# About The Best Formula In The World

Many people find the following formula shocking:

They think that I didn't learn mathematic at school, or that I am
becoming mad.

Well, honestly, I am not sure that they are wrong on the last point.

But there is another possibility:

Maybe your teachers didn't tell you **all** the truth?

After all, didn't they tell you that an atom is a kernel made of big balls named neutrons and protons,
with little balls orbiting around named electrons?

Here is another shocking formula:

1 + 2 + 3 + 4 + ... = -1/12

Hey! Don't leave! Come back! I will give you the explanation...

I will even give you

**two** explanations.

No, better, I will give you

**THREE** explanations!

## The Short Explanation

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

*e*^{iπ} = -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!

## The Long Explanation

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*) = ∑ x^{k}, *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

*e*^{iπ} = -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.

## The Geek Explanation

A geek does not need to learn mathematics.

He has a computer.

Run this in Gambas:

```
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 ...

Easy... My simple program shows that it converges to the -1 value.

Writing programs is so easier than mathematics... :-)

## Conclusion

After these three explanations, I hope you will be convinced.

For information,

*e*^{iπ} = -1 was the preferred Formula of

Richard Feynman.
I just added my little series because I am a computer scientist and I like powers of two. :-)