What Are Hyper Exponents and How Do They Extend Mathematical Operations?

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Discussion Overview

The discussion revolves around the concept of "hyper exponents," a term introduced by a participant to describe an extension of mathematical operations beyond traditional exponents. The scope includes theoretical exploration and potential applications in mathematics, particularly in relation to power towers and roots.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant proposes that hyper exponents can be understood as an extension of exponents, where multiplication relates to addition, exponents relate to multiplication, and hyper exponents relate to exponents, illustrated with examples.
  • The participant introduces the idea of hyper-hyper exponents and discusses their implications, including calculations involving large numbers.
  • There is a mention of hyper square roots and the challenge of defining a hyper hyper square root, raising questions about non-integer heights in power towers.
  • Another participant expresses interest in the concept and humorously suggests writing a PhD thesis on it.
  • A third participant references Knuth's arrow notation, indicating familiarity with related concepts but expressing intimidation by their complexity.
  • A suggestion is made to look up the Lambert W Function as a potential tool for addressing questions related to hyper exponents.

Areas of Agreement / Disagreement

Participants express interest in the concept of hyper exponents, but there is no consensus on its definitions or implications. Multiple viewpoints and references to existing mathematical notations indicate an ongoing exploration rather than agreement.

Contextual Notes

The discussion includes references to mathematical functions and notations that may require further exploration for clarity. The definitions and applications of hyper exponents remain unresolved, with participants suggesting various related concepts without reaching a definitive understanding.

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

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