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震惊:1 + 2 + 3 + ... = -1/12

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

告诉你令人震惊的结果
I’m gonna give you an astounding result
大吃一惊
Astounding?
对 令人震惊的结果
An astounding result
首先我会写下一个总和
So I’m gonna to write down a little sum
看看答案会是多少
and just gonna see what answer gives
1+2+3+4+…
1 + 2 + 3 + 4 + duh duh duh duh duh
加上所有的自然数
And I include all the natural numbers,
一直到无穷大
so all the way up to infinity
Brady 你觉得答案会是多少
So what do you reckon the answer to this is, Brady?
我觉得会趋向于无穷大
-Well I would say it would go to tend towards infinity. Yeah,
好像很在理
that makes sense, doesn’tit.
但是答案却是负十二分之一
The answer to this sum is — remarkably — minus a twelfth.
太不可思议
It’s amazing! I mean,
我第一次看到这个答案是在我刚开始学弦论的时候
I first saw this result when I start learning a bit of String Theory
更奇怪的是这个答案可以用在
And what’s even more bizarre is that this result is used
许多的物理学领域
in many areas of physics
这是一本Joe Polchinski 写的非常著名的弦论
This is a very well known string theory textbook by Joe Polchinski.
你看
As you can see,
在前半部分 书的22页
sort of quite early on, page 22,
这里注明了所有数的和
we have this statement here which is that the sum of all this is —
准确说所有的自然数之和是
– basically saying the sum of all the integers — – all natural numbers all the way up to infinity, is,
负十二分之一
minus a twelfth.
接下来我们证明一下
So we’re going to prove now,
不用让人难懂的黎曼函数
Without getting our knickers in a twist with Riemann zeta functions,
我们用一种
we gon na prove
十分简单的方法
in quite simple way,
证明所有的自然数之和是负十二分之一
why the sum of all the natural numbers is indeed minus a twelfth.
我们需要
to do that we’re gon na do this
几步来证明它
in a number of steps.
我们需要看看几个和
We’re gonna to look at a few different sums.
要看的第一个和
The first sum that I want to look at,
我将它称为S1
I’m gon na call it S1.
就是1-1+1-1+1-1一直加到无穷远
And it’s 1 -1 +1 -1 +1 -1 and so on.
原则上这就是我想看的第一组和
That’s the first sum I wanna look at in principle.
第二组我要看的和
The second sum I’m gon na look
是1-2+3-4然后一直
at is 1-2+3-4 and so on
如此以往持续下去
Carry on in that process all the way up.
而第三组当然就是
And the third one is of course going to be
我们很感兴趣的这个1+2+3+4+….
the one we’re really interested in which is 1+2+3+4… and so on.
我们要计算这三组不同的数的和
So we’re gonna evaluate all these 3 different sums.
那第一组的计算十分简单
Now the first one is really easy to evaluate.
我们需要添加一个数字
We need to attach a number.
现在这个答案是不是很清楚?
Clearly what is the answer to this now you take
你随便停在一个地方
You stop this at any point.
好吧 如果你停在
Okay if you stop it
奇数位置你得到的答案是1
at an odd point you’re going to get the answer 1.
偶数位置你得到的答案是0
You stop it at an even point you’ll get the answer 0, clearly.
很清楚对不对
That’s obvious right?
那当无穷远的时候答案会是多少呢
So what number are we gonna attachto this infinite sum?
最后是奇数位置还是偶数位置
Do we stop at an odd or an even point?
我们也不知道 所以我们取平均值
We don’t know so we take the average of the two.
答案就是二分之一
So the answer is a half.
也有其他方法可以证明
There’re other ways to prove
这个和是二分之一
that this sum is a half by the way,
如果你想..我们可以试试
which we can do if you want…
不 不 会有个链接
