User:Naruyoko/Thinking about Real Index Hyperoperations

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The Pattern in Whole Number Indexed Hyperoperations
The hyperoperations can be defined as follows:

\(\text{hyper}c(a,b)=\left\{\begin{array}{lr}a+b&c=1\\\left\{\begin{array}{lr}0&c=2\\1&c>2\end{array}\right.&b=0\\\text{hyper}c-1(a,\text{hyper}c(a,b-1))&b>0\land c>1\end{array}\right.\)

Alternatively, they could be defined in the following two ways:

\(\text{hyper}c(a,b)=\left\{\begin{array}{lr}a+1&c=0\\\left\{\begin{array}{lr}0&c=2\\1&c>2\end{array}\right.&b=0\\\text{hyper}c-1(a,\text{hyper}c(a,b-1))&b>0\land c>0\end{array}\right.\)

\(\text{hyper}c(a,b)=\left\{\begin{array}{lr}b+1&c=0\\\left\{\begin{array}{lr}a&c=1\\0&c=2\\1&c>2\end{array}\right.&b=0\\\text{hyper}c-1(a,\text{hyper}c(a,b-1))&b>0\land c>0\end{array}\right.\) --~

Linear Approximation of the Operands
Modifying the last interpretation of hyperoperations:

\(\text{hyper}c(a,b)=\left\{\begin{array}{lr}b+1&c=0\\\left\{\begin{array}{lr}a+b&c=1\\ab&c=2\\1+b&c>2\end{array}\right.&-1< b\leq0\\\text{hyper}c-1(a,\text{hyper}c(a,b-1))&b>0\land c>0\end{array}\right.\)

It might look pointless using addition and multiplication to define addition and multiplication. I don't care, it is there to generalize the rules. --~