51.34,0:07:56.54,English,,0,0,0,,They introduced a standard floating-point representation called IEEE floating point
Dialogue: 0,0:07:51.34,0:07:56.54,Chinese,,0,0,0,,他们引入了一个称为 IEEE 浮点数的标准浮点数表示法
Dialogue: 0,0:07:57.16,0:08:00.52,English,,0,0,0,,which was then adopted by all computer manufacturers
Dialogue: 0,0:07:57.16,0:08:00.52,Chinese,,0,0,0,,得到了所有计算机制造商的一致采用
Dialogue: 0,0:08:00.52,0:08:02.16,English,,0,0,0,,So now the situation is much better
Dialogue: 0,0:08:00.52,0:08:02.16,Chinese,,0,0,0,,所以现在情况要好得多
Dialogue: 0,0:08:02.60,0:08:06.96,English,,0,0,0,,Everybody supports the same kind of floating point representation
Dialogue: 0,0:08:02.60,0:08:06.96,Chinese,,0,0,0,,各方都支持相同类型的浮点数表示
Dialogue: 0,0:08:06.98,0:08:08.84,English,,0,0,0,,which is what we're going to learn about today
Dialogue: 0,0:08:06.98,0:08:08.84,Chinese,,0,0,0,,这就是我们今天要学习的内容
Dialogue: 0,0:08:14.66,0:08:18.38,English,,0,0,0,,So the floating point has the form
Dialogue: 0,0:08:14.66,0:08:18.38,Chinese,,0,0,0,,所以浮点数有如下形式
Dialogue: 0,0:08:19.34,0:08:24.26,English,,0,0,0,,It represents numbers in a... this is, like a scientific notation
Dialogue: 0,0:08:19.34,0:08:24.26,Chinese,,0,0,0,,它以一种类似科学计数法的方式来表示数字
Dialogue: 0,0:08:24.46,0:08:36.66,English,,0,0,0,,It represents numbers of the form (-1)^S*M*2^E,where S is a sign bit,times a significand M
Dialogue: 0,0:08:24.46,0:08:36.66,Chinese,,0,0,0,,它表示形式 (-1)^S*M*2^E 的数字，其中 S 是符号位，乘以尾数 M
Dialogue: 0,0:08:38.00,0:08:43.60,English,,0,0,0,,Ok times 2^E
Dialogue: 0,0:08:38.00,0:08:43.60,Chinese,,0,0,0,,然后乘以 2 的 E 次幂
Dialogue: 0,0:08:43.94,0:08:51.86,English,,0,0,0,,Ok so all of the numbers that we can represent in floating point have to be represented in this form
Dialogue: 0,0:08:43.94,0:08:51.86,Chinese,,0,0,0,,所有浮点数都必须以这种形式表示
Dialogue: 0,0:08:53.04,0:08:59.42,English,,0,0,0,,So the sign bit S determines whether the number is negative or positive obviously
Dialogue: 0,0:08:53.04,0:08:59.42,Chinese,,0,0,0,,显然，符号位 S 确定了该数字是负数还是正数
Dialogue: 0,0:08:59.86,0:09:04.02,English,,0,0,0,,The significand M and this is sometimes sometimes called the mantissa
Dialogue: 0,0:08:59.86,0:09:04.02,Chinese,,0,0,0,,尾数 M，有时叫做 mantissa
Dialogue: 0,0:09:04.02,0:09:05.68,English,,0,0,0,,where the M comes from
Dialogue: 0,0:09:04.02,0:09:05.68,Chinese,,0,0,0,,单词的首字母就是 M 的来源
Dialogue: 0,0:09:06.16,0:09:13.06,English,,0,0,0,,The significand M normally is a fractional value in the range between 1 and 2
Dialogue: 0,0:09:06.16,0:09:13.06,Chinese,,0,0,0,,尾数 M 通常是介于 1 和 2 之间的小数
Dialogue: 0,0:09:13.06,0:09:16.00,English,,0,0,0,,And we'll see why that is in a second
Dialogue: 0,0:09:13.06,0:09:16.00,Chinese,,0,0,0,,接下来，我们会很快看到这样设置的原因
Dialogue: 0,0:09:16.52,0:09:19.88,English,,0,0,0,,And then the exponent E weights the value by some power of 2
Dialogue: 0,0:09:16.52,0:09:19.88,Chinese,,0,0,0,,指数 E 会以 2 的 E 次幂形式扩大或或缩小尾数值
Dialogue: 0,0:09:20.48,0:09:25.76,English,,0,0,0,,Okay so this is encoded in either 32-bit or 64-bit
Dialogue: 0,0:09:20.48,0:09:25.76,Chinese,,0,0,0,,32 位或 64 位编码中都是这样
Dialogue: 0,0:09:26.28,0:09:30.42,English,,0,0,0,,In the following way the the sign bit S is the most significant bit
Dialogue: 0,0:09:26.28,0:09:30.42,Chinese,,0,0,0,,下面的表示中，符号位 S 是最重要的位
Dialogue: 0,0:09:31.26,0:09:34.62,English,,0,0,0,,And then this is followed by some number of
Dialogue: 0,0:09:31.26,0:09:34.62,Chinese,,0,0,0,,紧随其后的
Dialogue: 0,0:09:34.64,0:09:37.86,English,,0,0,0,,by some number of bits called the exp field
Dialogue: 0,0:09:34.64,0:09:37.86,Chinese,,0,0,0,,是一些被称为指数（exp）域的位（阶码）
Dialogue: 0,0:09:38.74,0:09:42.20,English,,0,0,0,,which encodes the exponent E
Dialogue: 0,0:09:38.74,0:09:42.20,Chinese,,0,0,0,,它编码了 E
Dialogue: 0,0:09:43.24,0:09:47.70,English,,0,0,0,,But it's not identical to E,it encodes E, we'll see how that works
Dialogue: 0,0:09:43.24,0:09:47.70,Chinese,,0,0,0,,但编码表示的值与 E的值不同，它只是编码了 E ，我们会看到它是如何起效的
Dialogue: 0,0:09:48.18,0:09:54.54,English,,0,0,0,,And then the frac. And then that's followed the remaining bits are the the frac field
Dialogue: 0,0:09:48.18,0:09:54.54,Chinese,,0,0,0,,后面剩下的比特位是尾数（frac）字段
Dialogue: 0,0:09:55.96,0:09:59.38,English,,0,0,0,,which encodes M but isn't identical to M
Dialogue: 0,0:09:55.96,0:09:59.38,Chinese,,0,0,0,,它编码了 M 但与 M 不完全相同
Dialogue: 0,0:10:02.02,0:10:09.46,English,,0,0,0,,Now we are...IEEE floating point has provides two different kinds of floating-point
Dialogue: 0,0:10:02.02,0:10:09.46,Chinese,,0,0,0,,IEEE 提供了两种不同的浮点数
Dialogue: 0,0:10:09.46,0:10:14.47,English,,0,0,0,,The single precision 32 bits and so-called double precision which is 64 bits
Dialogue: 0,0:10:09.46,0:10:14.47,Chinese,,0,0,0,,单精度 32 位和双精度 64 位
Dialogue: 0,0:10:15.12,0:10:20.38,English,,0,0,0,,There's this funky intel extended precision
Dialogue: 0,0:10:15.12,0:10:20.38,Chinese,,0,0,0,,还有这种特别的英特尔扩展精度
Dialogue: 0,0:10:20.38,0:10:22.84,English,,0,0,0,,which is non-standard and we won't talk about it anymore
Dialogue: 0,0:10:20.38,0:10:22.84,Chinese,,0,0,0,,由于这是非标准的，我们此处不再讨论
Dialogue: 0,0:10:25.18,0:10:27.82,English,,0,0,0,,And so in the single precision 32 bits
Dialogue: 0,0:10:25.18,0:10:27.82,Chinese,,0,0,0,,在 32 位单精度浮点数中
Dialogue: 0,0:10:27.82,0:10:30.30,English,,0,0,0,,You have one sign bit, there's always a sign bit
Dialogue: 0,0:10:27.82,0:10:30.30,Chinese,,0,0,0,,总有一个符号位
Dialogue: 0,0:10:30.30,0:10:34.68,English,,0,0,0,,You have 8 exp bits and you have 23 frac bits
Dialogue: 0,0:10:30.30,0:10:34.68,Chinese,,0,0,0,,还有 8 个阶码位和 23 个尾数位
Dialogue: 0,0:10:35.34,0:10:40.20,English,,0,0,0,,In double precision you have 11 exp and 52 frac bits
Dialogue: 0,0:10:35.34,0:10:40.20,Chinese,,0,0,0,,在双精度中，有 11 个阶码位和 52 个尾数位
Dialogue: 0,0:10:42.58,0:10:47.60,English,,0,0,0,,Now there's a number of different kinds of floating-point representations
Dialogue: 0,0:10:42.58,0:10:47.60,Chinese,,0,0,0,,现在有许多不同类型的浮点表示方法
Dialogue: 0,0:10:48.12,0:10:54.22,English,,0,0,0,,The sort of the default and sort of the common representation is called normalized
Dialogue: 0,0:10:48.12,0:10:54.22,Chinese,,0,0,0,,默认及通用的表示方法被称为规格化的（表示方法）
Dialogue: 0,0:10:55.38,0:10:57.17,English,,0,0,0,,So called normalized values
Dialogue: 0,0:10:55.38,0:10:57.17,Chinese,,0,0,0,,也就是所谓的规格化值
Dialogue: 0,0:10:58.26,0:11:00.28,English,,0,0,0,,And normalized values are
Dialogue: 0,0:10:58.26,0:11:00.28,Chinese,,0,0,0,,规格化值是指
Dialogue: 0,0:11:01.12,0:11:05.46,English,,0,0,0,,All the values where the exp field is not equal to 0
Dialogue: 0,0:11:01.12,0:11:05.46,Chinese,,0,0,0,,其中阶码字段的所有位的值
Dialogue: 0,0:11:06.06,0:11:10.78,English,,0,0,0,,And not equal to all zeros and not equal to all ones
Dialogue: 0,0:11:06.06,0:11:10.78,Chinese,,0,0,0,,不能全是 0 也不能全是 1
Dialogue: 0,0:11:11.46,0:11:17.48,English,,0,0,0,,So everything all zeros and all ones are special kinds of numbers we'll look at a second
Dialogue: 0,0:11:11.46,0:11:17.48,Chinese,,0,0,0,,全 0 和全 1 的情形都是特殊类型的数字，稍后我们会讲到
Dialogue: 0,0:11:18.58,0:11:30.54,English,,0,0,0,,So the exponent...this exponent E remember our numbers are (-1)^S*M*2^E
Dialogue: 0,0:11:18.58,0:11:30.54,Chinese,,0,0,0,,对于公式 (-1)^S*M*2^E 指数 E
Dialogue: 0,0:11:31.36,0:11:42.74,English,,0,0,0,,The exponent E is encoded as this biased value which is the called exp
Dialogue: 0,0:11:31.36,0:11:42.74,Chinese,,0,0,0,,指数 E 被解释为以「偏置（biased）」形式表示
Dialogue: 0,0:11:43.44,0:11:48.06,English,,0,0,0,,So the exponent E is exp-bias
Dialogue: 0,0:11:43.44,0:11:48.06,Chinese,,0,0,0,,所以指数 E 的实际值为 exp-bias
Dialogue: 0,0:11:48.50,0:11:50.88,English,,0,0,0,,And exp is the value
Dialogue: 0,0:11:48.50,0:11:50.88,Chinese,,0,0,0,,而 Exp 的值是
Dialogue: 0,0:11:51.10,0:11:57.56,English,,0,0,0,,Exp is the value of the exp bits, the bits in the exp field treated as a positive number
Dialogue: 0,0:11:51.10,0:11:57.56,Chinese,,0,0,0,,Exp 位的值，也就是把 exp 域的值（一直）作为一个正数
Dialogue: 0,0:11:57.96,0:12:09.72,English,,0,0,0,,Okay so Exp, you just look at it as an unsigned 7-bit in case of single precision 7-bit number
Dialogue: 0,0:11:57.96,0:12:09.72,Chinese,,0,0,0,,所以在单精度 7 位数字的情况下，你只需将它看作 7 位的无符号整数
Dialogue: 0,0:12:11.40,0:12:20.44,English,,0,0,0,,Okay now the Bias=2^(k-1)-1, where k is number of exponent bits
Dialogue: 0,0:12:11.40,0:12:20.44,Chinese,,0,0,0,,现在偏移值 Bias=2^(k-1)-1，其中 k 是阶码域的位数
Dialogue: 0,0:12:21.08,0:12:26.85,English,,0,0,0,,Okay so for single precision we have 8 exponent bits
Dialogue: 0,0:12:21.08,0:12:26.85,Chinese,,0,0,0,,那么，对于单精度，我们有 8 个指数位
Dialogue: 0,0:12:27.72,0:12:39.86,English,,0,0,0,,So 2^8... 2^(8-1=7) so 2^7=128-1 is 127
Dialogue: 0,0:12:27.72,0:12:39.86,Chinese,,0,0,0,,所以 2^(8-1)-1=2^7-1=128-1=127
Dialogue: 0,0:12:40.00,0:12:43.36,English,,0,0,0,,Ok so the Bias for single precision is 127
Dialogue: 0,0:12:40.00,0:12:43.36,Chinese,,0,0,0,,所以单精度的偏移值是 127
Dialogue: 0,0:12:43.88,0:12:47.00,English,,0,0,0,,And similarly for double precision it's 1023
Dialogue: 0,0:12:43.88,0:12:47.00,Chinese,,0,0,0,,同样，对于双精度，它是 1023
Dialogue: 0,0:12:49.82,0:12:54.94,English,,0,0,0,,Okay so you just have to memorize this formula
Dialogue: 0,0:12:49.82,0:12:54.94,Chinese,,0,0,0,,所以你只需要记住这个公式
Dialogue: 0,0:12:54.94,0:13:03.02,English,,0,0,0,,The exponent E is the exp field interpret it as an unsigned number minus the the bias
Dialogue: 0,0:12:54.94,0:13:03.02,Chinese,,0,0,0,,指数 E 的值是无符号 Exp 值减去偏置值 Bias
Dialogue: 0,0:13:04.24,0:13:10.58,English,,0,0,0,,This is kind of a strange thing the first time you encounter floating-point
Dialogue: 0,0:13:04.24,0:13:10.58,Chinese,,0,0,0,,当第一次遇到浮点时
Dialogue: 0,0:13:10.78,0:13:13.66,English,,0,0,0,,It seems kind of odd, right? So why not...
