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

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In summary, the conversation discusses a concept called 'hyper exponents' which is an extension of exponents and involves a power tower of non-integer height. The conversation also delves into the use of this concept in mathematics, specifically in differential equations and the Lambert W Function. The conversation ends with one person jokingly suggesting to write a PhD thesis on this topic.
  • #1
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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  • #2
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,
 
  • #4
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.
 
  • #5
For the Solution to your question, Look up Lambert W Function, that will help i think.
 

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

What are hyper exponents?

Hyper exponents, also known as tetration or power towers, are a type of mathematical operation that involves repeatedly exponentiating a number to itself. It is denoted by the symbol ^n.

How is a hyper exponent different from a regular exponent?

A regular exponent raises a number to a certain power, whereas a hyper exponent raises a number to the power of itself multiple times. For example, 3^3^3 is a hyper exponent, while 3^3 is a regular exponent.

What is the purpose of hyper exponents?

Hyper exponents are used to represent very large numbers that cannot be easily expressed using regular exponents. They can also be used in some mathematical and scientific calculations.

What is the value of a hyper exponent?

The value of a hyper exponent can vary greatly depending on the base and the number of iterations. For example, 3^3^3 is approximately equal to 7.6 trillion, while 2^2^2^2 is approximately equal to 1.34 x 10^154.

Are there any real-life applications of hyper exponents?

Hyper exponents are used in various fields such as physics, computer science, and cryptography. They can also be used to solve complex mathematical problems and equations.

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