# Tetration Forum

Full Version: Question about speed of growth
You're currently viewing a stripped down version of our content. View the full version with proper formatting.
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)
Hmm..

Ok, and if we have speed of growth of operation defined, can we say that it is a derivative of operation vs. something?

Like

d(x[4]n)/dx= w[3]n ? Or will it be d(x[4]n)/dx= (w+lnx/x )[3]n?
d(x[4]oo/dx= w[4]oo ? or d(x[4]oo/dx =(w+ln(x)/x)[3]oo ?

and my beloved:

d(h(e^(pi/2)))/d(e^pi/2) = (w+pi/2)* w[4]oo ?? ( the coefficient pi/2 may not be correct , it is just intuitive placement that if base is not e, somewhere we have to see it, the speed of growth has to be faster if x>e; it may be also (w+pi/2)[3]oo).

And again, since h(e^(pi/2))) = i

d(i)/d(e^pi/2) = (w+pi/2)[4]oo or (w+I*pi/2)[4]oo

Probably these questions have been solved in more satisfactory manner without differentiating imaginary unit and i^(1/i) in infinitary calculus mentioned by Conway in his Book of numbers , but I could not find accessible readable reference.

Excuse me for allowing myself to post such unchecked conjectures.

Ivars
Ivars Wrote: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?

Dont know what about you are speaking. If you have two positive real numbers $x$ and $y$ then $x^y$ is again a positive real number, hence the limit of $x[4]n$,$n\to\infty$ is a real number, if existing.
If you chose a negative number $x$ then of course you may get a complex number as the result of exponentiation.

Quote:My speculative sentence would be related to speeds of growth:

Tetration is already so fast operation that it turns real numbers into complex;

You see thats already a property of triation=exponentiation which turns to real numbers into a complex one.

Quote:d(x[4]n)/dx= w[3]n ? Or will it be d(x[4]n)/dx= (w+lnx/x )[3]n?

No need to go into speculation, thats an easy exercise which you can solve for yourself:

$\frac{dx[4]2}{dx}=\frac{dx^x}{dx}={x}^{x} \left( \ln \left( x \right) +1 \right)$
$\frac{dx[4]3}{dx}=\frac{dx^{x^x}}{d x}={x}^{{x}^{x}} \left( {x}^{x} \left( \ln \left( x \right) +1 \right)
\ln \left( x \right) +{\frac {{x}^{x}}{x}} \right)$

$\frac{dx[4]4}{dx}=\frac{dx^{x^{x^x}}}{dx}={x}^{{x}^{{x}^{x}}} \left( {x}^{{x}^{x}} \left( {x}^{x} \left( \ln
\left( x \right) +1 \right) \ln \left( x \right) +{\frac {{x}^{x}}{x
}} \right) \ln \left( x \right) +{\frac {{x}^{{x}^{x}}}{x}} \right)$

obviously $\frac{dx[4]n}{dx}\neq w^n$ and $\frac{d x[4]n}{dx}\neq \left(w+\frac{\ln(x)}{x}\right)^n$ whatever $w$ is. What is it btw?
bo198214 Wrote:obviously $\frac{dx[4]n}{dx}\neq w^n$ and $\frac{d x[4]n}{dx}\neq \left(w+\frac{\ln(x)}{x}\right)^n$ whatever $w$ is. What is it btw?

Thanks again. This will help to put it on better footing or dismiss. I was aware that differentiation by x is wrong here since speed is mentioned in relation to x-> infinity. So differentiation, if possible at all , must be either d/d(?) leaving it open, d/d (oo), d/d (i) or something else. Perhaps time calculus as it allows to pick out subsets from real numbers by using graininess function mju(t), so differentiation vs. mixed discrete continuous variable is possible. What if mju(t) would be w(t) of mju(w) ( speculation again).

w=$\omega$ is Cantors Ordinal number and its relation to speed of growth of functions f(x) as x->oo is mentioned in Conways Book of Numbers page 299.

It mentions Infinitary calculus which uses these notions-i have not been able to find a good reference yet how this infinitary calculus is constructed.

I thought that since cardinals/ordinals have well developed theoretic bacground up to continium hypothesis, applying them to the speed of growth of hyperoperations may simplify proofs of some basic identities and allow classification of hyperoperations as performing certain transformations of number types if speed is fast enough or within some limits.

Idea will be only applicable to certain subsets of reals and other numbers known today , but that will better illuminate differences between these subsets. e.g. why region x<e^(1/e) differs from x>e^(1/e) etc., why negative reals can be turned into complex easily while positive can not and in the region <e^(1/e) even infinite tetration is not fast enough.

For example, i strongly feel (wihout proof) that this will lead to finding of a speed that turns certain numbers ( e.g subset of integers) into surreal numbers, and maybe more.

Ivars
I stumbled upon idea of tetra factorial in another thread Tetra-factorial, which galathea kindly expanded there into infinity of options, related to rates of growth of functions.

tetra-factorial is in my language $n(4)!= 2^{3^{4^{...n}}}$

I was looking for combinatorial content of such factorial and found that according to quickfur here:

The Exploding tree function

It seems that tetra factorial could be related to branching tree chain combinatorics . Quickfur even has series similar to Ramanujan in his paper:

Quote:This in turn transforms into a chain of length $(m^{2m})^{2m^{2m}} + m^{2m} = m^{2m^{2m+1}} + m^{2m}$.
.

Further, in his thread, quickfur relates these growth rates to Cantor ordinals.

Tetration and higher order operations on transfinite ordinals

Now, despite all this, I have to confess I have problem understanding even the basic ideas in his branching tree paper. When searching on interenet, i also could not hit a BASIC source that would explain the terminology, at least, and also give introduction to the combinatorics of such trees.

Could someone please give some advice where to look or what to search for in internet?

Ivars