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.

# Questions tagged [hyperoperation]

132 questions

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### Why are addition and multiplication commutative, but not exponentiation?

We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not. My question is why.
As background there are plenty of mathematical schemes that can be used to define these operators. One of these…

rob levin

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### What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e.…

user180918

**24**

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### what operation repeated $n$ times results in the addition operator?

I had a difficult time in phrasing my question. But I was wondering if there is an operation that, when repeated n times, results in the addition operator. Same way as repeating addition n times results in the multiplication operator, and repeating…

Bert van den Bosch

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### Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one.
However, I was looking for the more general case where we find the continuum between the operators themselves -…

Dan W

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### Why do we stop at exponentiation stage in arithmetic of natural numbers?

In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. Also $\times$ is an iteration of $+$ and $\exp$ is…

user180918

**19**

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**2**answers

### Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, primarily by exploring various manipulations…

Archaick

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### Which digit occurs most often?

Is there any method to calculate, which digit occurs most often in the number
$$4 \uparrow \uparrow \uparrow \uparrow 4\ ,$$
the fourth Ackermann-number ?
Or would it be necessary to calculate the number digit by digit ?
I only know that the last n…

Peter

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### Can tetration 'escape' the complex plane?

Considering subtraction can break out of the natural numbers and into integers, division can break out of integers and into rational numbers, and exponentiation can break out of rational numbers and into irrational and complex numbers...
Can…

Graviton

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### Could someone tell me how large this number is?

Context: If you guys didn't know, I'm running a nice little contest to see who can program the largest number. More specific rules if you are interested may be found in my chat room (click here to join). If you are entering, do note that I am…

Simply Beautiful Art

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### Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$

So in the past few months when trying to learn about the properties of the fixed points in ordinals as I move from $0$ to $\epsilon_{\epsilon_0}$ I noticed when moving from $\epsilon_n$ to the next one $\epsilon_{n+1}$, the operation that does that…

Secret

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### How exactly does Knuth's Up-Arrow notation work?

I've done some research, and found this on Wikipedia.
\begin{matrix}a\uparrow b=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}
\begin{matrix}a\uparrow \uparrow b&={\ ^{b}a}=&\underbrace…

Hugh Chalmers

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### Superassociative operation

Background: Addition and multiplication are associative, but exponentiation is not.
Question: Does an operation $\circ_1:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ exist such that $$\circ_i(x,y)=\underset{y\text{…

Carucel

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### Example of Tetration in Natural Phenomena

Tetration is a natural extension of the concept of addition, multiplication, and exponentiation.
It is quite obvious that there are things in the physics world which can be modeled by these 3 lowest hyper-operations, as they are called.
For…

Ryan Stull

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### Why are tetrations not useful?

I've always wondered after learning addition, multiplication, and power facts (and their inverse operations) what the next higher level of facts I would need to memorize would be. However, instead of learning about the next higher operator, math…

Dynamic Light

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### Are points on the complex plane sufficient to solve every solvable equation composed of the hyperoperators, their inverses, and complex numbers?

Some background: I'm programming a maths environment. I'm computer science, so please excuse any probable ignorance and lack of precision in my question.
It seems $i$ and complex numbers were "invented" out of necessity to solve equations like
$$…

Max

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