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 A more consistent definition of tetration of tetration for rational exponents UVIR Junior Fellow Posts: 16 Threads: 2 Joined: Aug 2007 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. « Next Oldest | Next Newest »

 Messages In This Thread A more consistent definition of tetration of tetration for rational exponents - by UVIR - 09/29/2007, 11:56 PM RE: A more consistent definition of tetration of tetration for rational exponents - by Gottfried - 09/30/2007, 10:39 AM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 09/30/2007, 11:26 AM RE: A more consistent definition of tetration of tetration for rational exponents - by GFR - 09/30/2007, 11:38 AM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 09/30/2007, 04:41 PM RE: A more consistent definition of tetration of tetration for rational exponents - by bo198214 - 09/30/2007, 06:18 PM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 09/30/2007, 07:33 PM RE: A more consistent definition of tetration of tetration for rational exponents - by bo198214 - 09/30/2007, 08:06 PM RE: A more consistent definition of tetration of tetration for rational exponents - by Gottfried - 09/30/2007, 09:46 PM RE: A more consistent definition of tetration of tetration for rational exponents - by bo198214 - 09/30/2007, 10:23 PM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 09/30/2007, 10:29 PM RE: A more consistent definition of tetration of tetration for rational exponents - by bo198214 - 09/30/2007, 11:10 PM RE: A more consistent definition of tetration of tetration for rational exponents - by GFR - 10/01/2007, 12:55 AM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 10/01/2007, 05:56 PM RE: A more consistent definition of tetration of tetration for rational exponents - by bo198214 - 10/01/2007, 06:24 PM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 10/01/2007, 07:32 PM RE: A more consistent definition of tetration of tetration for rational exponents - by GFR - 10/02/2007, 01:40 PM RE: A more consistent definition of tetration of tetration for rational exponents - by andydude - 10/07/2007, 03:37 PM RE: A more consistent definition of tetration of tetration for rational exponents - by andydude - 10/20/2007, 07:57 PM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 10/20/2007, 09:05 PM RE: A more consistent definition of tetration of tetration for rational exponents - by andydude - 10/21/2007, 07:19 AM RE: A more consistent definition of tetration of tetration for rational exponents - by UVIR - 10/21/2007, 10:47 PM

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