09/30/2007, 07:33 PM
bo198214 Wrote:Nobody denies that the tetraroot is the inverse of \( {}^nx \) (by definition) however to define \( {}^{1/n}x \) as being the tetraroot is quite arbitrary
Why? What is "arbitrary" about it? And if it's "arbitrary", in what "sense" is it "arbitrary"? Again, if the definition of the n-th order tetraroot as \( {^{1/n}}x \) is "arbitrary", surely the tetraroot can be defined better as \( {^{m/n}}x \) for some m,n. Do you care to define m and n in some other consistent way for the tetraroot?
bo198214 Wrote:and additionally does not coincide with our other methods.
So? Has a divine judge decided on the validity of any of the proposed methods so far?
bo198214 Wrote:Sorry Ioannis, but this rather proves that it is the wrong definition. As \( {}^xe \) should be a function continuous in \( x \) it must
\( \lim_{n\to\infty}{}^{1/n}e={}^0e=1\neq e^{1/e} \)
Sorry, I am not convinced that \( {^{x}}e \) should even *be* continuous (even whether it exists), despite the agreement between all the current methods. All the methods so far (including mine), exhibit a certain "artificiality" if you wish, which is apparent from the complexity which reveals itself when one asks a very simple question: HOW do you define the tetration function for RATIONAL values.
If you cannot tell me how the tetration function is defined at the rationals, then you *cannot* tell me how it's defined at the reals.
If you want to debate the above, then I will ask you the following:
how do you define for example
\( {^{7/11}}e \)? or
\( {^{2/3}}e \)?
Sorry, definitions via decimal expansions won't cut it, because decimal expansions suffer from non-uniqueness. So, if you tell me for example, take Andrew's or Gottfried's or your method and "input" 0.666... or 0.6363..., and then see what the function outputs, this is already suffering badly as a definition.
