# another number group?

I noticed that for each basic increasing binary function (addition, multiplication, and exponentiation) its inverse (or just a inverse) of certain values adds more number types to the number line (or plane):

subtraction --> all integers
division --> rationals
roots --> complex numbers

so can this be extended even further onto the inverse tetration, making a new number group, or can the complex numbers accommodate this? Furthermore would these numbers break normal algebraic rules such as commutativity

• I would imagine that it depends how you define tetration on non-integers. If you choose a holomorphic function, though, then we stay in the complex numbers, since holomorphic functions are all surjective (with the possible exception of $1$ point), so we don't need to add more numbers get solutions. It also strikes me as a less natural construction - complex numbers are least defined as a field with a solution to $x^2=-1$. Given that tetration isn't "natural" in a field, it's hard to say what (algebraic) properties we want of its inverse (Also, the word is "commutativity", not "commutability") – Milo Brandt Sep 14 '15 at 23:26
• As you remember, complex exponentiation is a "clock-arithmetic", modular with $2 \pi î$, so we have "classical" problems when we want to invert the operation and especially when we want to do fractional iterates. What I've seen one time was an article of R. Corless et. al. (which also had main contributions to the theory of the Lambert-W-function) where they tried to introduce the concept of "winding-numbers" to keep track when repeatedly exponentiate/logarithmize and thus to overcome the limitations of the "clock arithmetic" (the article is online). I didn't really cought their result... – Gottfried Helms Sep 15 '15 at 14:55
• @GottfriedHelms could you please provide the link, thanks – tox123 Sep 15 '15 at 21:08
• – Gottfried Helms Sep 16 '15 at 2:38