• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 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

 Possibly Related Threads... Thread Author Replies Views Last Post Tetration research: 1986 - 1991 Daniel 0 175 03/20/2020, 02:47 AM Last Post: Daniel umbral and tetration tetration101 3 373 03/02/2020, 06:28 AM Last Post: Daniel tetration from alternative fixed point sheldonison 22 32,221 12/24/2019, 06:26 AM Last Post: Daniel Constructing real tetration solutions Daniel 4 996 12/24/2019, 12:10 AM Last Post: sheldonison Thoughts on hyper-operations of rational but non-integer orders? VSO 2 950 09/09/2019, 10:38 PM Last Post: tommy1729 I need somebody to help me clarifiy the elementary knowledge for tetration Ember Edison 13 5,060 08/26/2019, 01:44 PM Last Post: Ember Edison Complex Tetration, to base exp(1/e) Ember Edison 7 3,031 08/14/2019, 09:15 AM Last Post: sheldonison Minimum axioms to extend tetration Daniel 0 569 07/21/2019, 11:38 AM Last Post: Daniel Critic of Tetration.org Daniel 2 1,083 07/21/2019, 01:32 AM Last Post: Daniel Dimesional analysis and tetration tetration101 1 1,073 06/23/2019, 08:43 PM Last Post: Chenjesu

Users browsing this thread: 1 Guest(s)