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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Notation for Expressing Hyperoperations and their Inverses

This is a table for the new mathematics notation alongside its equivalent old mathematics rendition: \begin{align} a *_1 b & \equiv a + b & & a \; /_1 \; b \equiv a - b & & a \; \...
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Is $0$ the Exponential Inverse?

For a while, I've been wondering why this pattern seems to allude to the fact that $0$ is one of the inverses of exponentiation. \begin{align} & x \cdot -1 = -x & \text{Inverse of addition} \\ ...
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hyperidentities, unknown evaluation

I do not follow the notion of hyper-identities very clearly. In the last line in the snippet, how looks the substitution $$x_1x_2x_1$$ for $F$, what is the result and can I see that this is not ...
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Why does 2 result in the same value regardless of whether it is added to itself, multiplied by itself, or put to the power of itself? [closed]

I'm inferring that any hyperoperation you could apply here using two for every value would result in four. Why is this?
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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 ...
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Is anything significant known about inverting power towers?

This has to be something somebody has looked into, but I can't find it. In general, for finite expressions of the form $${a_1}^{{{a_2}^{{⋰}^{a_i}}}},$$ if one reverses the usual order of ...
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Closure under reverse tetration

The natural numbers are closed under addition, but not subtraction The integers are closed under multiplication, but not division The rationals are closed under exponentiation, but not roots The real ...
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What does $i\uparrow\uparrow i$ equal to?(or, it has no result?) [closed]

What does $i\uparrow\uparrow i$ (or, i↑↑i) equal to? $i+i=2i$ $ii=-1$ $i\uparrow i=i^i≈0.20787957635076193$
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Constant difference symbol

A simple notation question: I'm familiar with the notation $a \propto b$, which means $a$ and $b$ are constant multiples of each other, but is there an analogous symbol for when $a$ and $b$ differ by ...
116 views

How are the hyperoperations defined?

For the first few operations such as addition and multiplication they follow rules such as $$A(x,y+1)=A(x,y)+1$$ $$A(x,0)=x$$ For multiplication $$M(x,y+1)=M(x,y)+x$$ $$M(x,1)=x$$ So for a general ...
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Show the original Ackermann function is non-primitive recursive

There are a couple of questions here which show that the modern Ackermann function $A(i, x)$ is not primitive recursive. This new Ackermann function defined by Péter is a simplification of the ...
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Inverse hyperoperation code library

I hope you're having a nice day, I am an Electrical Engineer and I am trying to do a math paper, but I need a coding library in any coding language that can do inverse hyperoperations, do you know of ...
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"Shooting Room" ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
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We know that $2+2 = 2 \times 2 = 2^2$. Is this true for all successive hyperoperations?

Given that multiplication is repeated addition and exponentiation is repeated multiplication, if we were to continue onwards, what would the result be for successive hyperoperations? If we take the ...
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Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
The first hyperoperation $H_1(a, b) = a + b$ is addition, and is defined over the complex numbers. The second hyperoperation $H_2(a, b) = ab$ is multiplication, and is also defined over the complex ...