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我们需要用到π的多少位

How Many Digits of Pi Do We Really Need?

这期《真实工程》由Brilliant网站赞助
This episode of “Real Engineering” is brought you by “brilliant”.
Brilliant网站致力于解决问题 并且教你如何像工程师一样思考问题
A problem solving website that teaches you to think like an engineer.
π节就在上个月
It was π day last month,
我没有在π节发布这个视频真是太傻了
and call me an idiot for not releasing this video then
但我并不是个数学狂
but I’m just not a math nerd.
我只是一个工程师
I’m an engineer and all this hype
对这类数学常数的宣传有点超出我的能力范围
for a mathematical constant goes a little over my head.
网络剧中一直有许多支持或反对π的人
Internet drama has been born out of people for and against π.
只要在YouTube上快速搜索一下
A quick search of YouTube will yield tons,
就会有大量的视频
literally tons of videos, teaching you how
教你如何去记忆π的值
to memorize the digits of π.
关于背诵π的吉尼斯世界纪录
The Guinness World Record for reciting π,
已经被多次刷新
with the most digits has changed hands multiple
现在的冠军是印度的Rajveer Meena
times with the current champion, Rajveer Meena of India,
他在10小时内记住了π的70,000位数
managing 70,000 digits in 10 hours.
Rajveer干的非常棒
Fair play Rajveer, fair play.
你总算成功了
You have finally made it.
现在你可以用大脑思考些真正有用的事了吧
Now go use that brain for something actually useful.
不过玩笑之余 这种记忆力真是令人印象深刻
Jokes aside, this is a genuinely impressive act of memory.
π是一个迷人的常数 任何记忆技巧都对它束手无策
π is a fascinating mathematical constant, that defies any tricks for memorisation.
它会一直一直一直循环下去
It just keeps going and going and going, ( oh god ) just keeps going.
它有无限位 而且永不循环
It’s infinite and it doesn’t repeat itself.
所以你只能死记它
So you just have to brute force memorise it.
在这些小伙子花费了很多天
And while these lads are spending days learning
背尽可能多的位数的同时
as many digits off as possible.
数学家们也在努力计算最新 最长的数字串
Mathematicians are working away at calculating the new longest string of digits.
π的推导很简单
π is found fairly simply
只需将圆的周长除以其直径就可以
by dividing a circle’ s circumference by its diameter.
所以圆的周长与
So the diameter of a circle can fit
直径的比值为3.14159 26535
into its circumference 3.14159 26535
89793 23846 26433 83279 50588 4197……
89793 23846 26433 83279 50288 4197……
好吧
Okay.
你应该明白了 虽然纸上谈兵很简单
So you get the point, that is simple on paper,
但实际计算出π很难
but in practice it is far more difficult.
精确测量圆周长几乎是不可能的
Measuring a circle’s circumference accurately is practically impossible.
即使你尝试使用卷尺
Even your best attempts with a measuring tape will be off
也会受到卷尺厚度的影响
by the thickness of the measuring tape.
阿基米德是第一个能精确计算出
Archimedes was the first to calculate π with any level
π的任意一位数的人 他有一个独创的方法
of accuracy with an ingenious method
叫做“穷举法”
called the “ method of exhaustion ”,
“穷举法”和试图让这个视频显得风趣幽默
and yes it was nearly as exhausting as this video’s
一样令人精疲力竭
attempts at snarky humour.
起初他用正方形估算π
He started estimating π with squares,
虽然听上去不合常理
which sounds unconventional,
但是他的方法行得通
but it makes a lot of sense.
阿基米德的想法是
In the mind of archimedes a circle was simply a polygon
圆只是一个有无数条边的多边形
with an infinite number of sides,
所以从一个只有几条边的多边形开始
so by starting with a polygon
通过计算周长与直径的比值
with fewer sides we can get a very rough estimate of π by
我们就可以估算出一个非常粗略的π
calculating it’s ratio of circumferenceto diameter.
让我们从一个圆的内接正方形算起
Let’s start by placing a square inside
内接正方形的顶点都在圆上
of a circle with its corners touching the circles sides.
用正方形的边长之和除以对角线的长度
We can then find the ratio of this low estimate of π
我们就可以得出一个π的估计值
by dividing the sum of the squares sides
也就是4除以√2:约为2.828
by its diagonal diameter, which in this case is (4/root 2): 2.828
这样估算出的是π的下限
This is our lowest estimate of π.
现在我们做一个圆的外切正方形
Now let’s place a square with it’s side touching the circles side,
这一次我们用正方形的周长
and this time we will divide the sum of the squares sides
除以正方形一条边长
by the length of one of the squares sides,
得到4 也就是正方形的边数
which gives us 4, the number of sides a square has.
在这两种情况中
In boths of these cases we are
我们用来做除数的圆的直径 数值是准确的
dividing by the circles diameter, that figure is accurate.
但圆的周长没有被精确测量出来
What we are lacking in accuracy is the measurement of the circumference.
内接正方形的周长比圆的周长小
The inside squares perimeter is too small,
而外接正方形的周长又比圆周长大
and the outside squares perimeter is too big,
但现在我们知道 π的值介于这两个数之间
but now we know π lies between these two numbers.
接下来只需要缩小差距
Now we just need to narrow it down,
我们可以通过增加多边形边的数量
and we can do that by increasing the number of sides
来做到这一点
of the polygon.
每当我们多添一个边 π的值都会更加准确
Each time we add a side those two figures will get more accurate,
最终这两个数
and eventually the two numbers
将开始出现相同的数字
will start overlapping in their digits.
π值小数点的后几位最初就是这样计算出来的
This is how we got our first known digits of π.
这个方法沿用了几个世纪
This continued on for a couple of centuries,
使数学家们都筋疲力竭
with mathematicians out exhausting each other
直到最后有人开始使用电脑
until eventually someone started using computers
