Is there such a function??

Question

I always wondered this.

In math there is addition, multiplication, and powers, and they are reversed with subtraction, division, and logrithms, respectively. They are also connected like this:

in Powers, 4^5 is simply, 4*4*4*4*4,

and in multiplication, 4*5 is simply 4+4+4+4+4

So my question is...is there a function past powers, for example

4^4^4^4^4 can be simplified into a new type of function. Does this type of function exist? Obviously this is a ridiculously large number, but I thought that maybe there might be a theoretical use for it...

Is there a reverse function for "tetration"?

For example, logs revers powers so can you reverse a tetration?

Answer 1

Yes. There are actually an infinite number of them that go up faster and faster. But interestingly enough there is a function that grows faster than any of them, it is called the "Busy Beaver Function" (I kid you not). The Busy Beaver Function grows so fast that it is provably uncomputable.

http://en.wikipedia.org/wiki/Busy_beaver

I think I'm going to ask the conjugate to your question, that is, Is there a function that grows slower than addition?

Answer 2

Yes, absolutely! I've heard the names "tetration" and "hyperpowering", but there are others as well. Here are some references:

http://www.tetration.org/
http://mathworld.wolfram.com/PowerTower.html
http://tetration.itgo.com/
http://thinkzone.wlonk.com/MathFun/BigNum.htm

If you can get ahold of math magazines, Knoebel's "Exponentials Reiterated" is the standard work in the field.

Knuth's "arrow notation" is often used for notation:
http://mathworld.wolfram.com/ArrowNotation.html

Undoing tetration is extremely tricky. Geisler writes about this somewhere. When you have 2^(3*7) you can undo this "nicely" since (2^3)^7 = (2^7)^3, but the same does not hold for tetration. Futher, tetration hasn't been properly extended to the reals yet -- although Galidakis and others have tried. Geisler's site links to all the major attempts, some of which are more useful than others.

Answer 3

Yes. It is symbolized by two arrows pointing up. Please see link.