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meemoe_uk

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**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 ^^.

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