而现在 我们只需要麻烦电脑了
and now we are just exhausting them.
目前我们已经计算出了π的2.7万亿个小数
We now have 2.7 trillion digits of π calculated,
然而经过千年来的计算
and for some reason through these millenia,
从没有工程师站出来
an engineer never stood up and yelled: “stop.
向那些计算π的狂热爱好者叫停:
For the love of all that is holy stop.
“我们已经有足够的位数了!”
We have enough digits.
“我们不需要更多!”
We don’t need any more.
“我们没有足够大的 能证明这个值的准确性的圆!”
We don’t have any circles big enough to justify this level of accuracy,
“洗洗睡吧你们这群疯子!”
go to bed you lunatics.
因为最终来说
Because in the end of the day,
那就是π的用途
that’s what π is used for.
它是用来计算圆圈的周长和面积的
It’s for calculating the circumference and area of circles.
它是用来将角度转换为弧度的
It’s for converting degrees to radians,
这就是数学的艺术
and this is where the arts of mathematics
而工程学就不一样了
and engineering differ.
虽然数学家们对π的百万兆精确度十分痴迷
While mathematicians obsess over accuracy to the trillionth digit,
但工程师更注重 “实用”
engineers aim for “ good enough ”,
事实是3.141592653589793
and good enough turns out to be 3.141592653589793 for the people that
就足够那些在NASA计算最大的圆的人使用了
work with the biggest circles: NASA.
我们把旅行者一号与地球的距离
Let’s take the distance to Voyager 1, which is currently about 21.7 billion (21690753480.975746)
(现在约217亿千米)当做半径
kilometres away in interstellar space, as our radius.
如果我们想计算一个
Say we want to calculate the circumference of a circle
半径这么大的圆的周长
with a radius this large.
再精确一位的π有什么区别吗
What difference would adding one extra digit of π provide?
那是15位小数和16位小数的区别
That’s the difference between 15 decimal places and 16 decimal places.
实际上 无论是用计算器还是Excel
This is actually tough enough to calculate
这是都很难计算的
with a calculator or excel as both are limited
它们能够算到的小数位数都十分有限
in the number of decimal places they can calculate.
所以运用在线的高精度计算器
So using this online high precision
我们可以发现这个圆周长
calculator we can find that the circumference of this
大约1360亿公里
circle will be about 136 billion kilometres,
如果我们再精确一位
and if we use on extra digit it will be 8.67
就会比实际值再接近8.67毫米
millimeters closer to the actual value.
那太小了 而我们才仅仅走出太阳系
That’s tiny and we just travelled out ofthe solar system.
这就是为什么JPL和NASA
That’s why JPL and NASA don’t
不需要小数点后的更多位
need any more figures and the chances are you don’t
在你计算时 多数情况下3.1416就足够了
need anymore that 3.1416.
所以我会成为那个叫停的工程师
So I’m gonna be the engineer that yellsstop.
请停下来
Please stop.
我们有更多有用的事情需要担心呢
We have better things to be worrying about,
π是无限的
π is literally infinite and we are never
我们永远算不完
going to reach the end.
这就是重点
What is the point.
如果你不想浪费你的生活完成无意义的挑战
If you don’t want to waste your life completing pointless challenges.
你可以试试Brilliant上的挑战
You could try the challenges on Brilliant instead.
他们每天会提供一个挑战题目
Where they present you with daily challenges that you can solve
你可以用在Brilliant内学到的知识来完成
with Brilliant’s community.
Brilliant刚刚发布iOS版本的离线课程
Brilliant just released offline courses on iOS,
所以你可以在火车甚至飞机上
so you can work on learning new things
学习新的知识
even on an underground train or a plane.
Brilliant最近也发布了精彩的Python课程
Brilliant also recently released their fantastic course
课程名称叫Programming with Python
on Python coding called Programming with Python.
Python是最广泛运用的编程语言之一
Python is one of the most widely used programming languages,
对于新手程序员来说 这是很棒的第一语言
and it is an excellent first language for new programmers.
它可以用在电子游戏 数据可视化以及机器学习技术等一切事物上
It can be used for everything from video games to data visualization to machine learning.
我写硕士论文时就用到了Python
I used it in my own Master ’s
为我的有限元分析软件创建自定义插件
thesis to create custom plug-ins for my finite element analysis software,
但我不得不通过
but I had to teach it
不断出现的错误自学
to myself and work through constant errors.
本课程将为你展示
This course will show you how to use
如何使用Python创建复杂的图纸 信息加密
Python to create intricate drawings, coded messages
和美丽的数据图 同时教你一些基本的核心编程概念
and beautiful data plots, while teaching you some essential core programming concepts.
这只是Brilliant的许多课程之一
This is just one of many courses on Brilliant,
近期将发布更多课程
with more courses due to released soon on
比如关于自动化工程的课程
things like automotive engineering.
如果本视频对你有启发 或者你想自学
If I have inspired you and you want to educate yourself,
请访问brilliant.org/RealEngineering
then go to brilliant.org/RealEngineering
免费注册
and sign up for free.
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订阅年费会员 你就可以自由访问所有的课程
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以及参与所有的日常挑战
the entire daily challenges archive.
一如既往感谢您的观看 感谢所有在Patreon支持我的人
As always thanks for watching and thank you to all my Patreon supporters.
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视频概述

数学家已经算到π的百亿位,但我们用的到吗?我们需要用到π的多少位呢?

听录译者

收集自网络

翻译译者

一千九

审核员

审核员BY

视频来源

https://www.youtube.com/watch?v=l-vHGf4j90Y

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