I have a question about Wikipedia Tetration page.
Do You find it not proper yet to single out the "mysterious" property of infinite tetration to turn certain real numbers x in x[4]n into complex as n->oo?
My speculative sentence would be related to speeds of growth:
Tetration is already so fast operation that it turns real numbers into complex;
That might be used to define tetration since anything that does not do it in infinite limit is not tetration yet, while faster operations must produce even more interesting transformations of numbers.
What would be the speed of tetration in Conways notation (in Cantor ordinal numbers) if exp(x) has speed \( \omega \), exp(exp(x) = \( \omega^{\omega} \) (The Book of Numbers, p.299)?
Would it be \( \omega^{\omega}^{\omega}............. \)? For any base or only base e?
That means speed of operation:
x[4]n is \( \omega[3]n \)?
Then we can say that to turn certain range of Real numbers into complex, the speed of operation has to be at least:
\( \omega[3]\infty \).
Thus, it will not be related to convergence or divergence of operations, but to transformations of numbers it can perform, and would that be a safe enough information to mention in Wikipedia?
Another interesting thing is that infinitesimals has negative growth rates (accroding to the same page) , so if tetration would be applied to infinitesimal:
dx[4]n means its growth rate would be \( -\omega[3]n \)
and
dx[4]oo will have growth rate \( -\omega[3]\infty \)
Which is interesting as it links negative Cantor Ordinal numbers and negative growth rates in general with infinitesimals.
The question is what transformations under such or more negative growth rates are infinitesimals able to undergo? What do we get as a result, what type of number?
Ivars
Moderators note: Moved from "FAQ discussion", which is about discussing the forum's FAQ and not about asking questions (that not even occur frequently)
Do You find it not proper yet to single out the "mysterious" property of infinite tetration to turn certain real numbers x in x[4]n into complex as n->oo?
My speculative sentence would be related to speeds of growth:
Tetration is already so fast operation that it turns real numbers into complex;
That might be used to define tetration since anything that does not do it in infinite limit is not tetration yet, while faster operations must produce even more interesting transformations of numbers.
What would be the speed of tetration in Conways notation (in Cantor ordinal numbers) if exp(x) has speed \( \omega \), exp(exp(x) = \( \omega^{\omega} \) (The Book of Numbers, p.299)?
Would it be \( \omega^{\omega}^{\omega}............. \)? For any base or only base e?
That means speed of operation:
x[4]n is \( \omega[3]n \)?
Then we can say that to turn certain range of Real numbers into complex, the speed of operation has to be at least:
\( \omega[3]\infty \).
Thus, it will not be related to convergence or divergence of operations, but to transformations of numbers it can perform, and would that be a safe enough information to mention in Wikipedia?
Another interesting thing is that infinitesimals has negative growth rates (accroding to the same page) , so if tetration would be applied to infinitesimal:
dx[4]n means its growth rate would be \( -\omega[3]n \)
and
dx[4]oo will have growth rate \( -\omega[3]\infty \)
Which is interesting as it links negative Cantor Ordinal numbers and negative growth rates in general with infinitesimals.
The question is what transformations under such or more negative growth rates are infinitesimals able to undergo? What do we get as a result, what type of number?
Ivars
Moderators note: Moved from "FAQ discussion", which is about discussing the forum's FAQ and not about asking questions (that not even occur frequently)
