08/30/2025, 11:05 PM
Hi, I'm a new user and I wanted to make share with you some questions and stuff related to Zeration (rank 0 hyper-operation) and its inverse, Deltation.
On this forum I discovered GFR's post and I looked at the paper, and I also used the wayback machine to download an old paper from 1996 on hyper-operations from KAR's website.
If you are not familiar with Zeration and Deltation I suggest that you read the file that GFR attached in his post.
One question I was asking myself was what 0.5 * (△0) would be equal to, and if it could be reduced to a single number. In KAR's paper, this is resolved by introducing a new unit "j", like the imaginary unit for (-1)^(0.5). We then have 2*j = △0. The operation 0.5 * (△0) also has 2 solutions: △j and j.
We still don't know what 0.35801 * (△0), or with any other arbitrary coefficient, would equal to. I made a formula for any real number as a coefficient for what this would be equal to.
I came up with x * (△0) = j*2*(2n+1)*x with n being an integer. I verified this formula with (-1)^a = b^(△0 * a)
Verification: for (-1)^(1/2) we have b^(△0 * 1/2) = {b^j ; b^(△j)}, so we must have b^j = i
b is a real number that is not zero.
For (-1)^(1/3) we know that there will be 3 solutions, therefore, △0 * 1/3 will have 3 solutions. These are: △0 ; 2/3 * j ; -2/3 * j. All other solutions can be reduced to one of these 3.
We also have an interesting rule that completes one that already exists:
a^0 = 1
a^(△0) = -1
a^j = i
a^(△j) = -i
This makes sense because the absolute value is 1 in every case, and the exponent is linked in some way to the number 0.
If we have b = a*(△0), for any real a. c^b will always have an absolute value of 1.
I also discovered the rule △j = -j, which I think KAR has not mentioned in his paper.
Proof:
△j + △j = 2*△j = △0
△j + (-j) = △0 + j - j = △0
(-j) + (-j) = -(2*j) = -△0 = △0
This is as far as I went and there are many gaps in the algebraic structure, KAR used a new operation below Zeration for defining ln(△a) and created new units for other operations, and he mentioned that there is an infinite amount of new sets of numbers. It would be interesting to see if this can be generalized in some way.
I still have unanswered questions about this algebraic structure:
- How can we extend Zeration to the complex set? What would (3*i) ° 4 be equal to? What about j ° 2 ?
- What is the proof for Zeration's commutativity? I saw somewhere on the forum that there was a proof for that but I couldn't find it.
- What about other hypothetical extensions I didn't mention?
- Have I made any mistakes? What did other people discover?
Rayanso
On this forum I discovered GFR's post and I looked at the paper, and I also used the wayback machine to download an old paper from 1996 on hyper-operations from KAR's website.
If you are not familiar with Zeration and Deltation I suggest that you read the file that GFR attached in his post.
One question I was asking myself was what 0.5 * (△0) would be equal to, and if it could be reduced to a single number. In KAR's paper, this is resolved by introducing a new unit "j", like the imaginary unit for (-1)^(0.5). We then have 2*j = △0. The operation 0.5 * (△0) also has 2 solutions: △j and j.
We still don't know what 0.35801 * (△0), or with any other arbitrary coefficient, would equal to. I made a formula for any real number as a coefficient for what this would be equal to.
I came up with x * (△0) = j*2*(2n+1)*x with n being an integer. I verified this formula with (-1)^a = b^(△0 * a)
Verification: for (-1)^(1/2) we have b^(△0 * 1/2) = {b^j ; b^(△j)}, so we must have b^j = i
b is a real number that is not zero.
For (-1)^(1/3) we know that there will be 3 solutions, therefore, △0 * 1/3 will have 3 solutions. These are: △0 ; 2/3 * j ; -2/3 * j. All other solutions can be reduced to one of these 3.
We also have an interesting rule that completes one that already exists:
a^0 = 1
a^(△0) = -1
a^j = i
a^(△j) = -i
This makes sense because the absolute value is 1 in every case, and the exponent is linked in some way to the number 0.
If we have b = a*(△0), for any real a. c^b will always have an absolute value of 1.
I also discovered the rule △j = -j, which I think KAR has not mentioned in his paper.
Proof:
△j + △j = 2*△j = △0
△j + (-j) = △0 + j - j = △0
(-j) + (-j) = -(2*j) = -△0 = △0
This is as far as I went and there are many gaps in the algebraic structure, KAR used a new operation below Zeration for defining ln(△a) and created new units for other operations, and he mentioned that there is an infinite amount of new sets of numbers. It would be interesting to see if this can be generalized in some way.
I still have unanswered questions about this algebraic structure:
- How can we extend Zeration to the complex set? What would (3*i) ° 4 be equal to? What about j ° 2 ?
- What is the proof for Zeration's commutativity? I saw somewhere on the forum that there was a proof for that but I couldn't find it.
- What about other hypothetical extensions I didn't mention?
- Have I made any mistakes? What did other people discover?
Rayanso