Dialogue: 0,0:13:10.78,0:13:13.66,Chinese,,0,0,0,,不免觉得这似乎有点奇怪，对吧
Dialogue: 0,0:13:14.92,0:13:16.78,English,,0,0,0,,I mean we've already learned about two's complement
Dialogue: 0,0:13:14.92,0:13:16.78,Chinese,,0,0,0,,我的意思是，我们已经学会了二进制补码
Dialogue: 0,0:13:16.78,0:13:20.66,English,,0,0,0,,That's a perfectly fine way to represent positive and negative numbers
Dialogue: 0,0:13:16.78,0:13:20.66,Chinese,,0,0,0,,这是表示正数和负数的完美方式
Dialogue: 0,0:13:21.28,0:13:26.42,English,,0,0,0,,Okay we have exponents that are negative and positive, right
Dialogue: 0,0:13:21.28,0:13:26.42,Chinese,,0,0,0,,既然指数有负的也有正的
Dialogue: 0,0:13:26.42,0:13:29.04,English,,0,0,0,,So why not just use the exp field to represent
Dialogue: 0,0:13:26.42,0:13:29.04,Chinese,,0,0,0,,那为什么不在 exp 域中使用二进制补码
Dialogue: 0,0:13:29.98,0:13:37.12,English,,0,0,0,,Why not just use a two's complement in the exp field to represent those positive and negative exponents, right
Dialogue: 0,0:13:29.98,0:13:37.12,Chinese,,0,0,0,,来表示那些正的和负的指数呢
Dialogue: 0,0:13:37.88,0:13:42.14,English,,0,0,0,,So it turns out that the reason that they do this is
Dialogue: 0,0:13:37.88,0:13:42.14,Chinese,,0,0,0,,事实证明，这样做的原因是
Dialogue: 0,0:13:44.72,0:13:45.72,English,,0,0,0,,is that...
Dialogue: 0,0:13:44.72,0:13:45.72,Chinese,,0,0,0,,就是……
Dialogue: 0,0:13:46.28,0:13:49.72,English,,0,0,0,,And if you think about this and we'll come back to it
Dialogue: 0,0:13:46.28,0:13:49.72,Chinese,,0,0,0,,你不妨考虑考虑这个问题，我们会回过头来讨论
Dialogue: 0,0:13:49.74,0:13:56.90,English,,0,0,0,,But if we encode the the exponent E using this bias representation
Dialogue: 0,0:13:49.74,0:13:56.90,Chinese,,0,0,0,,但是如果我们使用这种偏置值的方式来编码指数 E
Dialogue: 0,0:13:57.82,0:14:02.40,English,,0,0,0,,The smallest negative exponent is represented by all zeros
Dialogue: 0,0:13:57.82,0:14:02.40,Chinese,,0,0,0,,最小的负指数由全 0 表示
Dialogue: 0,0:14:05.10,0:14:12.38,English,,0,0,0,,And the largest exponent is represented by 01...111
Dialogue: 0,0:14:05.10,0:14:12.38,Chinese,,0,0,0,,最大的指数由 01...111 表示
Dialogue: 0,0:14:15.22,0:14:20.64,English,,0,0,0,,So the smallest, the number with the smallest exponent
Dialogue: 0,0:14:15.22,0:14:20.64,Chinese,,0,0,0,,所以最小的指数......
Dialogue: 0,0:14:21.34,0:14:22.98,English,,0,0,0,,If we were just to compare the bits
Dialogue: 0,0:14:21.34,0:14:22.98,Chinese,,0,0,0,,为了比较二进制数
Dialogue: 0,0:14:22.98,0:14:30.12,English,,0,0,0,,using a just some kind of unsigned representation, just comparing the bits, treating it as an unsigned number
Dialogue: 0,0:14:22.98,0:14:30.12,Chinese,,0,0,0,,只需要采用无符号表达，比较各个位，把它当做无符号整数
Dialogue: 0,0:14:30.70,0:14:33.52,English,,0,0,0,,By using this biased representation
Dialogue: 0,0:14:30.70,0:14:33.52,Chinese,,0,0,0,,我们可以通过使用这种偏移值表示法
Dialogue: 0,0:14:33.52,0:14:38.40,English,,0,0,0,,we can just compare two floating-point numbers just as unsigned
Dialogue: 0,0:14:33.52,0:14:38.40,Chinese,,0,0,0,,来比较两个浮点数，就像无符号整数一样
Dialogue: 0,0:14:38.40,0:14:43.10,English,,0,0,0,,We can treat the whole floating-point number as an unsigned integer and compare two numbers
Dialogue: 0,0:14:38.40,0:14:43.10,Chinese,,0,0,0,,我们可以将整个浮点数视为无符号整数并比较这两个数字
Dialogue: 0,0:14:43.50,0:14:46.24,English,,0,0,0,,And get a true comparison
Dialogue: 0,0:14:43.50,0:14:46.24,Chinese,,0,0,0,,就可以得到真正的比较
Dialogue: 0,0:14:50.58,0:14:54.56,English,,0,0,0,,So the exponent uses this this sort of biasing encoding
Dialogue: 0,0:14:50.58,0:14:54.56,Chinese,,0,0,0,,所以指数使用这种偏置编码法
Dialogue: 0,0:14:55.22,0:15:00.98,English,,0,0,0,,And then the significand is encoded with an implied one
Dialogue: 0,0:14:55.22,0:15:00.98,Chinese,,0,0,0,,尾数采用隐含编码法
Dialogue: 0,0:15:01.52,0:15:06.52,English,,0,0,0,,So and this is where the... this is where the normalized
Dialogue: 0,0:15:01.52,0:15:06.52,Chinese,,0,0,0,,这就是
Dialogue: 0,0:15:07.04,0:15:09.04,English,,0,0,0,,... and the normalized representation comes from
Dialogue: 0,0:15:07.04,0:15:09.04,Chinese,,0,0,0,,规格化表示的来源
Dialogue: 0,0:15:09.70,0:15:11.84,English,,0,0,0,,That we're always going to represent our numbers
Dialogue: 0,0:15:09.70,0:15:11.84,Chinese,,0,0,0,,这是我们想表达数字的方法
Dialogue: 0,0:15:12.44,0:15:16.74,English,,0,0,0,,So M... we're always going to normalize M
Dialogue: 0,0:15:12.44,0:15:16.74,Chinese,,0,0,0,,所以我们总是要将 M 规格化
Dialogue: 0,0:15:16.74,0:15:19.24,English,,0,0,0,,no matter what the number we want to represent
Dialogue: 0,0:15:16.74,0:15:19.24,Chinese,,0,0,0,,无论我们想要表示什么数字
Dialogue: 0,0:15:19.24,0:15:23.26,English,,0,0,0,,We're always going to normalize M as 1.xxx...x
Dialogue: 0,0:15:19.24,0:15:23.26,Chinese,,0,0,0,,我们总是将 M 规格化为 1.xxx...x
Dialogue: 0,0:15:23.98,0:15:26.52,English,,0,0,0,,And then we adjust the exponent accordingly
Dialogue: 0,0:15:23.98,0:15:26.52,Chinese,,0,0,0,,我们相应地调整指数（来进行规格化）
Dialogue: 0,0:15:26.94,0:15:32.66,English,,0,0,0,,So if the value we want to represent is 100.01
Dialogue: 0,0:15:26.94,0:15:32.66,Chinese,,0,0,0,,因此，如果我们想要表示的值是 100.01
Dialogue: 0,0:15:33.56,0:15:37.68,English,,0,0,0,,We're going to shift it over and it will be 1.00
Dialogue: 0,0:15:33.56,0:15:37.68,Chinese,,0,0,0,,我们将把它的二进制小数点左移，使之成为 1.00
Dialogue: 0,0:15:40.60,0:15:45.04,English,,0,0,0,,Okay and then we'll adjust the exponent to represent that shift
Dialogue: 0,0:15:40.60,0:15:45.04,Chinese,,0,0,0,,然后我们调整指数来表示这种位移
Dialogue: 0,0:15:47.44,0:15:54.34,English,,0,0,0,,Okay so the bits in the frac field xxxx...x are
Dialogue: 0,0:15:47.44,0:15:54.34,Chinese,,0,0,0,,那么，尾数域 xxxx...x 中的位是
Dialogue: 0,0:15:54.86,0:15:58.10,English,,0,0,0,,All of the numbers to the right of the binary point right
Dialogue: 0,0:15:54.86,0:15:58.10,Chinese,,0,0,0,,二进制点右边的所有数字
Dialogue: 0,0:15:58.10,0:16:00.18,English,,0,0,0,,So there's always this implied one
Dialogue: 0,0:15:58.10,0:16:00.18,Chinese,,0,0,0,,这个1一直隐藏的
Dialogue: 0,0:16:01.48,0:16:03.52,English,,0,0,0,,But we don't even encode it we just throw it away
Dialogue: 0,0:16:01.48,0:16:03.52,Chinese,,0,0,0,,我们甚至没有对它进行编码，我们只是扔掉它（不对这个1进行存储）
Dialogue: 0,0:16:03.52,0:16:05.14,English,,0,0,0,,Because it's always there right
Dialogue: 0,0:16:03.52,0:16:05.14,Chinese,,0,0,0,,因为它总是在那里
Dialogue: 0,0:16:05.14,0:16:08.90,English,,0,0,0,,So this is a little trick just to get it one more bit for free
Dialogue: 0,0:16:05.14,0:16:08.90,Chinese,,0,0,0,,所以这是一个小技巧，只是为了免费获得一位
Dialogue: 0,0:16:10.18,0:16:14.40,English,,0,0,0,,Okay so the the xxx bits in the frac field
Dialogue: 0,0:16:10.18,0:16:14.40,Chinese,,0,0,0,,好吧，尾数域的 xxx 位
Dialogue: 0,0:16:15.06,0:16:21.72,English,,0,0,0,,All the bits to the right of the binary point after one point the one point
Dialogue: 0,0:16:15.06,0:16:21.72,Chinese,,0,0,0,,二进制小数点之后的所有位在 1. 之后
Dialogue: 0,0:16:22.66,0:16:27.54,English,,0,0,0,,And this is,the so this is the smallest fractal is all zeros
Dialogue: 0,0:16:22.66,0:16:27.54,Chinese,,0,0,0,,所以最小的尾数域全是 0
Dialogue: 0,0:16:27.56,0:16:31.10,English,,0,0,0,, Which corresponds to a significand of 1.0
Dialogue: 0,0:16:27.56,0:16:31.10,Chinese,,0,0,0,,这相当于尾数是 1.0
Dialogue: 0,0:16:34.26,0:16:39.64,English,,0,0,0,,A frac field of all ones corresponds to 2-ɛ
Dialogue: 0,0:16:34.26,0:16:39.64,Chinese,,0,0,0,,所有位都是 1 的 frac 对应于 2-ɛ
Dialogue: 0,0:16:39.64,0:16:43.04,English,,0,0,0,,So that's 1.111...
Dialogue: 0,0:16:39.64,0:16:43.04,Chinese,,0,0,0,,那是 1.111......
