Stupidly Large Numbers

Stupidly Large Numbers

Standard arithmetic:
It is common knowledge that addition, subtraction, multiplication, and division make up basic arithmetic. Specifically, multiplying is repetition of action adding. So, `2.3*5` means you start with an amount of 0, and add 2.3 to the amount repetitively (for a count of 5 times).

`2.3 * 5 = 2.3+(2.3+(2.3+(2.3+(2.3+(0)))))`

Imagine you repeated multying, like with [`2*2*2*2*...2`], this is called an exponent (actually, it has too many names). Exponents are commonly used in topics like interest banking and epidemiology (the study of widespread diseases such as COVID-19). In particular, an exponent like 1.2^7 means you start with an amount of 1, and multiply 1.2 by the amount repetitively (for a count of 7 times). 1.2^7 repeats the action of multiplying by 1.2.

`1.2 ^ 7 = 1.2*(1.2*(1.2*(1.2*(1.2*(1.2*(1.2*(1)))))))`
*Exponentiation makes multiplication seem like nothing!*

However, for some crazy mathematicians, a few exponents just aren't good enough, so they need to repeat them with something called tetration. Tetration does not have many practical applications, and results in numbers that grow ***far too fast***. Like with exponents, 3 tetrate 6 means you start with an amount of 1, and exponentiate 3 by the amount repetitively (for a count of 6 times).

`3 tetrate 6 = 3^(3^(3^(3^(3^(3^(1))))))`
*Tetration makes exponentiation seem like nothing!*

*But who said we have to stop there?* ***Nobody did.***

Pentation exists, and `9 pentate 4` means you start with an amount of 1, and tetrate it by 9 repetitively (for a count of 4 times). These numbers are so big that you could never even begin to comprehend them.

`9 pentate 4 = 9 tetrate (9 tetrate (9 tetrate (9 tetrate (1))))`
*Pentation makes tetration seem like nothing!*

In fact, there is no limit to how many times we can repeat these actions on themselves in this manner. If you are a keen pattern finder, you might have noticed that the same action is used to convert multiplication to tetration as is used to convert exponentiation to tetration. This action is {start with an amount of 1 (or 0 for multiplication), do the previous math action, upon the amount, by the 1st number, repetitively (for a count of the 2nd number)}, and we can repeat it many times to get big numbers. So, allow me to show you SUPERation.

SUPERation is a truly insane repetition of math actions that can result in numbers that are practically infinite in size. 7 SUPERate 3 means you start with the action of adding 1, and an amount of 0, then repetitively make a new action: `{start with an amount of 1 (or 0 for multiplication), do the previous math action, upon the amount, by the 1st `[7]` number, repetitively (for a count of the 1st `[7]` number)}`, for a count of the 2nd number [3] times. SUPERation results in 0 if the 1st number is 0 and the second number is 0, 1 or 2. SUPERation results in 1 if the 1st number is 0 and the 2nd number is more than 2. Here are some examples:

```
1 SUPERate 1 = 1 + 1
2 SUPERate 1 = 2 + 2
3 SUPERate 1 = 3 + 3
1 SUPERate 2 = 1 × 1
2 SUPERate 2 = 2 × 2
3 SUPERate 2 = 3 × 3
1 SUPERate 3 = 1 ^ 1
3 SUPERate 3 = 3 ^ 3
5 SUPERate 3 = 5 ^ 5
1 SUPERate 4 = 1 tetrate 1
3 SUPERate 4 = 3 tetrate 3
5 SUPERate 4 = 5 tetrate 5
10 SUPERate 10 = practically Infinity
```

And you thought ee2000 was a lot!

Let's imagine a magical world, far far away… a place where SUPERation is not enough. The people of this world decided to invent SUPER-DUPERation. SUPER-DUPERation is the repetition SUPERation. So, `4 SUPER-DUPERate 5` means you start with an amount of 1, and SUPERate it by 4 repetitively (a count of 5 times).