No, no,there’ll be a link, there’ll be a link there,
因为我们在这之前已经做过了 是的
We’ve done it before. ok,
如果不懂 看一下这链接
good…See the link there
是二分之一
Ok so this is a half
但我直觉上认为这是最简明的解释方式
but I’d think intuitively that’s the easiest way to say
要不是0要不是1
that you either get 0 or 1,
所以你只要取其平均
and therefore you just take the average…
所以这个数就是这些数字相加的和
So this is the natural number to attach to this sum.
当我们知道这个的时候我们高兴坏了
So once we know this we’re laughing, ok?
因为从这个结论我们可以得到想要的一切
Because from this we can achieve everything we want to achieve.
下一步就是证明这个和是多少
The next step is to find out what this sum is.
我将要做的就是……
So what I’m gonna do is…
我要算一下S2的两倍
I’m going to take two copies of this S2, ok.
S2+S2
So I’m gonna add it to itself.
那么两倍的S2就是等于……
So 2 times S2 is equal to…
让我直接写出来吧
Let me just write it out.
写两次S2
I will write it out twice
1-2+3-4……
1 -2 +3 -4 duh duh duh and so on.
然后我要把它们相加
And then I’m gonna add to it itself,
但是我要把它稍微改变一下位置
but I’m gonna shift it along a little bit
因此就是+1-2+3-4+……
So that’s +1-2+3-4 and so on
我已经把S2复制了一遍并且相加
I’ve just taken two copies and added them together.
用一种很棒的方法
In a particularly nice way
并且我将把它们整体稍微后移一位
and I just pushed this one along slightly on the bottom.
现在你可以看出来了
So now see what you get:
你把这个1和这个加起来 就是1
you take this 1 and this… This and this and I get 1.
那么1+空白=1 -2+1=-1
So 1 + nothing is 1 -2 +1=-1
3-2就是1
3 -2 is 1
-4+3是-1…
-4 +3 is -1 duh duh duh and so on.
以此推之
And I’m gonna keep getting this pattern.
稍等一下
So hang on a minute,
我就回到了最初的数字之和
I just got back to the sum that I started with.
我知道这个答案为二分之一
Which I know the answer to it: a half. Ok,
那么所以我知道这个和
so therefore I know this sum.
除以二 得到的答案是四分之一
So let’s divide through by this 2 and I get a quarter.
现在我们知道了这个和
So now I know that this sum,
第二组的和 那个符号经过换位的和
the second sum with somewhere the signs alternate
就等于四分之一
is actually equal to 1/4.
这个就是我们第二个等式的结果
So this is my second remarkable result.
现在准备就绪可以去证明负十二分之一了
Now I have everything I need to prove this crazy -1/12 thing, right. Ok,
开始吧
so let’s do it.
我将要拿这个减去这个
I’m going to take this one and I’m going to subtract this one. Ok,
那么我们将要用S-S2 先把它们写下来
so I’m going to subtract S2 from S. Let’s write them (?) Ok,
所以我先写S也就是
So I gonna write our S first which is 1 + 2
1+2+3+4+5…
+3 +4 +5 and so on. Ok.
接下来减去S2
And I’m going to subtract S2.
减号
So that’s minus
再加个括号 因为要减去所有
and let me just put a bracket. And now, because I’m gonna”minus” all of this
1-2+3-4…
1 -2 +3 -4 and so on.
1-1是多少呢
Ok 1 – 1 what’s that?
-0-0 的确就是0
-Zero-Zero, yeah, exactly. So I get nothing from that bit.
2减去-2等于4
2 – (-2) is… 4
3减去3是0
Ok 3 – 3… 0
4减去-4是8
4 – (-4) is… 8
等等
And so on.
接下来
And the next one,
5这得到的是零
I’m gon na get it from the 5, so I get nothing.
6这儿我们得到12
From the 6 here I get 12,
等等
and so on duh duh duh…
以此类推
And it proceeds in that way.
现在你看我们快算出来了
And now you can see we’re almost there now right.
看我得到的
Because look what I’ve got here.
4+8+12
I’ve got 4 + 8 + 12.
把4提出来
I’ll take a factor of 4 out.
它就是4乘以(1+2+3+…)
