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#### 我们需要用到π的多少位

How Many Digits of Pi Do We Really Need?

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,

and call me an idiot for not releasing this video then

but I’m just not a math nerd.

I’m an engineer and all this hype

for a mathematical constant goes a little over my head.

Internet drama has been born out of people for and against π.

A quick search of YouTube will yield tons,

literally tons of videos, teaching you how

to memorize the digits of π.

The Guinness World Record for reciting π,

with the most digits has changed hands multiple

times with the current champion, Rajveer Meena of India,

managing 70,000 digits in 10 hours.
Rajveer干的非常棒
Fair play Rajveer, fair play.

Now go use that brain for something actually useful.

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.

It just keeps going and going and going, ( oh god ) just keeps going.

It’s infinite and it doesn’t repeat itself.

So you just have to brute force memorise it.

And while these lads are spending days learning

as many digits off as possible.

Mathematicians are working away at calculating the new longest string of digits.
π的推导很简单
π is found fairly simply

by dividing a circle’ s circumference by its diameter.

So the diameter of a circle can fit

into its circumference 3.14159 26535
89793 23846 26433 83279 50588 4197……
89793 23846 26433 83279 50288 4197……

Okay.

So you get the point, that is simple on paper,

but in practice it is far more difficult.

Measuring a circle’s circumference accurately is practically impossible.

Even your best attempts with a measuring tape will be off

by the thickness of the measuring tape.

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

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

attempts at snarky humour.

He started estimating π with squares,

which sounds unconventional,

but it makes a lot of sense.

In the mind of archimedes a circle was simply a polygon

with an infinite number of sides,

so by starting with a polygon

with fewer sides we can get a very rough estimate of π by

calculating it’s ratio of circumferenceto diameter.

Let’s start by placing a square inside

of a circle with its corners touching the circles sides.

We can then find the ratio of this low estimate of π

by dividing the sum of the squares sides

by its diagonal diameter, which in this case is (4/root 2): 2.828

This is our lowest estimate of π.

Now let’s place a square with it’s side touching the circles side,

and this time we will divide the sum of the squares sides

by the length of one of the squares sides,

which gives us 4, the number of sides a square has.

In boths of these cases we are

dividing by the circles diameter, that figure is accurate.

What we are lacking in accuracy is the measurement of the circumference.

The inside squares perimeter is too small,

and the outside squares perimeter is too big,

but now we know π lies between these two numbers.

Now we just need to narrow it down,

and we can do that by increasing the number of sides

of the polygon.

Each time we add a side those two figures will get more accurate,

and eventually the two numbers

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.

This continued on for a couple of centuries,

with mathematicians out exhausting each other

until eventually someone started using computers

and now we are just exhausting them.

We now have 2.7 trillion digits of π calculated,

and for some reason through these millenia,

an engineer never stood up and yelled: “stop.

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.

Because in the end of the day,

that’s what π is used for.

It’s for calculating the circumference and area of circles.

It’s for converting degrees to radians,

and this is where the arts of mathematics

and engineering differ.

While mathematicians obsess over accuracy to the trillionth digit,

engineers aim for “ good enough ”,

and good enough turns out to be 3.141592653589793 for the people that

work with the biggest circles: NASA.

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.

Say we want to calculate the circumference of a circle

What difference would adding one extra digit of π provide?

That’s the difference between 15 decimal places and 16 decimal places.

This is actually tough enough to calculate

with a calculator or excel as both are limited

in the number of decimal places they can calculate.

So using this online high precision

calculator we can find that the circumference of this

circle will be about 136 billion kilometres,

and if we use on extra digit it will be 8.67

millimeters closer to the actual value.

That’s tiny and we just travelled out ofthe solar system.

That’s why JPL and NASA don’t

need any more figures and the chances are you don’t

need anymore that 3.1416.

So I’m gonna be the engineer that yellsstop.

We have better things to be worrying about,
π是无限的
π is literally infinite and we are never

going to reach the end.

What is the point.

If you don’t want to waste your life completing pointless challenges.

You could try the challenges on Brilliant instead.

Where they present you with daily challenges that you can solve

with Brilliant’s community.
Brilliant刚刚发布iOS版本的离线课程
Brilliant just released offline courses on iOS,

so you can work on learning new things

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

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,

and it is an excellent first language for new programmers.

It can be used for everything from video games to data visualization to machine learning.

I used it in my own Master ’s

thesis to create custom plug-ins for my finite element analysis software,

but I had to teach it

to myself and work through constant errors.

This course will show you how to use

Python to create intricate drawings, coded messages

and beautiful data plots, while teaching you some essential core programming concepts.

This is just one of many courses on Brilliant,

with more courses due to released soon on

things like automotive engineering.

If I have inspired you and you want to educate yourself,

then go to brilliant.org/RealEngineering

And the first 500 people that go to

so you can get full access to all their courses as well as

the entire daily challenges archive.

As always thanks for watching and thank you to all my Patreon supporters.

If you would like to see more from me the links to my instagram, twitter, subscribe it!
Discord服务器在视频简介中列出
and discord server are below.