Dialogue: 0,0:16:47.08,0:16:50.64,English,,0,0,0,,So let's look at an example
Dialogue: 0,0:16:47.08,0:16:50.64,Chinese,,0,0,0,,那么让我们看一个例子
Dialogue: 0,0:16:51.82,0:17:01.12,English,,0,0,0,,So let's look at how we would encode 15213 which our favorite number In two's complement in floating point
Dialogue: 0,0:16:51.82,0:17:01.12,Chinese,,0,0,0,,那么让我们来看看如何使用浮点数的方式来编码我们最喜欢的数字 15213
Dialogue: 0,0:17:02.52,0:17:05.22,English,,0,0,0,,So the if we represent that so it's an integer
Dialogue: 0,0:17:02.52,0:17:05.22,Chinese,,0,0,0,,如果我们使用这种发式表达它，它就是一个整数
Dialogue: 0,0:17:06.60,0:17:15.34,English,,0,0,0,,And we represent it using these bits take my word for it correspond to the integer 15213
Dialogue: 0,0:17:06.60,0:17:15.34,Chinese,,0,0,0,,我们使用这些位表示它，它对应于整数 15213
Dialogue: 0,0:17:16.22,0:17:18.56,English,,0,0,0,,So now we want to convert it we want to normalize it
Dialogue: 0,0:17:16.22,0:17:18.56,Chinese,,0,0,0,,现在我们要转换它，我们想要标准化它
Dialogue: 0,0:17:20.88,0:17:26.70,English,,0,0,0,, By expressing it as 1.110xxx * 2^13
Dialogue: 0,0:17:20.88,0:17:26.70,Chinese,,0,0,0,,通过将其表示为 1.110xxx*2^13
Dialogue: 0,0:17:27.22,0:17:29.50,English,,0,0,0,,Okay same number we just normalized it
Dialogue: 0,0:17:27.22,0:17:29.50,Chinese,,0,0,0,,这是相同的数字，我们只是将其规格化了
Dialogue: 0,0:17:29.78,0:17:32.26,English,,0,0,0,,So that there's always this leading 1
Dialogue: 0,0:17:29.78,0:17:32.26,Chinese,,0,0,0,,所以总是有这个开头的 1
Dialogue: 0,0:17:36.76,0:17:45.62,English,,0,0,0,, So the significant m is 1.11011xxx
Dialogue: 0,0:17:36.76,0:17:45.62,Chinese,,0,0,0,,所以尾数 m 是 1.11011xxx
Dialogue: 0,0:17:46.74,0:17:50.78,English,,0,0,0,,The frac field is everything to the right of the binary point
Dialogue: 0,0:17:46.74,0:17:50.78,Chinese,,0,0,0,,二进制小数点右侧的内容对应着尾数域
Dialogue: 0,0:17:50.94,0:17:54.60,English,,0,0,0,,So we just,we throw this one away
Dialogue: 0,0:17:50.94,0:17:54.60,Chinese,,0,0,0,,所以我们可以把这个 1 忽略掉
Dialogue: 0,0:17:54.60,0:17:59.44,English,,0,0,0,,And we just take the the values to the right of the binary point
Dialogue: 0,0:17:54.60,0:17:59.44,Chinese,,0,0,0,,我们只需取得二进制点的右侧的值
Dialogue: 0,0:17:59.74,0:18:04.62,English,,0,0,0,,And then we pat it out with zeros to get the 23 bits that we need for single precision
Dialogue: 0,0:17:59.74,0:18:04.62,Chinese,,0,0,0,,然后我们用零来补全它以获得我们需要的单精度浮点数的 23 位
Dialogue: 0,0:18:07.40,0:18:09.10,English,,0,0,0,, Ok so now we've got our frac field
Dialogue: 0,0:18:07.40,0:18:09.10,Chinese,,0,0,0,,好的，现在我们有了尾数域
Dialogue: 0,0:18:11.14,0:18:12.22,English,,0,0,0,, The exponent
Dialogue: 0,0:18:11.14,0:18:12.22,Chinese,,0,0,0,,指数
Dialogue: 0,0:18:13.12,0:18:15.74,English,,0,0,0,,So in this case right E is
Dialogue: 0,0:18:13.12,0:18:15.74,Chinese,,0,0,0,,所以在这种情况下，E 是
Dialogue: 0,0:18:16.98,0:18:20.72,English,,0,0,0,,We're multiplying by 2 ^13.So E is 13
Dialogue: 0,0:18:16.98,0:18:20.72,Chinese,,0,0,0,,我们乘以 2^13， 所以 E 是 13
Dialogue: 0,0:18:21.26,0:18:25.36,English,,0,0,0,,So we want to represent the value 13 using this bias encoding
Dialogue: 0,0:18:21.26,0:18:25.36,Chinese,,0,0,0,,因此，我们希望使用此偏置编码来表示值 13
Dialogue: 0,0:18:27.84,0:18:29.86,English,,0,0,0,,So the bias is 127
Dialogue: 0,0:18:27.84,0:18:29.86,Chinese,,0,0,0,,所以偏置值是 127
Dialogue: 0,0:18:32.26,0:18:41.38,English,,0,0,0,,And so the exponent the value...the value of the exp field should be 140
Dialogue: 0,0:18:32.26,0:18:41.38,Chinese,,0,0,0,,所以 exp 域的值应该是 140
Dialogue: 0,0:18:41.58,0:18:49.32,English,,0,0,0,,Because if we take 140-127 we get 13
Dialogue: 0,0:18:41.58,0:18:49.32,Chinese,,0,0,0,,因为如果我们拿 140-127 就会得到 13
Dialogue: 0,0:18:50.98,0:18:54.52,English,,0,0,0,,Okay so 140 is represented using these 8 bits
Dialogue: 0,0:18:50.98,0:18:54.52,Chinese,,0,0,0,,好的，所以使用这 8 位表示 140
Dialogue: 0,0:18:55.58,0:18:58.02,English,,0,0,0,,And that becomes our exp field
Dialogue: 0,0:18:55.58,0:18:58.02,Chinese,,0,0,0,,这就成了我们的阶码字段
Dialogue: 0,0:19:00.02,0:19:01.92,English,,0,0,0,, Okay so the resulting floating-point number
Dialogue: 0,0:19:00.02,0:19:01.92,Chinese,,0,0,0,,好的，所以得到的浮点数
Dialogue: 0,0:19:01.92,0:19:05.08,English,,0,0,0,, And of course it's a positive number so the significand bit is zero
Dialogue: 0,0:19:01.92,0:19:05.08,Chinese,,0,0,0,,符号位为零，因此是一个正数
Dialogue: 0,0:19:06.96,0:19:09.72,English,,0,0,0,, So the resulting floating point number has these eight
Dialogue: 0,0:19:06.96,0:19:09.72,Chinese,,0,0,0,,因此得到的浮点数有 8 位 exp
Dialogue: 0,0:19:10.30,0:19:19.30,English,,0,0,0,,The 8 bits and exp field that we computed by subtracting 127 from 140
Dialogue: 0,0:19:10.30,0:19:19.30,Chinese,,0,0,0,,我们通过从 140 减去 127 来计算阶码的值
Dialogue: 0,0:19:20.00,0:19:27.68,English,,0,0,0,,And then the frack field are these bits that we got here by just taking everything to the right of the binary point
Dialogue: 0,0:19:20.00,0:19:27.68,Chinese,,0,0,0,,我们将所有内容都放在二进制点右侧而得到的这些位就是小数字段
Dialogue: 0,0:19:32.12,0:19:33.10,English,,0,0,0,, So this...
Dialogue: 0,0:19:32.12,0:19:33.10,Chinese,,0,0,0,,所以这...
Dialogue: 0,0:19:35.88,0:19:38.18,English,,0,0,0,,So let's look at this exponent just a little bit more
Dialogue: 0,0:19:35.88,0:19:38.18,Chinese,,0,0,0,,让我们再看一下这个指数吧
Dialogue: 0,0:19:38.18,0:19:41.80,English,,0,0,0,,Because I think can be a little confusing
Dialogue: 0,0:19:38.18,0:19:41.80,Chinese,,0,0,0,,因为我觉得可能有点令人困惑
Dialogue: 0,0:19:46.90,0:19:54.94,English,,0,0,0,,So in our example
Dialogue: 0,0:19:46.90,0:19:54.94,Chinese,,0,0,0,,所以在我们的例子中
Dialogue: 0,0:19:55.02,0:20:00.32,English,,0,0,0,,If we want to look at the range of Exp
Dialogue: 0,0:19:55.02,0:20:00.32,Chinese,,0,0,0,,如果我们想看看 Exp 的范围
Dialogue: 0,0:20:00.32,0:20:07.14,English,,0,0,0,,So what's the smallest value of exp
Dialogue: 0,0:20:00.32,0:20:07.14,Chinese,,0,0,0,,那么 Exp 的最小值是多少
Dialogue: 0,0:20:07.26,0:20:09.04,English,,0,0,0,,What's the smallest value of exp?
Dialogue: 0,0:20:07.26,0:20:09.04,Chinese,,0,0,0,,Exp 的最小值是多少？
Dialogue: 0,0:20:10.82,0:20:14.86,English,,0,0,0,,Don't confuse the encoding with the value it represents
Dialogue: 0,0:20:10.82,0:20:14.86,Chinese,,0,0,0,,不要将编码值与它的实际值混淆
Dialogue: 0,0:20:15.38,0:20:17.60,English,,0,0,0,, Exp is just the encoding
Dialogue: 0,0:20:15.38,0:20:17.60,Chinese,,0,0,0,,Exp 只是编码
Dialogue: 0,0:20:17.70,0:20:21.14,English,,0,0,0,,So what's... what's the smallest value of exp?
Dialogue: 0,0:20:17.70,0:20:21.14,Chinese,,0,0,0,,那么......exp 的最小值是什么？
Dialogue: 0,0:20:22.02,0:20:26.08,English,,0,0,0,, 0 that's right
Dialogue: 0,0:20:22.02,0:20:26.08,Chinese,,0,0,0,,0 那是对的
Dialogue: 0,0:20:26.18,0:20:29.82,English,,0,0,0,,Okay and the largest value is? is what?
Dialogue: 0,0:20:26.18,0:20:29.82,Chinese,,0,0,0,,好的，最大的值是什么？
Dialogue: 0,0:20:29.82,0:20:34.56,English,,0,0,0,,It's... what's the largest value?
Dialogue: 0,0:20:29.82,0:20:34.56,Chinese,,0,0,0,,这是......最大的值是什么？
Dialogue: 0,0:20:34.76,0:20:35.62,English,,0,0,0,,Yes
Dialogue: 0,0:20:34.76,0:20:35.62,Chinese,,0,0,0,,是
Dialogue: 0,0:20:35.62,0:20:38.22,English,,0,0,0,,[student speaking]
Dialogue: 0,0:20:35.62,0:20:38.22,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:20:38.22,0:20:43.88,English,,0,0,0,,Well the largest value would be remember it can't be all ones
Dialogue: 0,0:20:38.22,0:20:43.88,Chinese,,0,0,0,,那么最大的值就是记住它不可能每个位全是 1
Dialogue: 0,0:20:45.72,0:20:48.94,English,,0,0,0,,Right so the largest value would be 2^8-1
Dialogue: 0,0:20:45.72,0:20:48.94,Chinese,,0,0,0,,正确，因此最大值为 2^8-1
Dialogue: 0,0:20:51.04,0:20:55.32,English,,0,0,0,,So okay so the largest value would be 01111...
Dialogue: 0,0:20:51.04,0:20:55.32,Chinese,,0,0,0,,好吧所以最大的值是 01111......
Dialogue: 0,0:20:55.32,0:20:57.20,English,,0,0,0,,0 followed by 7 ones
Dialogue: 0,0:20:55.32,0:20:57.20,Chinese,,0,0,0,,0 后跟 7 个 1
Dialogue: 0,0:20:59.88,0:21:03.62,English,,0,0,0,,So that's a 2^8-1 is ...
Dialogue: 0,0:20:59.88,0:21:03.62,Chinese,,0,0,0,,那是 2^8-1 是......
Dialogue: 0,0:21:03.64,0:21:13.06,English,,0,0,0,,255 good
Dialogue: 0,0:21:03.64,0:21:13.06,Chinese,,0,0,0,,255
Dialogue: 0,0:21:13.12,0:21:16.42,English,,0,0,0,,Okay so then what's the range the of E
Dialogue: 0,0:21:13.12,0:21:16.42,Chinese,,0,0,0,,那么那么 E 的范围是什么呢？
Dialogue: 0,0:21:16.56,0:21:19.90,English,,0,0,0,,Which is the actual value that we want to encode
Dialogue: 0,0:21:16.56,0:21:19.90,Chinese,,0,0,0,,也就是我们想要编码的实际值
Dialogue: 0,0:21:24.28,0:21:26.94,English,,0,0,0,,Okay member use this we have this definition
Dialogue: 0,0:21:24.28,0:21:26.94,Chinese,,0,0,0,,我们有这个定义
Dialogue: 0,0:21:27.54,0:21:29.98,English,,0,0,0,,So we have the range of values for exp
Dialogue: 0,0:21:27.54,0:21:29.98,Chinese,,0,0,0,,所以我们有 Exp 的值范围
Dialogue: 0,0:21:30.46,0:21:32.52,English,,0,0,0,,We have our bias
Dialogue: 0,0:21:30.46,0:21:32.52,Chinese,,0,0,0,,我们有偏置值
Dialogue: 0,0:21:32.52,0:21:34.64,English,,0,0,0,,So what is the range of values for E
Dialogue: 0,0:21:32.52,0:21:34.64,Chinese,,0,0,0,,那么 E 的值范围是多少？
Dialogue: 0,0:21:34.64,0:21:37.66,English,,0,0,0,,This is the actual exponents that we're encoding
Dialogue: 0,0:21:34.64,0:21:37.66,Chinese,,0,0,0,,这是我们编码的实际指数
Dialogue: 0,0:21:38.74,0:21:47.20,English,,0,0,0,,So what's the smallest exponent,-127 good
Dialogue: 0,0:21:38.74,0:21:47.20,Chinese,,0,0,0,,那么什么是最小的指数，-127，对
Dialogue: 0,0:21:47.36,0:21:53.66,English,,0,0,0,,Right because that's zero is our smallest exp-bias which is -127
Dialogue: 0,0:21:47.36,0:21:53.66,Chinese,,0,0,0,,因为零是我们最小的 Exp-bias，即 -127
Dialogue: 0,0:21:53.66,0:21:59.06,English,,0,0,0,,Okay so the smallest value of E is 127
Dialogue: 0,0:21:53.66,0:21:59.06,Chinese,,0,0,0,,好的，所以 E 的最小值是 127（应该是 -127）
Dialogue: 0,0:21:59.06,0:22:02.42,English,,0,0,0,,And the largest value of E is ?
