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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, 10:29 PM bo198214 Wrote:It is in the same sense arbitrary as say that I define ${}^{1/n}x=1$. No rule justifies that ${}^{1/n}x$ should be the inverse of ${}^nx$. Let me try to explain the "obvious" rule behind such a definition. Call the inverse operator of multiplication @t. Then the inverse of multiplication must satisfy: $(x*n)@t=x$ consequently it is easily seen that $@t=*(1/n)$, which is the only operator which fits the bill. Similarly, call the inverse operator of exponentiation @@k. Then the inverse of exponentiation must satisfy: $(x^n)@@k=x$ consequently it is esily seen that $@@k={\^}(1/n)$, which is the only operator which fits the bill. Again similarly, call the inverse operator of tetration @@@m. Then the inverse of tetration must satisfy: $(@@@m)({^n}x)=x$ from which it follows that $@@@m=$tetraroot of order n of x, since the tetraroot is the only operator which satisfies: $(tetraroot-n)(^{n}x)=x$ Now, the very subtle problem which I guess nobody sees (for some strange reason) is that if the operator for tetrating to (1/n) *is NOT* the same operator as that of the tetraroot of order n, then we have an operator discrepancy at a very low level in the hierarchy of operators: ${^{1/n}}({^{n}x)=/= x$ I am not going to argue more about it. Whoever "sees" it, great. Whoever doesn't, great again. bo198214 Wrote:If you not even demand that ${}^xe$ is continuous, what will then remain?Sorry, "demanding" and "constructing" are not the same as "existing". Besides, there's no teling whether tetration as defined using tetraroots is or is not continuous. « 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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