# 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.

93 questions
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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 ...
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### Why do we have the present order of operations, and how do hyperoperations fit in?

Something that's been bugging me for a fairly decent while is the order of operations - not so much using it, however, as to understanding where it comes from. Typically we're introduced to it in the ...
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### Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...
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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 ...
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### How much bigger is 3↑↑↑↑3 compared to 3↑↑↑3?

3↑↑↑3 is already mind-bogglingly large, but how much larger is 3↑↑↑↑3? Is it so large that it is simply around 3↑↑↑↑3 times larger than 3↑↑↑3? Or is there another way to express its magnitude in terms ...
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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 ...
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### What is the name for this: “x^^y” (as in 2^^2=2^2 and 3^^3=3^3^3 and so on)

How do you call / is there any specific name for the following: x^^y e.g.: 2^^2 = 2^2 3^^3 = 3^3^3 4^^4 = 4^4^4^4 ...
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### Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
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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 ...
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### Tetration by a Non-Integer

Does anyone think that tetration by a non-integer will ever be defined ... really properly? Great mathematicians struggled with finding an implementation of the non-integer factorial for a long time ...
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### Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher order ...
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### Hyperidentities and Clones, Trivial observation, commutativity

In the book Hyperidentities and Clones they (Denecke and Wismath) write: $xy \approx yx$, in other words $$F(x,y)=F(y,x)$$ considered as a hyperidentity implies $$x\approx y.$$ I would like to ...
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### Derivative of super square root [closed]

What is the derivative of $y=^{1/2}x$? I tried finding the derivative of $x^{x}$ and then finding the inverse of that, but that didn't work.
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### Semigorup variety, hyperassociativity,idempotentunclear proof of $x^4\approx x^2$

Let $V$ be a hyperasociative semigroup variety. For hyperasociativiy see below. Then $V$ satisfies the following identity: $$x^2 \approx x^4.$$ A proof attempt is given here: If $V$ is idempotent (i.e....
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### Is there a notation for the repetition of basic operations?

I mean... multiplication is the repetition of addition: $2*2 = 2+2$ $3*3 = 3+3+3$ exponential is the repetition of multiplication: $2^2 = 2 * 2$ $3^3 = 3*3*3$ .. it is an obvious pattern. I ...
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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 ...
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