So called 'hyper exponents

[meemoe_uk]

so called 'hyper exponents...

...or powers or indicies'

Hi,
The other day I was thinking about a maths concept I've called 'hyper exponents'. adlib symbol ^^

Description of a hyper exponent.

Mutiplication is to addition, as
Exponents are to multiplication, as
Hyper exponents are to exponents.

e.g.
2*4 = 2+2+2+2 = 8
2^4 = 2*2*2*2 = 16
2^^4 = 2^2^2^2 = 65536

Of course, this hyperizing is unbounded, e.g. you can have hyper-hyper exponents

2^^^4 = 2^^2^^2^^2 = 2^65536 = 2*10^19728

Interesting I think.

roots.

The stadard square root ( exponent root ) of 2 is such that...
x*x = 2
x= 1.412 to 3 d.p.

The hyper square root of 2 is such that...
x^x = 2
x= 1.560 to 3.d.p

The hyper hyper square root of 2 is such that...
x^^x = 2

Or phrased awkwardly in english,
Which number, when put in as all the components of a 'power tower' , and also is the height of the 'power tower' makes the power tower equal to 2?
I don't know. There's an interesting concept of a power tower of non integer height, which at first seems silly, but likewise, exponents, which represent repeated mutiplication, can be non integer.

I feel this hyper power stuff is a little known, but large branch of maths.

e.g. differential stuff.
We know that e^x ' = e^x
But what is the equivalent e for hyper exponents? Also, I think hyper exponents might need separate logarithms as well.

hehe, we know working out that i^i = 0.208 to 3 d.p. is interesting, but what is i^^i ?

Does anyone know anything about 'hyper exponents' ?

edit Note : I've since seen the symbol ^^ used for power tower, which is sensible, so I've changed my original hyper exponent symbols to ^^.

 

[theperthvan]

That's cool, and now I have something to think about for awhile.

I hope you don't mind but I'm going to steal your idea and write a PhD thesis on it.
Just kidding.

Cheers,

 

[DeadWolfe]

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

 

[meemoe_uk]

ah, I've seen Knuth's arrow notation before. I didn't realize it could be expressed in power towers like that. That hyper4 operator link on the wiki link was good, but sufficiently complicated to scare me off ever going back to this idea.

 

[Gib Z]

For the Solution to your question, Look up Lambert W Function, that will help i think.