Dialogue: 0,0:21:59.06,0:22:02.42,Chinese,,0,0,0,,E的最大值是？
Dialogue: 0,0:22:08.64,0:22:25.16,English,,0,0,0,,So if we take the largest value of exp that's 255-127=128
Dialogue: 0,0:22:08.64,0:22:25.16,Chinese,,0,0,0,,因此，如果我们取 Exp 的最大值为 255-127=128
Dialogue: 0,0:22:25.16,0:22:30.80,English,,0,0,0,,Okay so the numbers that we can represent using this bias notation
Dialogue: 0,0:22:25.16,0:22:30.80,Chinese,,0,0,0,,好的，我们可以使用这种偏置表示法表示的数字
Dialogue: 0,0:22:31.70,0:22:35.40,English,,0,0,0,,We have the smallest exponent will be 2^ -127
Dialogue: 0,0:22:31.70,0:22:35.40,Chinese,,0,0,0,,我们有最小的指数将是 -127
Dialogue: 0,0:22:36.12,0:22:41.65,English,,0,0,0,,And the largest exponent will be 2 to the raised to the power 128
Dialogue: 0,0:22:36.12,0:22:41.65,Chinese,,0,0,0,,并且最大指数将是 128
Dialogue: 0,0:22:42.40,0:22:46.24,English,,0,0,0,,Okay so you see what's going on here
Dialogue: 0,0:22:42.40,0:22:46.24,Chinese,,0,0,0,,好的，所以你看看这里发生了什么
Dialogue: 0,0:22:46.24,0:22:49.78,English,,0,0,0,,So we have a range of values for E
Dialogue: 0,0:22:46.24,0:22:49.78,Chinese,,0,0,0,,所以我们有一系列的 E 值
Dialogue: 0,0:22:49.96,0:22:54.74,English,,0,0,0,,Here's the largest value,here's the smallest value,here's 0 right in the middle
Dialogue: 0,0:22:49.96,0:22:54.74,Chinese,,0,0,0,,这是最大的值，这里是最小值，中间是 0
Dialogue: 0,0:22:55.30,0:22:58.52,English,,0,0,0,,These are the values we want to represent this biasing is like a shift
Dialogue: 0,0:22:55.30,0:22:58.52,Chinese,,0,0,0,,这些是我们想要表示这种偏置的值
Dialogue: 0,0:22:59.08,0:23:03.40,English,,0,0,0,, We've just now we're shifting up by 127
Dialogue: 0,0:22:59.08,0:23:03.40,Chinese,,0,0,0,,我们刚才将 E 的范围移动 127
Dialogue: 0,0:23:03.62,0:23:07.32,English,,0,0,0,,So that now the smallest exponent is represented by 0
Dialogue: 0,0:23:03.62,0:23:07.32,Chinese,,0,0,0,,所以现在最小的指数用 0 表示
Dialogue: 0,0:23:09.50,0:23:13.34,English,,0,0,0,,Okay any questions about that
Dialogue: 0,0:23:09.50,0:23:13.34,Chinese,,0,0,0,,好的，对此有任何疑问
Dialogue: 0,0:23:23.20,0:23:30.66,English,,0,0,0,,Now these normalized values always have this implied one
Dialogue: 0,0:23:23.20,0:23:30.66,Chinese,,0,0,0,,现在这些规格化值尾数总是有一个隐含的 1
Dialogue: 0,0:23:31.68,0:23:36.92,English,,0,0,0,,Okay which... when we want to represent numbers closer to zero
Dialogue: 0,0:23:31.68,0:23:36.92,Chinese,,0,0,0,,好的...当我们想要表示接近零的数字时
Dialogue: 0,0:23:37.22,0:23:40.22,English,,0,0,0,,That limits us right
Dialogue: 0,0:23:37.22,0:23:40.22,Chinese,,0,0,0,,这（个隐含的 1 ）限制了我们
Dialogue: 0,0:23:40.22,0:23:46.40,English,,0,0,0,,So there's another type of floating-point number called the denormalized value
Dialogue: 0,0:23:40.22,0:23:46.40,Chinese,,0,0,0,,所以还有另一种称为非规格化的浮点数
Dialogue: 0,0:23:47.18,0:23:50.94,English,,0,0,0,,Which is characterized by an exp field of all 0
Dialogue: 0,0:23:47.18,0:23:50.94,Chinese,,0,0,0,,其特征在于 exp 域全为 0
Dialogue: 0,0:23:52.72,0:23:58.42,English,,0,0,0,,And in a denormalized number or denorm there's no implied one
Dialogue: 0,0:23:52.72,0:23:58.42,Chinese,,0,0,0,,在非标准化值中，没有隐含的数字 1
Dialogue: 0,0:23:59.20,0:24:09.24,English,,0,0,0,,Okay so the the significant m is represented exactly is encoded identically in the frac field
Dialogue: 0,0:23:59.20,0:24:09.24,Chinese,,0,0,0,,所以尾数 M 的值 frac 域编码值完全等价
Dialogue: 0,0:24:09.98,0:24:11.38,English,,0,0,0,,So there's no implied one
Dialogue: 0,0:24:09.98,0:24:11.38,Chinese,,0,0,0,,所以没有隐藏的 1
Dialogue: 0,0:24:12.08,0:24:18.56,English,,0,0,0,,And all of the numbers are the form zero point something something something after them
Dialogue: 0,0:24:12.08,0:24:18.56,Chinese,,0,0,0,,并且所有数字都是 0.xxx 形式的东西
Dialogue: 0,0:24:21.18,0:24:28.58,English,,0,0,0,,And the exponent value E instead of you remember for denorms we subtracted bias from exp
Dialogue: 0,0:24:21.18,0:24:28.58,Chinese,,0,0,0,,并且指数值 E 不是规格化的值那样等于 Exp 减去偏置 bias
Dialogue: 0,0:24:29.18,0:24:33.64,English,,0,0,0,, For denorm we subtract the bias from one
Dialogue: 0,0:24:29.18,0:24:33.64,Chinese,,0,0,0,,对于非规格化值，E=1-bias
Dialogue: 0,0:24:33.68,0:24:37.12,English,,0,0,0,,Okay now if we just use the definition
Dialogue: 0,0:24:33.68,0:24:37.12,Chinese,,0,0,0,,好的，现在我们使用这个定义
Dialogue: 0,0:24:37.12,0:24:40.86,English,,0,0,0,,We use for normalized numbers that would be...exp would be zero
Dialogue: 0,0:24:37.12,0:24:40.86,Chinese,,0,0,0,,我们使用的规格化数字将是...... exp 将为 0
Dialogue: 0,0:24:41.88,0:24:45.18,English,,0,0,0,,So we don't do that instead
Dialogue: 0,0:24:41.88,0:24:45.18,Chinese,,0,0,0,,所以我们不这样做
Dialogue: 0,0:24:45.92,0:24:47.72,English,,0,0,0,,E = 1 - bias
Dialogue: 0,0:24:45.92,0:24:47.72,Chinese,,0,0,0,,（非规格化数中）E=1-bias
Dialogue: 0,0:24:49.18,0:24:54.78,English,,0,0,0,,And then the significant is coded with an implied leading 0 instead of a leading one like a mentioned
Dialogue: 0,0:24:49.18,0:24:54.78,Chinese,,0,0,0,,然后，符号用隐含的前导 0 编码，而不是如前所述的前导 1
Dialogue: 0,0:24:57.98,0:25:02.86,English,,0,0,0,,So when exp is zero and also I mean
Dialogue: 0,0:24:57.98,0:25:02.86,Chinese,,0,0,0,,因此当 exp 为零时，我的意思是
Dialogue: 0,0:25:02.86,0:25:08.06,English,,0,0,0,,We think about it. If we only had normalized numbers
Dialogue: 0,0:25:02.86,0:25:08.06,Chinese,,0,0,0,,我们考虑一下。如果我们只有规格化的值（老师口误，说成denormalized）
Dialogue: 0,0:25:08.06,0:25:12.58,English,,0,0,0,, We wouldn't be able to represent zero right because there'd always be a 1 in there
Dialogue: 0,0:25:08.06,0:25:12.58,Chinese,,0,0,0,,我们无法代表零，因为总有一个隐含的 1
Dialogue: 0,0:25:13.72,0:25:16.92,English,,0,0,0,, So the denorms allow us to represent zero
Dialogue: 0,0:25:13.72,0:25:16.92,Chinese,,0,0,0,,因此，这些非标准化的值使我们能够代表零
Dialogue: 0,0:25:16.92,0:25:20.96,English,,0,0,0,,And they allow us to represent values that are very close to zero
Dialogue: 0,0:25:16.92,0:25:20.96,Chinese,,0,0,0,,它们允许我们表示非常接近零的值
Dialogue: 0,0:25:22.70,0:25:24.26,English,,0,0,0,,So when the exponent is zero
Dialogue: 0,0:25:22.70,0:25:24.26,Chinese,,0,0,0,,所以当指数为零时
Dialogue: 0,0:25:24.52,0:25:29.74,English,,0,0,0,, And the exp field is zero and the frac field is zero then this is the zero value
Dialogue: 0,0:25:24.52,0:25:29.74,Chinese,,0,0,0,,exp 字段为零，frac 为零，则为零值
Dialogue: 0,0:25:30.60,0:25:35.08,English,,0,0,0,,And so this is nice because all zeros represents zero
Dialogue: 0,0:25:30.60,0:25:35.08,Chinese,,0,0,0,,所以这很好，因为全零代表零
Dialogue: 0,0:25:35.12,0:25:37.22,English,,0,0,0,,Which is you know sort of which one
Dialogue: 0,0:25:35.12,0:25:37.22,Chinese,,0,0,0,,你知道哪一种
Dialogue: 0,0:25:38.56,0:25:40.42,English,,0,0,0,,There is the sort of weird thing though that
Dialogue: 0,0:25:38.56,0:25:40.42,Chinese,,0,0,0,,尽管如此，有一种奇怪的东西
Dialogue: 0,0:25:41.08,0:25:46.48,English,,0,0,0,,If you have a the sign bit is 1
Dialogue: 0,0:25:41.08,0:25:46.48,Chinese,,0,0,0,,如果你有一个符号位是 1
Dialogue: 0,0:25:46.48,0:25:52.32,English,,0,0,0,,And then your frac and exp is 0 then you have -0
Dialogue: 0,0:25:46.48,0:25:52.32,Chinese,,0,0,0,,然后你的 frac 和 exp 为 0 然后你有 -0
Dialogue: 0,0:25:52.32,0:25:55.58,English,,0,0,0,, So you have this kind of odd situation where there's a positive and negative zero
Dialogue: 0,0:25:52.32,0:25:55.58,Chinese,,0,0,0,,所以你有这种奇怪的情况，那里有正负零
Dialogue: 0,0:26:00.42,0:26:04.34,English,,0,0,0,,Okay and other values if exp is zero and you have a nonzero frac
Dialogue: 0,0:26:00.42,0:26:04.34,Chinese,,0,0,0,,好吧和其他值，如果 exp 为零，你有一个非零小数
Dialogue: 0,0:26:04.34,0:26:08.98,English,,0,0,0,,And those are the equally spaced numbers that are closest to zero
Dialogue: 0,0:26:04.34,0:26:08.98,Chinese,,0,0,0,,那些是最接近零的等间距数字
Dialogue: 0,0:26:12.14,0:26:14.86,English,,0,0,0,,Now there's a couple of other special values
Dialogue: 0,0:26:12.14,0:26:14.86,Chinese,,0,0,0,,现在还有其他几个特殊的值
Dialogue: 0,0:26:15.04,0:26:21.14,English,,0,0,0,,When exp is all ones and frac is all zeros
Dialogue: 0,0:26:15.04,0:26:21.14,Chinese,,0,0,0,,当 exp 为全 1 时，frac 全为 0
Dialogue: 0,0:26:21.14,0:26:23.08,English,,0,0,0,,This represents the value infinity
Dialogue: 0,0:26:21.14,0:26:23.08,Chinese,,0,0,0,,这代表无穷大的值
Dialogue: 0,0:26:24.78,0:26:27.44,English,,0,0,0,,So this is a value that that we get
Dialogue: 0,0:26:24.78,0:26:27.44,Chinese,,0,0,0,,这是当
Dialogue: 0,0:26:27.44,0:26:31.66,English,,0,0,0,,When is some kind of addition or multiplication and we overflow
Dialogue: 0,0:26:27.44,0:26:31.66,Chinese,,0,0,0,,因为加法或乘法而且溢出时得到的值（无穷大）
Dialogue: 0,0:26:32.46,0:26:38.48,English,,0,0,0,, So in floating point always overflows to the special value called infinity there positive or negative infinity
Dialogue: 0,0:26:32.46,0:26:38.48,Chinese,,0,0,0,,所以在浮点上总是溢出到称为无穷大的特殊值，有正或负无穷大
Dialogue: 0,0:26:38.74,0:26:41.02,English,,0,0,0,,So it's different from two's complement right where
Dialogue: 0,0:26:38.74,0:26:41.02,Chinese,,0,0,0,,所以它与二进制补码的不同之处在于
Dialogue: 0,0:26:41.44,0:26:48.52,English,,0,0,0,,You sort of overflowed into in this modular way back just back it round into the range of numbers
Dialogue: 0,0:26:41.44,0:26:48.52,Chinese,,0,0,0,,你可以用这种模块化的方式回推，然后回到数字范围内
Dialogue: 0,0:26:48.52,0:26:56.54,English,,0,0,0,, In floating point we just overflow to the sticky value called infinity and then everything we do on that remains infinity
Dialogue: 0,0:26:48.52,0:26:56.54,Chinese,,0,0,0,,在浮点数中，我们会溢出到称为无穷大，然后我们对其做任何操作结果仍然是无穷大
Dialogue: 0,0:26:59.80,0:27:01.62,English,,0,0,0,, So for example if we did...
