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I have a couple of ideas about the hyper 0 operator.

I have see a couple of different definitions of it that don't agree with each other and nobody seems to have a definite solution.
Some people say that a[0]b= b+1. Others have sort of a piece wise approach that is discontinuous and frankly doesn't make much sense.
I have found a different approach (that may be completely wrong).

The solution that i have found is that a[0]b= (a+b)/a + a
The initial conditions that i worked with were as follows:
1. a[0]a=a+2
2. 2[0]4=5 (2[x]4 = 2[x+1]3)
3. This operation should be something that is fundamental (this is not really an initial condition but rather something i figured to be true)

Now what is operation really means is something that may not be inherently obvious from the the definition a[0]b= (a+b)/a + a. I am going to explain using fingers. To start you would put a fingers in one hand and b fingers in the other. Next you would figure out how many groups of a fingers you had total. Then you would perform the sum (# of groups) + (# in each group ) which is the same as (# of groups) + a.

For example 3[0]3
You have 6 fingers total. So you have 2 groups of 3 fingers. So the answer is 2+3 = 5

3[0]2
You have 5 fingers total.That is 1 and 2/3s groups of 3. So the answer is 5/3+3 = 14/3

An error that some people may see is that a[0](a[0]a) DOES NOT= a+3
however, I feel that a[0]a[0]a = a+3.
a[0]b[0]c using "the finger method" would equal (a+b+c)/a + a.

I hope that you guys will not have to struggle too hard to understand what I am saying and I also hope that all of this is not completely wrong

Thanks.
I see another error. I have 10 fingers, not 6.