It’s 4 times 1 + 2 + 3 duh duh duh
好的这是我想要的和
Ok It’s my sum I want to…
现在我已经得到了这个公式
So now I’ve got a formula
所以这是4倍的S
So this is just 4 time S,
这就是我要的和
which is my sum.
现在我只需要解这个等式
Now I just solve this equation, right,
因为我知道S2的值
because I know what S2 is.
现在S减去已知的S2也就是1/4
So I have now the expression S minus… I know what S2 is: S2 is 1/4
等于4S
is equal to 4S. Ok,
把这个S放到另一边
let’s take S from either sides,
可得到负四分之一等于3S.
so I get -1/4 is equal to 3S.
也就是S等于负十二分之一
Which implies that S equals -1/12.
这样你信我了吧
Do that you believe me?
太令人震惊了 我太喜欢它了
That’s amazing, I love it… It’s so nice…
托尼
(?) Tony,
如果现在我拿一个计算器
if I’ve got a calculator out,
算1+2+3+4+5……
and wrote 1 + 2 + 3 + 4 +
而我一直坐在这里加到地老天荒
5 and I sat here until the end of everything,
然后按等于键
and then press =
我会得到负十二分之一吗?
-Will I get -1/12?
你的意思是到无穷大吗
-What do you mean by the end of everything?
但你做不到加到无穷大呀
You can’t do it till the end of everything, can you? So,
我知道它看起来
the point is I know it looks
有点数学骗局或者其它 但是
like a bit of mathematical hocus-pocus or thing, but…
但它不是
I tell you the truth it’s not,
我告诉你为什么我知道它不是
and I’ll tell you why we know it’s not:
我知道你认为我想太多物理了
And I know you think I’ve gone about physics too much.
但是我们知道它不是 是因为物理
But we know it’s not because of physics.
因为这种和在物理学中出现过
Because these kinds of sums appear in physics.
而在物理学上我们不会得到无穷大的答案
And in physics we don’t get infinite answers.
不可思议了 它仅……
It’s amazing. It’s amazing and it’s just…
你也知道 我在尝试想出
You know, I was trying to come up
一个主观的推论
with an intuitive reason for this,
但我做不到
and I just couldn’t.
哈哈哈 老实的说
*laughing*to be honest
你必须得做那种数学骗局才能看出来
You have to do the mathematicalhocus-pocus really to see it.
然后你必须得去相信
And then you just have to believe
你不是在测量物理意义上的无穷大
that you’re not measuring physical infinities in nature
依靠这个两个事实你应该可以相信这个结论了
and those two facts, I think, give you confidence in this result. But…
但主观上它明显有悖常理
it is clearly counter-intuitive.It’s counter-intuitive because intuitively,
因为主观上你想停止这个序列
you just want to stop the sequence,
而当你停止这个序列
and in the minute you stop the sequence…
然后你对这个结果的所有主观推论都没用了
then all your intuition for this result goes out the window
倘若我把1+2+3+4+5+…
So what if I do 1 + 2 + 3 + 4 + 5…
去Google上搜索
and I go up to a googleplex?
你得到了很大的数字…你不得到负十二分之一
You get a big number… you won’tget anything like -1/12.
你必须得加到无穷大 Brady
You’ll have to get to infinity, Brady
它是负数
It’s negative
我把所有的正数相加直到无穷大
I’ve added all these positive numbers together up to infinity
得到负十二分之一
and I’ve got -1/12.
但是它确实在很多不同的领域里有涉及到
But it does play a role in lots of different things
数字十二
that the number 12 and and it’s so
比如我说过预测出
for example as I said the calculations of
弦理论中的关键维度 也就是26个维度
the critical dimension in string theory, the 26 dimensions
来自于这个计算
comes from this calculation.

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视频概述

所有的自然数相加之和为负十二分之一

听录译者

收集自网络

翻译译者

11路

审核员

审核员_BZ

视频来源

https://www.youtube.com/watch?v=w-I6XTVZXww

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