Dialogue: 0,0:26:59.80,0:27:01.62,Chinese,,0,0,0,,例如，如果我们做了......
Dialogue: 0,0:27:06.70,0:27:14.84,English,,0,0,0,, If we did 1/0 or -1/-0 we'd get positive infinity
Dialogue: 0,0:27:06.70,0:27:14.84,Chinese,,0,0,0,,如果我们做了 1/0 或 -1/-0，我们就得到正无穷大
Dialogue: 0,0:27:14.84,0:27:19.46,English,,0,0,0,, If we divide 1 by 0 negative 0 we get negative infinity
Dialogue: 0,0:27:14.84,0:27:19.46,Chinese,,0,0,0,,如果我们将 1/-0，我们得到负无穷大
Dialogue: 0,0:27:20.72,0:27:33.06,English,,0,0,0,,The fourth kind of special case special number is the case where the exponent again is all once but the frac is nonzero in a nonzero value
Dialogue: 0,0:27:20.72,0:27:33.06,Chinese,,0,0,0,,第四种特殊数字是指数全部是 1 但小数字段非零值的情况
Dialogue: 0,0:27:33.26,0:27:36.56,English,,0,0,0,,And this represents what's called not a number so this is
Dialogue: 0,0:27:33.26,0:27:36.56,Chinese,,0,0,0,,这代表了非数字（NaN）的情况
Dialogue: 0,0:27:37.22,0:27:41.92,English,,0,0,0,,This handles cases where there's no,there's no answer
Dialogue: 0,0:27:37.22,0:27:41.92,Chinese,,0,0,0,,这个值代表了没有答案的情况
Dialogue: 0,0:27:41.92,0:27:46.52,English,,0,0,0,,There's no feasible answer for example if we you know try to take the square root of -1
Dialogue: 0,0:27:41.92,0:27:46.52,Chinese,,0,0,0,,例如，如果我们尝试取 -1 的平方根，就没有可行的答案
Dialogue: 0,0:27:46.84,0:27:49.10,English,,0,0,0,,What we'll get back is not a number
Dialogue: 0,0:27:46.84,0:27:49.10,Chinese,,0,0,0,,我们得到的不是数字
Dialogue: 0,0:27:55.26,0:28:02.10,English,,0,0,0,,So I think it's helpful to to visualize these the number line now so the
Dialogue: 0,0:27:55.26,0:28:02.10,Chinese,,0,0,0,,我认为将这些数字可视化是有帮助的
Dialogue: 0,0:28:02.86,0:28:11.48,English,,0,0,0,,We have +0,-0 sort of right next to each other and then to the right of that are the positive denorms
Dialogue: 0,0:28:02.86,0:28:11.48,Chinese,,0,0,0,,我们有 0，-0 彼此相邻，然后在右边是正的非标准化值
Dialogue: 0,0:28:13.14,0:28:19.54,English,,0,0,0,,And then those sort of transition into the normalized numbers
Dialogue: 0,0:28:13.14,0:28:19.54,Chinese,,0,0,0,,然后过渡到标准化数字
Dialogue: 0,0:28:20.32,0:28:23.90,English,,0,0,0,, And then you have infinity positive infinity
Dialogue: 0,0:28:20.32,0:28:23.90,Chinese,,0,0,0,,然后你有正无穷大
Dialogue: 0,0:28:24.44,0:28:29.20,English,,0,0,0,,You have a similar similar situation on the for the negative numbers
Dialogue: 0,0:28:24.44,0:28:29.20,Chinese,,0,0,0,,对于负数，你有类似的类似情况
Dialogue: 0,0:28:30.36,0:28:34.48,English,,0,0,0,,And then negative and positive not a numbers
Dialogue: 0,0:28:30.36,0:28:34.48,Chinese,,0,0,0,,然后就分别是正数和负数情况下的非数字（NaN）
Dialogue: 0,0:28:39.58,0:28:43.28,English,,0,0,0,,So I think the you know you've probably discovered
Dialogue: 0,0:28:39.58,0:28:43.28,Chinese,,0,0,0,,所以我想你知道你可能已经发现了
Dialogue: 0,0:28:43.50,0:28:50.38,English,,0,0,0,,The really good way to try to understand these numeric and Coatings is just to do examples with only a few number of bits
Dialogue: 0,0:28:43.50,0:28:50.38,Chinese,,0,0,0,,举一个仅包含几位的例子是尝试理解这些数字和规则的有效方法
Dialogue: 0,0:28:50.86,0:28:54.02,English,,0,0,0,,You know if you want to understand like for example in two's complement
Dialogue: 0,0:28:50.86,0:28:54.02,Chinese,,0,0,0,,你知道如果你想要理解，比如二进制补码
Dialogue: 0,0:28:54.60,0:28:59.20,English,,0,0,0,,If you want to understand Tmin and Tmax and how they relate to each other
Dialogue: 0,0:28:54.60,0:28:59.20,Chinese,,0,0,0,,如果你想了解 Tmin 和 Tmax 以及它们之间的相互关系
Dialogue: 0,0:28:59.56,0:29:03.88,English,,0,0,0,,Just do a three bit two's complement system
Dialogue: 0,0:28:59.56,0:29:03.88,Chinese,,0,0,0,,只做三位二进制补码系统
Dialogue: 0,0:29:03.88,0:29:07.08,English,,0,0,0,,You know imagine just have a system with two's complement of three bits
Dialogue: 0,0:29:03.88,0:29:07.08,Chinese,,0,0,0,,你想象一下，系统有二进制三位补码
Dialogue: 0,0:29:07.68,0:29:11.68,English,,0,0,0,,And then you can just enumerate all those all the possibilities and you can see
Dialogue: 0,0:29:07.68,0:29:11.68,Chinese,,0,0,0,,然后你可以列举所有可能性，你可以看到
Dialogue: 0,0:29:12.38,0:29:17.96,English,,0,0,0,,And then you see where you know the Tmin is actually one greater than Tmax
Dialogue: 0,0:29:12.38,0:29:17.96,Chinese,,0,0,0,,然后你会看到你知道 Tmin 实际上比 Tmax 更大的地方
Dialogue: 0,0:29:17.96,0:29:22.58,English,,0,0,0,,You know so if you add one to Tmax it actually overflows into Tmin
Dialogue: 0,0:29:17.96,0:29:22.58,Chinese,,0,0,0,,你知道如果你在 Tmax 中添加 1，它实际上会溢出到 Tmin
Dialogue: 0,0:29:25.20,0:29:29.98,English,,0,0,0,,Similar thing for floating point it's...I would really encourage you to do this
Dialogue: 0,0:29:25.20,0:29:29.98,Chinese,,0,0,0,,浮点数也有类似的事情......我真的很鼓励你们这样做
Dialogue: 0,0:29:29.98,0:29:36.80,English,,0,0,0,,Just try some simple examples with a sort of tyent with a small number of exp and frack bits
Dialogue: 0,0:29:29.98,0:29:36.80,Chinese,,0,0,0,,只需尝试一些带有少量 exp 和 frac 位的简单示例
Dialogue: 0,0:29:37.32,0:29:43.48,English,,0,0,0,,So in this case we're going to look at a floating point system based on IEEE
Dialogue: 0,0:29:37.32,0:29:43.48,Chinese,,0,0,0,,所以在这种情况下，我们将看到一个基于 IEEE 的浮点系统
Dialogue: 0,0:29:43.50,0:29:47.58,English,,0,0,0,,That has 4 exp bits and three frack bits
Dialogue: 0,0:29:43.50,0:29:47.58,Chinese,,0,0,0,,它有 4 个 exp 位和 3 个 frac 位
Dialogue: 0,0:29:48.64,0:29:51.48,English,,0,0,0,,As always the sign bit is the most significant bit
Dialogue: 0,0:29:48.64,0:29:51.48,Chinese,,0,0,0,,一如既往，符号位是最重要的位
Dialogue: 0,0:29:52.04,0:29:54.20,English,,0,0,0,,Followed by exp followed by frac
Dialogue: 0,0:29:52.04,0:29:54.20,Chinese,,0,0,0,,其次是 exp，然后是 frac
Dialogue: 0,0:29:55.92,0:30:03.08,English,,0,0,0,,Okay so this is identical to IEEE floating point except for the size of that exp in fractals
Dialogue: 0,0:29:55.92,0:30:03.08,Chinese,,0,0,0,,好的，除了exp，frac 的位数外，这与 IEEE 浮点相同
Dialogue: 0,0:30:04.42,0:30:10.72,English,,0,0,0,,So it's actually small enough that we can now just enumerate these values
Dialogue: 0,0:30:04.42,0:30:10.72,Chinese,,0,0,0,,所以它实际上足够小，小到我们可以枚举这些值
Dialogue: 0,0:30:11.56,0:30:22.96,English,,0,0,0,,And now you can, you can start to see why these weird definitions for the exponent in the denominator
Dialogue: 0,0:30:11.56,0:30:22.96,Chinese,,0,0,0,,现在你可以，你可以开始看到分母中指数的这些奇怪定义的原因
Dialogue: 0,0:30:23.84,0:30:27.02,English,,0,0,0,,So let's just look at...
Dialogue: 0,0:30:23.84,0:30:27.02,Chinese,,0,0,0,,那么我们来看......
