# Questions tagged [hyperoperation]

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

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### Hyperoperation notation

Given that $H_1(x,y)=x+y$ $H_2(x,y)=x*y$ $H_3(x,y)=x^y$ $H_4(x,y)= {^y x}$ I want to represent $(((p^{\frac{p}{1+p}}+\ln(p))^{\frac{p}{1+p}}+\ln(p))^{\frac{p}{1+p}}+\ln(p))$ with the $H$ notation. Is ...
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### Definition of tetration [duplicate]

We all know that many years ago we invented powers. e.g. $3^4$ meant how many times we multiply 3. i.e. $3^4=3\cdot 3\cdot 3\cdot 3$. But then, people started asking questions like what is the ...
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### How to use $\uparrow$ to define an explicit bijective mapping $f:\varepsilon_{1}\rightarrow\mathbb{N}$?

The map $f:\varepsilon_{1}\rightarrow\mathbb{N}$ which I am trying to define has to send $\varepsilon_{0}$ to some natural number. Since $\varepsilon_{0}=\omega\uparrow^{2}\omega$, a potential ...
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### Is it possible to express logarithm with tetration?

Subtraction and division can be expressed with multiplication and exponentiation, as follows: a - b = a + (b * -1) a / b = a * (b ^ -1) My question is: does this ...
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### What $\uparrow^{\frac{1}{2}}$ should mean, knowing that $\uparrow$ is the Knuth's up-arrow?

I was thinking about generalizing $\uparrow^n$ past the integers so the first problem that came to my mind was what would be $\uparrow^{\frac{1}{2}}$. Firstly, that would be the operation between ...
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### Possible Tetration extension for a specific interval (part 1)

My friend and I have been developing an extension of tetration for non integer values. We managed to get definitions of extensions for : ${^r}x$. $x$>0. $r$ not equal to any whole number below -1. ...
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### Is there a way to have non-integer hyperoperations?

I'm pretty interested into hyper operators, but they are only defined for integers : $H_0(a,b)=b+1$ succession $H_1(a,b)=a+b$ addition $H_2(a,b)=ab$ multiplication $H_3(a,b)=a^b$ exponentiation ... I ...
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