natural numbers

– conflictmeditationcoach

Dear Me,

The following entry is a conjunction of thoughts from a confused mind; exploring his mind palace as he tries to come up with something to write about. Somehow, he succeeds. This “success” lead to the following text, filled with weird concepts and random mathematical facts.

Usually, I am full of ideas. However, sometimes I get stuck. My thoughts fixate on boring topics which often are not worth writing about. A so-called writers-block. Sometimes it is a number. E.g. pi, tau or even Avogadro’s number. This week it is the sum of all natural numbers. Integers from one to infinity. The answer to this is incredibly indefinite. I have seen proofs of the answer in textbooks, videos and movies. I have, however, never found an in-depth explanation to why this is. Only proofs.

On top of my head, here are some of the documentation that I have read/watched:

I have proven it to several of my peers. I even proved it to some of my teachers, including my maths teacher. Since neither of them understood Reimann Zeta functions, I explained it in a simpler way. Instead of using complex functions, I taught them the concept using only a bit of algebra, addition, subtraction, multiplication and division. Regardless of the simplicity, I was asked the same question multiple times: “How can the sum of positive numbers be negative? I mean, surely, if I was to type it out on my calculator I would never reach a negative number.” Although this seems right, it is not the case. The truth is that you would not be able to do this. Since this number occurs in several areas of physics, infinity is not a possibility. Also, according to modern laws of physics nothing can be infinite. This is probably a good time to reveal the answer to reveal the answer. The number is essentially not that complicated. It is -1/12. This result is as Numberphile puts it:

“Astounding”.

In order to prove this similarly to how I taught my peers, I will use three variables/constants.

X = 1 - 1 + 1 - ... Y = 1 - 2 + 3 - ... Z = 1 + 2 + 3 + ...

All of these are continuous, as shown by the three dots. They resemble three different values. Two of which I will use to prove the third. Firstly, if we look at X: how can we find the value of this?

X = 1 - 1 + 1 - ... ^ ^ 1 0

If we stop at any given point, the result will either be one or zero; Two values. In order to get the result of the continuous function, we simply calculate the average of all possible values. Then, we get the result:

1 + 0 ----- 2

Now that we know the value of X, we can use this to calculate Y.

X = 1 - 1 + 1 - ... = 1/2 Y = 1 - 2 + 3 - ... = ? Z = 1 + 2 + 3 + ... = ?

In order to do this, we will observe what happens when we multiply Y by two. We know that 2Y is the same as Y + Y. Let us use this information and calculate the value of 2Y:

2Y = 1 - 2 + 3 - ... + [ 1 - 2 + ... ] = 1 - 1 + 1 - ...

By doing so we discover that 2Y is equal to X. We can now use the value of X to calculate Y:

2Y = X -> 2Y = 1/2 -> Y = 1/4

We have now calculated the value of Y. We will use the result to prove our initial theorem.

X = 1 - 1 + 1 - ... = 1/2 Y = 1 - 2 + 3 - ... = 1/4 Z = 1 + 2 + 3 + ... = ?

Our statement is symbolized with the letter, Z. Finally, to prove that the result is -1/12, we will observe what happens when we subtract Y from Z:

1 + 2 + 3 + 4 + 5 + ... - [ 1 - 2 + 3 - 4 + 5 - ... ] = 0 + 4 + 0 + 8 + 0 + ... (12, 16 and so on.)

The result shows us that the value has a common factor of four. Therefore, it is the equivalent of four Z since: one times four is four, two times four is eight and so on. We can, then, isolate Z by subtracting Z from both sides. Afterwards, we divide by three to get the value of one Z.

Z - Y = 4Z -> Z - 1/4 = 4Z -> -1/4 = 3Z -> Z = -1/12

We have now proven the sum of all natural numbers to be -1/12. This shows us how entangled maths can be. However, it also shows us that there always is an intuitive explanation to even the most complicated of maths. As I said to one of my sceptic friends:

“I agree with you. The result may not be logical, but the justification is.”

For the sake of randomness here is a poll:

Mind palaces. I like my mind palace. It is a comfortable place. In there I keep my best memories. I attach important information to these memories. More precise: locations in my memories. They make it possible for me to remember digits of pi, physics-related formulas and other information that I enjoy thinking about. It is basically what I do when I am bored. I sit down and go through my hotel. The special thing about my hotel is that it has an infinite amount of rooms. Behind each door is a tightly memory. Within each memory are several pieces of information. There is even a café. Guess what I store there? I remember going to room 217 one day while talking with my teacher about programming. He had made a mistake so I corrected him. Of course this memory is now stored behind a door too, or rather a key. I keep memories of memories in a special place. The lobby. It is a nice place with sofas, plants, a counter and the keys to each room. Again, mind palaces are great. You should definitely get one.

I am now at the conclusion of this weird abstraction of thoughts; I have no idea of what I was thinking while writing this, but somehow it ended up describing a bit of maths, science and psychology. Thank you for reading this, possibly, confusing article. I hope you enjoyed it. I promise that the overall topic will be more precise next time.

Sincerely, Me.

TOG

The Theory of Grammar

natural numbers