Dialogue: 0,0:30:27.76,0:30:32.52,English,,0,0,0,,So we will stand here,we've just
Dialogue: 0,0:30:27.76,0:30:32.52,Chinese,,0,0,0,,我们只是
Dialogue: 0,0:30:33.46,0:30:40.92,English,,0,0,0,,All of 2^7 possibilities possible combinations of exp and frac fields starting with zero
Dialogue: 0,0:30:33.46,0:30:40.92,Chinese,,0,0,0,,列出从零开始所有 2^7 种可能的 exp 和 frac 字段的组合
Dialogue: 0,0:30:42.50,0:30:45.16,English,,0,0,0,,Okay so when exp is all zeros
Dialogue: 0,0:30:42.50,0:30:45.16,Chinese,,0,0,0,,好的，所以当 exp 全部为零时
Dialogue: 0,0:30:45.52,0:30:48.38,English,,0,0,0,,That's a denorm right by definition
Dialogue: 0,0:30:45.52,0:30:48.38,Chinese,,0,0,0,,根据定义，这是一种非规格化值
Dialogue: 0,0:30:49.72,0:30:59.34,English,,0,0,0,,And then we have the three frac bits are all zero
Dialogue: 0,0:30:49.72,0:30:59.34,Chinese,,0,0,0,,然后我们有三个小数字段位全零
Dialogue: 0,0:30:59.66,0:31:03.04,English,,0,0,0,,Okay so remember the exp is all zeros
Dialogue: 0,0:30:59.66,0:31:03.04,Chinese,,0,0,0,,好的，所以记住 exp 全是零
Dialogue: 0,0:31:03.44,0:31:13.20,English,,0,0,0,,Right so the for denorm the exponent value is 1- bias, bias here is 7
Dialogue: 0,0:31:03.44,0:31:13.20,Chinese,,0,0,0,,因此，对于非标准化数指数值是 1-bias，这里的 bias 是7（bias 即偏置值）
Dialogue: 0,0:31:14.12,0:31:18.71,English,,0,0,0,,Okay so 1-7 gives us a value for the exponent of -6
Dialogue: 0,0:31:14.12,0:31:18.71,Chinese,,0,0,0,,好的，1-7 得到指数值是 -6
Dialogue: 0,0:31:19.56,0:31:21.42,English,,0,0,0,,So we're going to take the frac
Dialogue: 0,0:31:19.56,0:31:21.42,Chinese,,0,0,0,,接着我们要取得小数字段
Dialogue: 0,0:31:23.20,0:31:28.70,English,,0,0,0,, The bits in the frac field times
Dialogue: 0,0:31:23.20,0:31:28.70,Chinese,,0,0,0,,frac 表示的值乘以 m
Dialogue: 0,0:31:30.02,0:31:31.24,English,,0,0,0,,So m is 0
Dialogue: 0,0:31:30.02,0:31:31.24,Chinese,,0,0,0,,m 是 0
Dialogue: 0,0:31:32.06,0:31:38.42,English,,0,0,0,,So m in this case is 0*2^-6 gives us 0
Dialogue: 0,0:31:32.06,0:31:38.42,Chinese,,0,0,0,,所以在这种情况下是 M 是 0，0*2^-6 得到 0
Dialogue: 0,0:31:38.76,0:31:42.12,English,,0,0,0,,So this is our representation of 0
Dialogue: 0,0:31:38.76,0:31:42.12,Chinese,,0,0,0,,所以这是我们 0 的表达方式
Dialogue: 0,0:31:42.12,0:31:44.72,English,,0,0,0,,And like I said it's really it's nice that it actually is
Dialogue: 0,0:31:42.12,0:31:44.72,Chinese,,0,0,0,,就像我说的那样，实际上真的很棒
Dialogue: 0,0:31:45.16,0:31:50.12,English,,0,0,0,,If we just treated this as an unsigned integer, it would be 0
Dialogue: 0,0:31:45.16,0:31:50.12,Chinese,,0,0,0,,如果我们只是将其视为无符号整数，那么它表示 0
Dialogue: 0,0:31:50.50,0:31:52.08,English,,0,0,0,,Okay so that's kind of pleasing
Dialogue: 0,0:31:50.50,0:31:52.08,Chinese,,0,0,0,,好的，这样很愉快
Dialogue: 0,0:31:53.20,0:31:54.94,English,,0,0,0,,Now let's look at the very next number right
Dialogue: 0,0:31:53.20,0:31:54.94,Chinese,,0,0,0,,现在让我们看看下一个数字吧
Dialogue: 0,0:31:54.96,0:32:00.10,English,,0,0,0,,The exp is still 0 and we've just bumped up frac by 1
Dialogue: 0,0:31:54.96,0:32:00.10,Chinese,,0,0,0,,exp 仍为 0，我们刚刚将 frac 加了 1
Dialogue: 0,0:32:01.40,0:32:06.74,English,,0,0,0,,Ok so again it's a 1-bias which is -6
Dialogue: 0,0:32:01.40,0:32:06.74,Chinese,,0,0,0,,好的，依然 E=1-bias，即 -6
Dialogue: 0,0:32:08.22,0:32:18.62,English,,0,0,0,,And now this,this frac field represents this is 1/2,1/4,1/8
Dialogue: 0,0:32:08.22,0:32:18.62,Chinese,,0,0,0,,现在这个，这个 frac 域各位依次表示 1/2, 1/4, 1/ 8
Dialogue: 0,0:32:19.12,0:32:23.62,English,,0,0,0,,So it represents,so this is a value of amaz it is 1/8
Dialogue: 0,0:32:19.12,0:32:23.62,Chinese,,0,0,0,,所以是 1/8
Dialogue: 0,0:32:24.72,0:32:31.08,English,,0,0,0,, So 1/8*2^-6(1/64)=1/512
Dialogue: 0,0:32:24.72,0:32:31.08,Chinese,,0,0,0,,所以 1/8*2^-6（1/64）= 1/512
Dialogue: 0,0:32:31.46,0:32:34.14,English,,0,0,0,,So in this floating point system the smallest nonzero number
Dialogue: 0,0:32:31.46,0:32:34.14,Chinese,,0,0,0,,这个数是这个浮点数系统中最小的非零数
Dialogue: 0,0:32:34.14,0:32:38.58,English,,0,0,0,,We can represent is the denormalized representation of 1/512
Dialogue: 0,0:32:34.14,0:32:38.58,Chinese,,0,0,0,,我们只能用非规格化表示 1/512
Dialogue: 0,0:32:39.42,0:32:41.12,English,,0,0,0,, The next smallest number is 0
Dialogue: 0,0:32:39.42,0:32:41.12,Chinese,,0,0,0,,下一个最小的数字是 0
Dialogue: 0,0:32:44.74,0:32:49.24,English,,0,0,0,,Okay now let's look at, so let's take, let's bump this up by one
Dialogue: 0,0:32:44.74,0:32:49.24,Chinese,,0,0,0,,好的，现在让我们来看看，让我们把它再增大 1
Dialogue: 0,0:32:49.24,0:32:54.16,English,,0,0,0,, So now we get a frac field of 010
Dialogue: 0,0:32:49.24,0:32:54.16,Chinese,,0,0,0,,所以现在小数字段区域是 010
Dialogue: 0,0:32:54.16,0:32:57.88,English,,0,0,0,,So this represents 1/4
Dialogue: 0,0:32:54.16,0:32:57.88,Chinese,,0,0,0,,所以这代表 1/4
Dialogue: 0,0:32:59.46,0:33:05.12,English,,0,0,0,,So m is 1/4 that significant is 1/4*2^-6
Dialogue: 0,0:32:59.46,0:33:05.12,Chinese,,0,0,0,,所以 m 是 1/4，明显（结果）是 1/4*2^-6
Dialogue: 0,0:33:05.26,0:33:10.06,English,,0,0,0,,So 2/8*1/64 is 2/512
Dialogue: 0,0:33:05.26,0:33:10.06,Chinese,,0,0,0,,所以 2/8*1/64 是 2/512
Dialogue: 0,0:33:11.08,0:33:19.74,English,,0,0,0,,Okay so the the numbers were just it by incrementing this by incrementing are significant
Dialogue: 0,0:33:11.08,0:33:19.74,Chinese,,0,0,0,,这些数字通过递增 frac（尾数）来不断递增是有意义的
Dialogue: 0,0:33:20.12,0:33:23.78,English,,0,0,0,,We're increasing the numbers by 1/512
Dialogue: 0,0:33:20.12,0:33:23.78,Chinese,,0,0,0,,我们将这个数字增加了 1/512
Dialogue: 0,0:33:26.32,0:33:33.34,English,,0,0,0,,Now the largest denorm in this number system is the case where the frac field is all ones
Dialogue: 0,0:33:26.32,0:33:33.34,Chinese,,0,0,0,,现在，这个数字系统中最大的变形就是 frac 是全 1
Dialogue: 0,0:33:34.48,0:33:44.68,English,,0,0,0,,And so this encodes 1/2+1/4+1/8 which is 7/8
Dialogue: 0,0:33:34.48,0:33:44.68,Chinese,,0,0,0,,所以这编码为1/2 + 1/4 + 1/8，即7/8
Dialogue: 0,0:33:44.68,0:33:52.90,English,,0,0,0,,And then our value is 7/8*2^-6(1/64) =7/512
Dialogue: 0,0:33:44.68,0:33:52.90,Chinese,,0,0,0,,然后我们的值是 7/8*2^-6（1/64）= 7/512
Dialogue: 0,0:33:54.94,0:34:01.10,English,,0,0,0,,So this 7/512 is the largest denormalized number we can represent in this system
Dialogue: 0,0:33:54.94,0:34:01.10,Chinese,,0,0,0,,所以这个 7/512 是我们在这个系统中可以表示的最大的非规格化的值
Dialogue: 0,0:34:02.50,0:34:09.70,English,,0,0,0,,And now if we bump our number up by one the the frac field resets to zero
Dialogue: 0,0:34:02.50,0:34:09.70,Chinese,,0,0,0,,现在如果我们将数字增加 1，则 frac 重置为零
Dialogue: 0,0:34:09.72,0:34:13.48,English,,0,0,0,,And now the exp field goes from 0 to 1
Dialogue: 0,0:34:09.72,0:34:13.48,Chinese,,0,0,0,,现在 exp 域从 0 变为 1
Dialogue: 0,0:34:17.02,0:34:19.86,English,,0,0,0,,Now because the exp field is nonzero
Dialogue: 0,0:34:17.02,0:34:19.86,Chinese,,0,0,0,,现在因为 exp 域非全零
Dialogue: 0,0:34:19.86,0:34:23.38,English,,0,0,0,,This is no longer a denormalized number,It's a normalized number
Dialogue: 0,0:34:19.86,0:34:23.38,Chinese,,0,0,0,,这不再是非规格化的值，而是一个规格化数字
Dialogue: 0,0:34:25.06,0:34:33.44,English,,0,0,0,,Okay so now we're changing the definition of E from 1-bias to exp - bias
Dialogue: 0,0:34:25.06,0:34:33.44,Chinese,,0,0,0,,好的，现在我们将 E 的定义从 1-bias 改为 Exp-bias
Dialogue: 0,0:34:35.34,0:34:38.08,English,,0,0,0,,So what's exp in this case ?
Dialogue: 0,0:34:35.34,0:34:38.08,Chinese,,0,0,0,,那么在这种情况下指数字段 exp 是什么？
Dialogue: 0,0:34:40.50,0:34:46.50,English,,0,0,0,,1-7,the bias is -6
Dialogue: 0,0:34:40.50,0:34:46.50,Chinese,,0,0,0,,1-7，偏置为 -6
Dialogue: 0,0:34:46.52,0:34:57.08,English,,0,0,0,, So lower in hold,it's the same exponent as the previous denormalized number
Dialogue: 0,0:34:46.52,0:34:57.08,Chinese,,0,0,0,,因此指数保持不变，它与先前非标规格化的值的指数相同
Dialogue: 0,0:34:57.50,0:35:00.72,English,,0,0,0,,So this is actually very clever by the IEEE folks
Dialogue: 0,0:34:57.50,0:35:00.72,Chinese,,0,0,0,,所以说 IEEE 的家伙们实际上非常聪明
Dialogue: 0,0:35:01.10,0:35:06.54,English,,0,0,0,,Because you get the spirit,you get this nice smooth transition from denorm to normalize
Dialogue: 0,0:35:01.10,0:35:06.54,Chinese,,0,0,0,,因为你领略了其中的精髓，你会明白从非规格化数到规格化数的平滑过渡
Dialogue: 0,0:35:07.44,0:35:13.30,English,,0,0,0,,So you can see that our exponent E is -6 still
Dialogue: 0,0:35:07.44,0:35:13.30,Chinese,,0,0,0,,所以你可以看到我们的指数 E 仍为 -6
Dialogue: 0,0:35:15.08,0:35:29.54,English,,0,0,0,,And now,because we're taking 1-bias(7) And are significant now remember this is a normalized number
Dialogue: 0,0:35:15.08,0:35:29.54,Chinese,,0,0,0,,现在，因为我们正在使用 1-bias（7），记住这是一个规格化的数字
Dialogue: 0,0:35:29.54,0:35:32.92,English,,0,0,0,,Because the exp field is nonzero
Dialogue: 0,0:35:29.54,0:35:32.92,Chinese,,0,0,0,,因为 exp 域非全零
Dialogue: 0,0:35:35.74,0:35:41.26,English,,0,0,0,,They are a frac field of all zeros represents a significant of 1.000
Dialogue: 0,0:35:35.74,0:35:41.26,Chinese,,0,0,0,,值得注意的是，小数字段全 0 代表 1.000
Dialogue: 0,0:35:43.00,0:35:47.96,English,,0,0,0,,Okay so this normalized number this the smallest normalized number
Dialogue: 0,0:35:43.00,0:35:47.96,Chinese,,0,0,0,,好的，这个标准化的数字是最小的标准化数字
Dialogue: 0,0:35:49.06,0:35:59.70,English,,0,0,0,,Represents is 1.000 which is 8/8*2^-6(1/64) which gives us 8/512
Dialogue: 0,0:35:49.06,0:35:59.70,Chinese,,0,0,0,,代表是 1.000，这是 8/8*2^-6（1/64），它给我们 8/512
Dialogue: 0,0:35:59.80,0:36:05.72,English,,0,0,0,,So we've made this nice smooth transition from 7/512 to 8/512
Dialogue: 0,0:35:59.80,0:36:05.72,Chinese,,0,0,0,,所以我们已经实现了从 7/512 到 8/512 的平滑过渡
Dialogue: 0,0:36:07.28,0:36:08.82,English,,0,0,0,,And how do one more
Dialogue: 0,0:36:07.28,0:36:08.82,Chinese,,0,0,0,,又如何做一个
Dialogue: 0,0:36:08.82,0:36:09.90,English,,0,0,0,, I don't want to
Dialogue: 0,0:36:08.82,0:36:09.90,Chinese,,0,0,0,,我不想
Dialogue: 0,0:36:10.60,0:36:15.46,English,,0,0,0,,So if we bump keeping the exp field 1
Dialogue: 0,0:36:10.60,0:36:15.46,Chinese,,0,0,0,,因此，如果我们保持 exp 字段 1
Dialogue: 0,0:36:15.46,0:36:17.94,English,,0,0,0,,If we bump our frac field from 0 to 1
Dialogue: 0,0:36:15.46,0:36:17.94,Chinese,,0,0,0,,如果我们将我们的 frac 从 0 增加到 1
Dialogue: 0,0:36:18.86,0:36:31.12,English,,0,0,0,,Now this represents the the significant of 1.0, 1.001
Dialogue: 0,0:36:18.86,0:36:31.12,Chinese,,0,0,0,,现在这代表了 1.001 的重要性
Dialogue: 0,0:36:32.92,0:36:36.66,English,,0,0,0,,Which is 8/8+1/8
Dialogue: 0,0:36:32.92,0:36:36.66,Chinese,,0,0,0,,这是 8/8+1/8
Dialogue: 0,0:36:38.48,0:36:48.10,English,,0,0,0,,So that's 9/8 times...and then the value is 9/8*2^-6(1/64) which is equal to 9/512
Dialogue: 0,0:36:38.48,0:36:48.10,Chinese,,0,0,0,,最终值为 9/8*2^-6（1/64），等于 9/512
Dialogue: 0,0:36:48.54,0:36:52.07,English,,0,0,0,,Okay
Dialogue: 0,0:36:48.54,0:36:52.07,Chinese,,0,0,0,,好的
Dialogue: 0,0:36:52.44,0:36:53.68,English,,0,0,0,,Any questions about that?
