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 A more consistent definition of tetration of tetration for rational exponents bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 10/01/2007, 06:24 PM Quote:*IF* we assume that tetration is defined as ${^0x}=1$. If we don't assume that (in particular if we assume that ${^0x}=x^{1/x}$) ... ... then we have a more serious problem. If you define ${^0x}$ to be different from 1, say ${^0x}=a$, then ${^1x}=x^a$. And in common use of the word tetration ${^1x}=x$ (hopefully you dont want to change this for the sake of continuity of your construction). So if you accept this then any (real) value of $a$ different from 1 poses the contradiction $x^a=x$. You see that it is necessary to define ${^0x}=1$. I mean there is no need to define it for ${^0x}=1$, it rather shows the pattern. Your solution is also not continuous at exponent 1. Consider the sequence $1+1/n$, by your definition ${^{1+1/n}x}=x^{^{1/n}x}$. Then $\lim_{n\to\infty} {^{1+1/n}x}=x^{x^{1/x}}\neq x={^1x}$. « 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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