关于世上最好的公式
很多人发现下面的公式不正常：
他们觉得我在学校没学过数学，或者我变蠢了。好，说实话，我不确保他们的后一个观点是错误的。
但是另一个却是可能的：
也许你的老师没有告诉你*全部的*事实？
难道他们没有告诉你一个原子是由叫做中子和质子的大球组成的核心和外面环形轨道上名为电子的小球组成的？
这是另一个不正常的公式：
1 + 2 + 3 + 4 + ... = 1/12
嗨！别离开！回来！我来给你解释...
我甚至将给你*两种*解释。
不，还是给你*三种*解释比较好！
简短的解释
Your confusion comes from the fact that you supposed that this formula is calculated in
R, the set of real numbers.
But were you told about the
p-adic numbers?
In a few words, the
p/-adic numbers are an extension of **Q**, the set of rational numbers, but not done
in the same way that leads to
R.
The difference is in the way the absolute value is computed, and that implies that the right part
of the formula is a convergent series in the
p/-adic numbers set when /p = 2. And the limit of this series is...
-1.
And as
ei**π = -1, the formula is correct.
As for the left part of the formula, you may ask if it has sense with p-adic numbers?
To be honest, I am not sure, but if I understand the Wikipedia article above correctly, it seems that it is ok.
If you can confirm that, please tell me!
长的解释
In this explanation, your confusion comes from the fact that in mathematics
you often write an algorithm and the value of a function the same way when the value of the
function is calculated with that algorithm
most of the time.
I know, that is not very clear. I want to talk about
analytic functions
and their
continuation.
In a few word, the right part of the formula is not just an infinite sum. It is actually the value
of a function defined this way:
f(x) = ∑ xk, k:0→+∞
You will tell me, this function is defined only for
x ∈ ]0,1[!
I will agree.
But if
f has a specific form, i.e. if
f is an
analytic function, which is the case there,
then we can define a function
g that is the
continuation of
f.
That function
g takes the same value as
f in the interval where
f
is defined, and takes other values on area where
f is not defined.
Moreover, this continuation is unique!
See the Wikipedia articles above for more details.
So, in our specific case, our infinite sum is actually the continuation of
f(
x), and that continuation takes the value -1 when
x = 2!
And as
ei**π = -1, the formula is correct.
As for the second shocking formula, this is the result of a continuation too. That formula was discovered by
Riemann, and discovered
again a few years later by
Ramanujan.
低级解释
低级解释不需要学习数学。
需要有一台计算机。
在Gambas中运行下面程序：
DIM S AS Integer
DIM P AS Integer
P = 1
DO
S += P
PRINT S;;
P += P
LOOP
1 3 7 15 31 63 127 ... 1073741823 2147483647 -1 -1 -1 -1 -1 -1 ...
很笨的问题。
我的简单程序显示，它的值趋于-1。
编写程序比数学更简单...:-)
结论
这三个解释之后，我希望你不再有怀疑。
For information,
ei**π = -1 is the prefered Formula of
Richard Feynman.
I just added my little series because I am a computer scientist and I like powers of two. :-)
下一次，也许我会告诉你
0/
0的值。:-)
Or I will explain that void and light do not exist, and that time is a pure illusion like temperature. :-p