Dialogue: 0,0:36:52.44,0:36:53.68,Chinese,,0,0,0,,对此有任何疑问？
Dialogue: 0,0:36:59.42,0:37:00.44,English,,0,0,0,,Question back here
Dialogue: 0,0:36:59.42,0:37:00.44,Chinese,,0,0,0,,回答这里
Dialogue: 0,0:37:00.74,0:37:04.68,English,,0,0,0,,[student speaking]
Dialogue: 0,0:37:00.74,0:37:04.68,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:37:04.68,0:37:09.68,English,,0,0,0,,Okay the question is why do you choose the bias that to be 2^(k-1) - 1
Dialogue: 0,0:37:04.68,0:37:09.68,Chinese,,0,0,0,,好的问题是为什么你选择偏置为 2^(k-1)-1
Dialogue: 0,0:37:10.04,0:37:13.16,English,,0,0,0,,Well go back to my finger diagram here
Dialogue: 0,0:37:10.04,0:37:13.16,Chinese,,0,0,0,,那么回到我的手指图
Dialogue: 0,0:37:13.78,0:37:18.46,English,,0,0,0,, so this is the range of exp fields that you want to
Dialogue: 0,0:37:13.78,0:37:18.46,Chinese,,0,0,0,,所以这是你想要的 exp 域的范围
Dialogue: 0,0:37:18.82,0:37:22.14,English,,0,0,0,,This is the range of E that you want to represent with 0 in the middle
Dialogue: 0,0:37:18.82,0:37:22.14,Chinese,,0,0,0,,你希望 E 范围的中位数是 0
Dialogue: 0,0:37:23.54,0:37:26.96,English,,0,0,0,,Okay and so you want to bias
Dialogue: 0,0:37:23.54,0:37:26.96,Chinese,,0,0,0,,所以你想要偏置值
Dialogue: 0,0:37:26.96,0:37:32.62,English,,0,0,0,,So that this is halfway，the smallest exponent is halfway to 0
Dialogue: 0,0:37:26.96,0:37:32.62,Chinese,,0,0,0,,因此，这是中间，最小的指数是 0 的一半
Dialogue: 0,0:37:34.36,0:37:36.20,English,,0,0,0,,So you want the bias to shift it up
Dialogue: 0,0:37:34.36,0:37:36.20,Chinese,,0,0,0,,所以你希望偏置值能够改变它
Dialogue: 0,0:37:36.20,0:37:42.86,English,,0,0,0,,So it's the smallest value of exp corresponds to the smallest exponent
Dialogue: 0,0:37:36.20,0:37:42.86,Chinese,,0,0,0,,因此，exp 的最小值对应于最小的指数
Dialogue: 0,0:37:43.56,0:37:51.70,English,,0,0,0,,Right so 2^(k-1)-1 is just giving you a value that's halfway between the smallest exponent and the largest exponent
Dialogue: 0,0:37:43.56,0:37:51.70,Chinese,,0,0,0,,对，所以 2^(k-1)-1 只给你一个在最小指数和最大指数之间的值
Dialogue: 0,0:37:53.40,0:37:53.76,English,,0,0,0,,Okay
Dialogue: 0,0:37:53.40,0:37:53.76,Chinese,,0,0,0,,好的
Dialogue: 0,0:37:56.58,0:37:57.84,English,,0,0,0,,That's good.Any other questions?
Dialogue: 0,0:37:56.58,0:37:57.84,Chinese,,0,0,0,,那很好。还有其他问题吗？
Dialogue: 0,0:38:05.20,0:38:10.74,English,,0,0,0,,Okay if we take a little smaller six bit format same thing as we did before
Dialogue: 0,0:38:05.20,0:38:10.74,Chinese,,0,0,0,,好的，如果我们采用与之前相同的六位格式
Dialogue: 0,0:38:10.74,0:38:14.20,English,,0,0,0,,Just different numbers of bits in exp and frac
Dialogue: 0,0:38:10.74,0:38:14.20,Chinese,,0,0,0,,exp 和 frac 有不同的位数
Dialogue: 0,0:38:14.84,0:38:18.66,English,,0,0,0,,And then we plot these on the number line
Dialogue: 0,0:38:14.84,0:38:18.66,Chinese,,0,0,0,,然后我们在数轴上绘制这些图
Dialogue: 0,0:38:18.66,0:38:24.74,English,,0,0,0,,You can see that the values are very dense around zero
Dialogue: 0,0:38:18.66,0:38:24.74,Chinese,,0,0,0,,你可以看到值在零附近非常密集
Dialogue: 0,0:38:24.74,0:38:27.50,English,,0,0,0,,So they're very spaced very closely together
Dialogue: 0,0:38:24.74,0:38:27.50,Chinese,,0,0,0,,所以他们非常紧密地在一起
Dialogue: 0,0:38:28.12,0:38:33.32,English,,0,0,0,,And then every time you increase the exponent by one
Dialogue: 0,0:38:28.12,0:38:33.32,Chinese,,0,0,0,,然后每次将指数增加一
Dialogue: 0,0:38:33.94,0:38:40.38,English,,0,0,0,,The numbers are spaced twice as far apart as the previous
Dialogue: 0,0:38:33.94,0:38:40.38,Chinese,,0,0,0,,这些数字的距离是前一个的两倍
Dialogue: 0,0:38:40.38,0:38:45.18,English,,0,0,0,,The numbers that were represented by the previous exponent
Dialogue: 0,0:38:40.38,0:38:45.18,Chinese,,0,0,0,,由前一个指数表示的数字
Dialogue: 0,0:38:46.26,0:38:48.66,English,,0,0,0,, So you can see like in this little example
Dialogue: 0,0:38:46.26,0:38:48.66,Chinese,,0,0,0,,所以你可以在这个小例子中看到
Dialogue: 0,0:38:50.72,0:38:55.92,English,,0,0,0,,You see these...these values here are all equally spaced
Dialogue: 0,0:38:50.72,0:38:55.92,Chinese,,0,0,0,,你看到这些......这里的这些值都是等间距的
Dialogue: 0,0:38:56.82,0:39:00.38,English,,0,0,0,,These four values are equally spaced and they're right next to each other
Dialogue: 0,0:38:56.82,0:39:00.38,Chinese,,0,0,0,,这四个值间隔相等，它们彼此相邻
Dialogue: 0,0:39:01.06,0:39:07.48,English,,0,0,0,,And then you see this jump in the spacing that represents an increase in the exp field by one
Dialogue: 0,0:39:01.06,0:39:07.48,Chinese,,0,0,0,,你会看到这些值之间是有间隔的，表示 exp 域增加了 1
Dialogue: 0,0:39:09.52,0:39:14.02,English,,0,0,0,,And then that spacing doubles as we go out
Dialogue: 0,0:39:09.52,0:39:14.02,Chinese,,0,0,0,,然后，随着点向外扩展，间距加倍
Dialogue: 0,0:39:16.54,0:39:21.25,English,,0,0,0,,And you can see it even better here this if we zoom in
Dialogue: 0,0:39:16.54,0:39:21.25,Chinese,,0,0,0,,如果我们放大，你可以更清楚的看到它
Dialogue: 0,0:39:21.68,0:39:27.36,English,,0,0,0,,These blue values or denorms
Dialogue: 0,0:39:21.68,0:39:27.36,Chinese,,0,0,0,,这些蓝色值或非规格化值
Dialogue: 0,0:39:28.36,0:39:34.20,English,,0,0,0,,And then we have the the next set of normalized values where the frac field is 1
Dialogue: 0,0:39:28.36,0:39:34.20,Chinese,,0,0,0,,然后我们得到下一组规格化值，其中 frac 为 1
Dialogue: 0,0:39:34.20,0:39:39.32,English,,0,0,0,,They're spaced the same right we saw this when we kept increasing by 1/512
Dialogue: 0,0:39:34.20,0:39:39.32,Chinese,,0,0,0,,当我们不停的增加 1/512 时，它们与我们看到的相同
Dialogue: 0,0:39:41.42,0:39:45.36,English,,0,0,0,,And then we increase the the exponent by one and the spacing doubles
Dialogue: 0,0:39:41.42,0:39:45.36,Chinese,,0,0,0,,然后我们将指数增加 1，间距增加一倍
Dialogue: 0,0:39:51.46,0:39:55.46,English,,0,0,0,,Okay so this like I mentioned this encoding has some pretty nice properties
Dialogue: 0,0:39:51.46,0:39:55.46,Chinese,,0,0,0,,好的，所以这就像我提到的这个编码有一些非常好的属性
Dialogue: 0,0:39:55.82,0:40:02.66,English,,0,0,0,,The floating point zero is encoded as integer zero which is nice
Dialogue: 0,0:39:55.82,0:40:02.66,Chinese,,0,0,0,,浮点零被编码为整数零
Dialogue: 0,0:40:04.50,0:40:14.48,English,,0,0,0,,And with the exception of not a number,with the exception of not a number
Dialogue: 0,0:40:04.50,0:40:14.48,Chinese,,0,0,0,,除了非数字之外
Dialogue: 0,0:40:14.78,0:40:16.98,English,,0,0,0,,You can compare any floating-point numbers
Dialogue: 0,0:40:14.78,0:40:16.98,Chinese,,0,0,0,,你可以比较任何浮点数
Dialogue: 0,0:40:17.28,0:40:22.60,English,,0,0,0,,Including infinity by just treating it as doing an unsigned comparison
Dialogue: 0,0:40:17.28,0:40:22.60,Chinese,,0,0,0,,通过将其视为无符号比较来包括无穷大
Dialogue: 0,0:40:23.16,0:40:24.78,English,,0,0,0,,So that's really nice too
Dialogue: 0,0:40:23.16,0:40:24.78,Chinese,,0,0,0,,所以这也很好
Dialogue: 0,0:40:30.90,0:40:35.42,English,,0,0,0,,Um now if you, when you add and multiply floating point numbers
Dialogue: 0,0:40:30.90,0:40:35.42,Chinese,,0,0,0,,嗯，当你对浮点数进行相加或相乘操作时
Dialogue: 0,0:40:35.42,0:40:38.22,English,,0,0,0,,Right we're dealing with a limited set of bits here
Dialogue: 0,0:40:35.42,0:40:38.22,Chinese,,0,0,0,,对，我们在这里处理一组有限的位
Dialogue: 0,0:40:39.14,0:40:44.12,English,,0,0,0,,So if we take two numbers encoded as floating-point
Dialogue: 0,0:40:39.14,0:40:44.12,Chinese,,0,0,0,,所以如果我们把两个数字编码为浮点数
Dialogue: 0,0:40:44.12,0:40:48.26,English,,0,0,0,,And we add them together there's no guarantee that we'll be able to fit all those bits
Dialogue: 0,0:40:44.12,0:40:48.26,Chinese,,0,0,0,,两个数加在一起并不能保证结果能用这些有限位表示
Dialogue: 0,0:40:48.80,0:40:52.64,English,,0,0,0,,In the bits we have available to us
Dialogue: 0,0:40:48.80,0:40:52.64,Chinese,,0,0,0,,我们有限的位
Dialogue: 0,0:40:52.64,0:40:55.74,English,,0,0,0,,There's no guarantee that we'll be able to fit the the bits and the result
Dialogue: 0,0:40:52.64,0:40:55.74,Chinese,,0,0,0,,不能保证结果能用这些有限位表示
Dialogue: 0,0:40:56.64,0:40:59.84,English,,0,0,0,,So what we do in this case is
Dialogue: 0,0:40:56.64,0:40:59.84,Chinese,,0,0,0,,所以我们在这种情况下做的是
Dialogue: 0,0:41:00.14,0:41:06.58,English,,0,0,0,,You can imagine the model you want to have a floating-point operations addition to multiplication
Dialogue: 0,0:41:00.14,0:41:06.58,Chinese,,0,0,0,,浮点数加法你可以类比成乘法
Dialogue: 0,0:41:06.58,0:41:12.42,English,,0,0,0,,It's that you add or multiply you do the addition or multiplication
Dialogue: 0,0:41:06.58,0:41:12.42,Chinese,,0,0,0,,你自己做加法或乘法
Dialogue: 0,0:41:12.78,0:41:14.34,English,,0,0,0,,And you compute the exact result
Dialogue: 0,0:41:12.78,0:41:14.34,Chinese,,0,0,0,,然后你计算出确切的结果
Dialogue: 0,0:41:14.34,0:41:18.12,English,,0,0,0,, You just assume that you have an infinite amount of number of bits
Dialogue: 0,0:41:14.34,0:41:18.12,Chinese,,0,0,0,,你只是假设你拥有无限数量的位数
Dialogue: 0,0:41:19.02,0:41:25.24,English,,0,0,0,,And then you round use the technique called rounding to get it to fit into the of the available bits
Dialogue: 0,0:41:19.02,0:41:25.24,Chinese,,0,0,0,,然后你使用称为舍入的技术使有限的位数能够表示
Dialogue: 0,0:41:25.64,0:41:27.48,English,,0,0,0,,So the the effect
Dialogue: 0,0:41:25.64,0:41:27.48,Chinese,,0,0,0,,所以效果是
Dialogue: 0,0:41:28.20,0:41:35.22,English,,0,0,0,,Is like you doing say x + y is getting the precise value of x + y at an infinite number of bits
Dialogue: 0,0:41:28.20,0:41:35.22,Chinese,,0,0,0,,就像你说的那样，x + y 在无穷多的位数上得到 x + y 的精确值
Dialogue: 0,0:41:35.22,0:41:38.60,English,,0,0,0,,And then rounding into the available bits
Dialogue: 0,0:41:35.22,0:41:38.60,Chinese,,0,0,0,,然后四舍五入到可用位
Dialogue: 0,0:41:39.74,0:41:42.54,English,,0,0,0,,So let's look at this idea of rounding
Dialogue: 0,0:41:39.74,0:41:42.54,Chinese,,0,0,0,,那么让我们来看看这种舍入的想法
Dialogue: 0,0:41:42.54,0:41:47.46,English,,0,0,0,,This can be confusing the first you see it requires
Dialogue: 0,0:41:42.54,0:41:47.46,Chinese,,0,0,0,,一开始你可能会感到困惑
Dialogue: 0,0:41:48.02,0:41:52.62,English,,0,0,0,,You need to think about it carefully to get it right
Dialogue: 0,0:41:48.02,0:41:52.62,Chinese,,0,0,0,,你需要认真思考才能想明白
Dialogue: 0,0:41:52.76,0:41:55.54,English,,0,0,0,, Now in IEEE there's four different kinds of rounding
Dialogue: 0,0:41:52.76,0:41:55.54,Chinese,,0,0,0,,现在在 IEEE 中有四种不同的舍入方式
Dialogue: 0,0:41:55.54,0:42:05.88,English,,0,0,0,,There's round towards zero, round down route or round to minus infinity, Round up to which is rounding it to positive infinity
Dialogue: 0,0:41:55.54,0:42:05.88,Chinese,,0,0,0,,有向零舍入，向下舍入，向上舍入
Dialogue: 0,0:42:06.34,0:42:07.58,English,,0,0,0,, And then the default case
Dialogue: 0,0:42:06.34,0:42:07.58,Chinese,,0,0,0,,然后是默认情况
Dialogue: 0,0:42:07.58,0:42:09.94,English,,0,0,0,,Which is the only one route we're really going to consider
Dialogue: 0,0:42:07.58,0:42:09.94,Chinese,,0,0,0,,这是我们唯一一种需要去思考的方式
Dialogue: 0,0:42:10.24,0:42:12.92,English,,0,0,0,,Which is round to the nearest even
Dialogue: 0,0:42:10.24,0:42:12.92,Chinese,,0,0,0,,也就是向偶数舍入
Dialogue: 0,0:42:14.30,0:42:17.12,English,,0,0,0,,And this is what IEEE does by default
Dialogue: 0,0:42:14.30,0:42:17.12,Chinese,,0,0,0,,这就是 IEEE 制定的默认舍入规则
Dialogue: 0,0:42:17.98,0:42:18.84,English,,0,0,0,,You can,you can...
