Zeration

[marraco]

The interesting thing about this, is that if we then construct inverse elements w.r.t. to #, then we must admit new numbers that lie "before" \( -\infty \). This seems quite reminiscient of how constructing the inverse of addition created the negative numbers, the inverse of multiplication created the rational numbers

and numbers like \( \frac{1}{x} \) lie beyond the neutral number of product \( 1=\sqrt[\infty]{n} \)

, and the (radical) inverse of exponentiation created the real numbers (due to such constructs as \( \sqrt{2} \)). Combining the (radical) inverse of exponentiation with the negative numbers gave us the complex numbers. One can hardly wonder that the inverse of zeration would yield new numbers too. (It makes one wonder if the inverse of tetration would also create new numbers... I suspect it must've come up in this forum before, right?)

\( ^{\frac{1}{x}a} \) should have a special meaning, and also \( ^{\sqrt[x]{n}}a \) , \( ^{\sqrt[n]{x}}a \) , \( ^{\ln_nx}a \) and \( ^{\ln_xn}a \)