A Pattern I Noticed: -1 Inverts Everything?

[NegativeOne]

Hey, I was thinking about math (Which I do a lot), and I noticed a pattern:

Applying -1 to any number through an operation changes it to the inverse of the operation it repeats:

Adding -1 changes to _____________ inverse
Multiplying by -1 changes to additive inverse
Raising to the power of -1 changes to multiplicative inverse
Raising to the hyper-power of -1 changes to exponential inverse
Raising to the hyper-hyper-power of -1 changes to the hyper-exponential inverse

The most basic operation is adding 1. Adding -1 inverts adding 1.

Adding numbers is repeated additions of 1. Multiplying by -1 inverts it to it's additive inverse such that adding it simplifies it to it's additive identity, 0.

Multiplying numbers is repeated additions of numbers. Raising to the -1 power inverts it to it's multiplicative inverse such that multiplying it simplifies to the multiplicative identity, 1.

Exponents is repeated multiplying of numbers. Raising to the -1 hyper-power inverts it to it's (What I call) exponential inverse such that raising X to X^^-1 you would get the exponential identity, _________.

And so on and so fourth.
I apologize if it's difficult to understand (And it's probably impossible to understand the exponents part, because addition and multiplication have the commutative property, so I didn't know whether it would be x^(x^^-1) or (x^^-1)^x).

I just wanted to show a pattern I saw, and I apologize if it is in the wrong section. Number patterns was the most similar thing I could find to this (Operational patterns).

Does anyone know of some sort of theory about this? I'm usually pessimistic (Thus my name is a double entendre), so I doubt I discovered some new pattern...?

 

[HallsofIvy]

It is true that -x is the additive inverse of x and that $x^{-1}$ is the multiplicative inverse. The use of "-1" for other "inverses" is just convention.

 

[NegativeOne]

Well, I figured out that x^^-1 = 1. Of course I could be wrong, but...

You can extract x^6 to xx^5, and you can extract x^^6 to x^(x^^5).

You can change x^6 to (x^7)/x, without changing value. Just the same, you can change x^^6 to (I meant this to mean x root) x√(x^^7)

Why? We know that exponents is repeated multiplication, and when we multiply numbers, we add to the exponents. We know that when we divide (Inverse of multiplying), we subtract from the exponents. We can apply the same logic to hyper-exponents if we change multiplying to exponents and division to roots (Since roots are the opposite of exponents).

We know that H-exponents (Hyper-Exponents) is repeated exponentiation, which means when raising (x^^4)^(X^^5), we get x^^9. If instead of exponentiation, we could take its root (Inverse), in which case we should subtract: (x squared root) (x^2)√x^^5 = x^^3

That being said, x^^-1 = x√(x^^0) = x√1 = 1

1 can be said to be the exponential identity, because x^1 = x

 

[HallsofIvy]

What do you mean by "^^"? That is not standard notation.

You seem to be making a big deal out of what is, basically, arithmetic.

The symbols used for operations, how we write identities and inverses, is, as I said before, convention. Yes, the convention for inverse functions if $f^{-1}$. There is no "deep" mathematics involved.

 

[M65]

I can tell you this: you might be interested in abstract algebra.