Dialogue: 0,0:42:17.98,0:42:18.84,Chinese,,0,0,0,,你可以，你可以......
Dialogue: 0,0:42:19.22,0:42:22.14,English,,0,0,0,,If you go into assembly language you can change the rounding
Dialogue: 0,0:42:19.22,0:42:22.14,Chinese,,0,0,0,,如果使用汇编语言，则可以更改舍入
Dialogue: 0,0:42:22.24,0:42:25.26,English,,0,0,0,,But we're not going to consider it any more.Yes question
Dialogue: 0,0:42:22.24,0:42:25.26,Chinese,,0,0,0,,但我们不会再考虑它了。好的，请提问
Dialogue: 0,0:42:25.26,0:42:30.18,English,,0,0,0,,[student speaking]
Dialogue: 0,0:42:25.26,0:42:30.18,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:42:30.18,0:42:33.54,English,,0,0,0,,I'm sorry
Dialogue: 0,0:42:30.18,0:42:33.54,Chinese,,0,0,0,,抱歉
Dialogue: 0,0:42:33.54,0:42:35.02,English,,0,0,0,,There is right because there's one
Dialogue: 0,0:42:33.54,0:42:35.02,Chinese,,0,0,0,,是的，因为有一个
Dialogue: 0,0:42:35.18,0:42:40.62,English,,0,0,0,,So the question is there a distinct negative infinity and a distinct positive infinity
Dialogue: 0,0:42:35.18,0:42:40.62,Chinese,,0,0,0,,所以问题是存在独特的负无穷大和独特的正无穷大
Dialogue: 0,0:42:40.92,0:42:47.42,English,,0,0,0,,And there is they have the exact same bit representation except the most significant bit for negative infinity is one
Dialogue: 0,0:42:40.92,0:42:47.42,Chinese,,0,0,0,,并且它们具有完全相同的位表示，除了负无穷大的符号位是 1
Dialogue: 0,0:42:48.30,0:42:50.30,English,,0,0,0,,And for positive infinity it's zero
Dialogue: 0,0:42:48.30,0:42:50.30,Chinese,,0,0,0,,对于正无穷大，它是零
Dialogue: 0,0:42:52.96,0:42:54.00,English,,0,0,0,, Yes question
Dialogue: 0,0:42:52.96,0:42:54.00,Chinese,,0,0,0,,请提问
Dialogue: 0,0:42:54.00,0:42:58.14,English,,0,0,0,,[student speaking]
Dialogue: 0,0:42:54.00,0:42:58.14,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:42:58.16,0:42:59.08,English,,0,0,0,,Why is nearest even?
Dialogue: 0,0:42:58.16,0:42:59.08,Chinese,,0,0,0,,为什么距离最近？
Dialogue: 0,0:42:59.08,0:43:00.20,English,,0,0,0,,[student speaking]
Dialogue: 0,0:42:59.08,0:43:00.20,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:43:00.20,0:43:02.04,English,,0,0,0,,Oh I'll show you,let me go over the others
Dialogue: 0,0:43:00.20,0:43:02.04,Chinese,,0,0,0,,哦，我会告诉你，让我们听听别人的问题
Dialogue: 0,0:43:02.22,0:43:04.12,English,,0,0,0,,The question was about nearest even.yes
Dialogue: 0,0:43:02.22,0:43:04.12,Chinese,,0,0,0,,问题是关于最近的偶数。请继续
Dialogue: 0,0:43:04.12,0:43:08.42,English,,0,0,0,,[student speaking]
Dialogue: 0,0:43:04.12,0:43:08.42,Chinese,,0,0,0,,[学生说话]
Dialogue: 0,0:43:08.42,0:43:10.48,English,,0,0,0,,Well that was the exception right
Dialogue: 0,0:43:08.42,0:43:10.48,Chinese,,0,0,0,,那是例外
Dialogue: 0,0:43:11.22,0:43:15.76,English,,0,0,0,,That's right so aside from infinity you can't do that comparison
Dialogue: 0,0:43:11.22,0:43:15.76,Chinese,,0,0,0,,这是正确的，除了无限大之外你不能做那种比较
Dialogue: 0,0:43:17.72,0:43:21.52,English,,0,0,0,, Positive infinity works right but not negative so you're right
Dialogue: 0,0:43:17.72,0:43:21.52,Chinese,,0,0,0,,正无穷有效，负无穷不可以，所以你是对的
Dialogue: 0,0:43:21.52,0:43:22.88,English,,0,0,0,,But good point
Dialogue: 0,0:43:21.52,0:43:22.88,Chinese,,0,0,0,,但这是个好问题
Dialogue: 0,0:43:24.02,0:43:29.08,English,,0,0,0,,Let's look at these rounding modes and i'll explain why the neuro steven works
Dialogue: 0,0:43:24.02,0:43:29.08,Chinese,,0,0,0,,让我们看看这些舍入模式，我将解释这些舍入模式的原理
Dialogue: 0,0:43:29.46,0:43:32.12,English,,0,0,0,,But let's look at so rounding towards zero what we're going to do
Dialogue: 0,0:43:29.46,0:43:32.12,Chinese,,0,0,0,,但是让我们看看向零舍入，我们要做些什么
Dialogue: 0,0:43:32.68,0:43:35.98,English,,0,0,0,,We'll use dollars,so we want to round to the nearest dollar
Dialogue: 0,0:43:32.68,0:43:35.98,Chinese,,0,0,0,,我们将使用美元举例，因此我们希望舍入到最接近的美元
Dialogue: 0,0:43:37.64,0:43:43.38,English,,0,0,0,,Towards zero just anything less than you just round towards zero
Dialogue: 0,0:43:37.64,0:43:43.38,Chinese,,0,0,0,,向零舍入，不到 1 你就舍去为 0
Dialogue: 0,0:43:43.38,0:43:45.14,English,,0,0,0,,So dollar forty would go to one
Dialogue: 0,0:43:43.38,0:43:45.14,Chinese,,0,0,0,,所以 1.4 美元会变成 1 美元
Dialogue: 0,0:43:45.42,0:43:47.22,English,,0,0,0,, A dollar sixty would go down to one
Dialogue: 0,0:43:45.42,0:43:47.22,Chinese,,0,0,0,,1.60 美元将舍为 1 美元
Dialogue: 0,0:43:47.52,0:43:48.86,English,,0,0,0,,A dollar fifty to one
Dialogue: 0,0:43:47.52,0:43:48.86,Chinese,,0,0,0,,1.50 美元舍为 1 美元
Dialogue: 0,0:43:49.30,0:43:50.70,English,,0,0,0,,Two fifty would go to two
Dialogue: 0,0:43:49.30,0:43:50.70,Chinese,,0,0,0,,2.50 美元会舍为 2 美元
Dialogue: 0,0:43:51.16,0:43:53.66,English,,0,0,0,,-1.50 you round towards zero
Dialogue: 0,0:43:51.16,0:43:53.66,Chinese,,0,0,0,,-1.50 你向零舍入
Dialogue: 0,0:43:53.68,0:43:56.27,English,,0,0,0,,So that would be negative one
Dialogue: 0,0:43:53.68,0:43:56.27,Chinese,,0,0,0,,所以这将是 -1 美元
Dialogue: 0,0:43:56.70,0:44:06.40,English,,0,0,0,,Round down or towards negative infinity It's similar the only difference is that for negative one point
Dialogue: 0,0:43:56.70,0:44:06.40,Chinese,,0,0,0,,向下舍入或朝向负无穷大舍入类似，唯一的区别是 -1.50
Dialogue: 0,0:44:07.04,0:44:12.47,English,,0,0,0,,-1.50 you drown towards negative infinity which would be -2
Dialogue: 0,0:44:07.04,0:44:12.47,Chinese,,0,0,0,,-1.50 你趋向负无穷大，即 -2
Dialogue: 0,0:44:12.84,0:44:18.56,English,,0,0,0,,Rounding up is symmetric to round down
Dialogue: 0,0:44:12.84,0:44:18.56,Chinese,,0,0,0,,向上舍入和向下舍入正好相反
Dialogue: 0,0:44:18.78,0:44:24.22,English,,0,0,0,, The interesting case is round to nearest even
Dialogue: 0,0:44:18.78,0:44:24.22,Chinese,,0,0,0,,有趣的情况是向偶数舍入
Dialogue: 0,0:44:25.24,0:44:29.58,English,,0,0,0,,Interesting because it's what IEEE floating point actually uses
Dialogue: 0,0:44:25.24,0:44:29.58,Chinese,,0,0,0,,因为它是 IEEE 浮点实际使用的舍入方式
Dialogue: 0,0:44:30.22,0:44:37.66,English,,0,0,0,,So the idea here if you have a value that's less than half then you round down
Dialogue: 0,0:44:30.22,0:44:37.66,Chinese,,0,0,0,,所以这里的想法是，如果你的值小于一半，那么你向下舍入
Dialogue: 0,0:44:37.94,0:44:43.61,English,,0,0,0,,If you have a value that's more than halfway between two dollar amounts you round up
Dialogue: 0,0:44:37.94,0:44:43.61,Chinese,,0,0,0,,如果你的值超过两美元的一半，那么你就会向上舍入
Dialogue: 0,0:44:44.38,0:44:50.60,English,,0,0,0,,The route that nearest even part comes when you have something that's exactly halfway
Dialogue: 0,0:44:44.38,0:44:50.60,Chinese,,0,0,0,,向偶数舍入可以区分正好在中间的部门
Dialogue: 0,0:44:51.50,0:44:54.28,English,,0,0,0,, Then what you do is you round towards the nearest even number
Dialogue: 0,0:44:51.50,0:44:54.28,Chinese,,0,0,0,,然后你要做的是你向最近的偶数舍入
Dialogue: 0,0:44:56.44,0:44:59.76,English,,0,0,0,, Okay so in this example a dollar forty
Dialogue: 0,0:44:56.44,0:44:59.76,Chinese,,0,0,0,,好的，在这个例子中 1.40 美元
Dialogue: 0,0:45:00.38,0:45:05.14,English,,0,0,0,,Because it's less than half rounds down to one dollar
Dialogue: 0,0:45:00.38,0:45:05.14,Chinese,,0,0,0,,因为它不到一半舍入到一美元
Dialogue: 0,0:45:05.44,0:45:08.74,English,,0,0,0,,A dollar sixty is greater than half so we round up to two dollars
Dialogue: 0,0:45:05.44,0:45:08.74,Chinese,,0,0,0,,1.60 美元大于一半，所以舍入为 2 美元
Dialogue: 0,0:45:10.16,0:45:16.24,English,,0,0,0,,A dollar fifty right in the middle we round up to the nearest even
Dialogue: 0,0:45:10.16,0:45:16.24,Chinese,,0,0,0,,1.50 美元我们四舍五入到最近的偶数
Dialogue: 0,0:45:16.50,0:45:18.48,English,,0,0,0,,Dollar amount which is two dollars
Dialogue: 0,0:45:16.50,0:45:18.48,Chinese,,0,0,0,,也就是舍入到两美元
Dialogue: 0,0:45:20.04,0:45:25.26,English,,0,0,0,, Two dollars and fifty cents we round down to the nearest even