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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 09/30/2007, 11:10 PM UVIR Wrote:$(x*n)@t=x$ consequently it is easily seen that $@t=*(1/n)$ As I already explained before it is not "consequently easily seen" but a consequence of the law $(xa)b=x(ab)$ and similarly for exponentiation of the law $(x^a)^b=x^{ab}$. This law holds for natural $a$ and $b$ and if we demand it to hold for fractional $a$ and and $b$ too, then necessarily $\frac{1}{n} x$ is the inverse of $nx$ and $x^{1/n}$ is the inverse of $x^n$ by: $(x\frac{1}{n})n=x(\frac{1}{n}n)=x1=x$ and $(x^{1/n})^n=x^{(1/n)n}=x^1=x$ but we have not ${^a(^bx)}={^{ab}x}$ for natural a and b and hence can not generally demand it for fractional a and b. Quote: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: ${^{1/n}}({^{n}x)\neq x$ Yes, well we have to live with it. But except that we have to deal with more different operations on the tetra level, I see no problems arising from the inequality. We also have to live with for example: ${^2}({^3 x})\neq {^6x}$. Quote:Besides, there's no teling whether tetration as defined using tetraroots is or is not continuous. There is a telling. It is not continuous at (the exponent) 0 (if we assume that tetration is defined on natural numbered exponents in the usual way, particularly ${^0x}=1$). You showed already that $\lim_{n\to\infty} {^{1/n}x}=x^{1/x}\neq 1={^0x}$ for $x>1$ in your 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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