`4 SUPER-DUPERate 5 = 4 SUPERate (4 SUPERate (4 SUPERate (4 SUPERate (4 SUPERate (1)))))`

*We can go deeper*

Let's imagine a yet more magical world where SUPER-DUPERation is not enough, and so they invent GIGA-DUPERation. GIGA-DUPERation is the repetition SUPER-DUPERation. So, `11 GIGA-DUPERate 5` means you start with an amount of 1, and SUPERate it by 4 repetitively (a count of 5 times).

`11 GIGA-DUPERate 4 = 11 SUPER-DUPERate (11 SUPER-DUPERate (11 SUPER-DUPERate (11 SUPER-DUPERate (1))))`

This pattern is pretty clear, and we could continue on with TERA-DUPERation, PETA-DUPERation, and even EXA-DUPERation

The mountain to infinity is so tall that in the time it takes **light** from the top to reach the base, this universe would have died and been reborn an almost unenumerable number of times. I present to you:

**ALPHAtation**
ALPHAtation is the repetition of this action: `{start with an amount of 1, do the previous math action, upon the amount, by the 1st number, repetitively (for a count of the 1st number)}`, where the 1st action is SUPERation, and the 0th action is simply returning 1.

Here are some examples of possible outputs:

```
1 APLHAtate 0 = 1
0 ALPHAtate 0 = 1
1 APLHAtate 1 = 1 SUPERate 1
2 APLHAtate 1 = 2 SUPERate 2
3 APLHAtate 1 = 3 SUPERate 3
1 APLHAtate 2 = 1 SUPER-DUPERate 1
3 APLHAtate 2 = 3 SUPER-DUPERate 3
5 APLHAtate 2 = 5 SUPER-DUPERate 5
1 APLHAtate 3 = 1 GIGA-DUPERate 1
5 APLHAtate 3 = 5 GIGA-DUPERate 5
9 APLHAtate 3 = 9 GIGA-DUPERate 9
1000 APLHAtate 1000 = This must be Infinity, right? *Not quote*
```

The repetition of this APHAtation, or BETAtation as we shall call it, is the next logical step up the mountain. If you are a keen or smart pattern finder you will have already guessed that `13 BETAtate 7` achieved by starting with 1 and ALPHAtating it by 7 repetitively (for a count of 7 times).

`13 BETAtate 7 = 13 ALPHAtate (13 ALPHAtate (13 ALPHAtate (13 ALPHAtate (13 ALPHAtate (13 ALPHAtate (13 ALPHAtate (1)))))))`, which is a number that I doubt even the most intelligent AI could even understand.

***Onwards!*** We can repeat BETAtation in the same fashion to get GAMMAtation, and then repeat GAMMAtation in the same manner once again to get DELTAtation. There is no limit to how many times things can be repeated in this manner, so we can probably create some kind of ULTRAtation. *Let's go ahead and do that.*

**ULTRAtation**
ULTRAtation is the repetition of the following action: `{start with an amount of 1, do the previous math action, upon the amount, by the 1st number, repetitively (for a count of the 1st number)}`, where the 1st action is ALPHAtation and the 0th action is just returning 1.

Here are some examples of ULTRAtation:
```
1 ULTRAtate 0 = 1
0 ULTRAtate 0 = 1
1 ULTRAtate 1 = 1 ALPHAtate 1
2 ULTRAtate 1 = 2 ALPHAtate 2
3 ULTRAtate 1 = 3 ALPHAtate 3
1 ULTRAtate 2 = 1 BETAtate 1
3 ULTRAtate 2 = 3 BETAtate 3
5 ULTRAtate 2 = 5 BETAtate 5
1 ULTRAtate 3 = 1 GIGAtate 1
5 ULTRAtate 3 = 5 GIGAtate 5
9 ULTRAtate 3 = 9 GIGAtate 9
1000 APLHAtate 1000 = **Does this ever end?!** *I don't think so, my friend.*
```

**The final action**, *for today*

ULTRAtation is the same to ALPHAtation as ALPHAtation is to SUPERtation, and *(mostly)* as SUPERtation is to addition.

With that in mind, we can think of going from SUPERtation to ALPHAtation as an action, and going from SUPERtation to ULTRAtation as a repetition of that same action. By repeating this action a certain number of times, we get INFINITATION.