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The power is to the multiplication what the multiplication is to the addition.

We can put those this way:

  • Addition
  • Multiplication
  • Exponentiation
  • Tetration

What comes after tetration? What comes whatever comes after tetration? Is this generalized and easily understandale for non-mathematician?

Note: I'm aware of this question, but it doesn't go deeper. I'm asking about the generalization of this process.

Side question:

We know that

2 + 2 = 4
2 * 2 = 4
2 ^ 2 = 4

Is 2 <sign> 2 always 4 ?

Community
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Olivier Grégoire
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3 Answers3

29

What comes after tetration ?

Pentation.

And after ?

Hexation.

And after ?

Heptation.

etc.

Take the Greek numerals in order. Tetra means four, penta means five, hexa means six, etc.

Is 2 <sign> 2 always 4 ?

Yes.

Is this generalized ?

Yes. $a\uparrow^nb$ is the consecrated notation.

P.S.: The operation of order $0$, coming right before addition, is incrementation.

Lucian
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  • It seems that you forgot the hexadecation ! – Claude Leibovici Mar 28 '14 at 13:41
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    This answers all my questions. Concisely. I have yet to properly understand the implications of the arrow notation, but thanks for clarifying this already :) – Olivier Grégoire Mar 28 '14 at 13:46
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    What are you refering to when you say that "The operation of order 0, coming right before addition, is incrementation."? It does not seem to fit in the sequence in my eyes (it is a unary operator and not binary like the others). – example Mar 28 '14 at 16:33
  • @example: The P.S. is almost but not quite true. Let $x\star y=y+1$. Then $$\underbrace{a\star(a\star(\cdots(a\star(a\star a}_{b\text{ times}}))\cdots)) = a+b-1$$ rather than $a+b$. – hmakholm left over Monica Mar 28 '14 at 19:52
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Knuth's up-arrow notation is the usual generalized notation of this. It is used the following way:

  • If there are only two numbers with only one arrow between them, then the arrow means "Raised to the power of", e.g. $3\uparrow 4 = 3^4 = 81$.
  • If there are only two numbers with arrows between them (like $3\uparrow\uparrow\uparrow 4$), then you take the first of the two numbers (in this case $3$), you repeat it a number of times signified by the latter number (in this case $4$), and then between them all you put arrows, one less than what you had. So we have $3\uparrow\uparrow\uparrow 4 = 3\uparrow\uparrow 3 \uparrow\uparrow 3 \uparrow\uparrow 3$
  • Lastly, if there are more than two numbers, you read it from right to left. Continuing on our example, that means $$ 3\uparrow\uparrow\uparrow 4 = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3 \uparrow\uparrow 3}\\ = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3 \uparrow 3 \uparrow 3} = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3^{3^3}}\\ = 3\uparrow \uparrow \color{blue}{3\uparrow \uparrow 3^{3^3}} = 3\uparrow\uparrow \color{blue}{3\uparrow 3 \uparrow 3 \cdots \uparrow 3} = 3\uparrow \uparrow 3^{3^{\cdots^3}} $$ which gets large. That is, it's a "power tower" of threes so tall you'd need a power tower of threes that's seven trillion tall to describe how tall it is ($3^{3^3} \approx 7\vphantom{\dfrac{1}{2}}$ trillion).
  • When there are too many arrows to practically write down, you use exponentiation. So $3\uparrow\uparrow\uparrow4 = 3\uparrow^34$.
Arthur
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I'll answer the question about $2 ? 2=4$. It is true that it always gives $4$, and you can prove it by induction. We define $+_1=\cdot$, $+_2$ is the power, etc. For two natural numbers $a$ and $b$ we define, $a+_{n+1}b=\overbrace{a+_na+_na+_n\cdots+_na}^{b\text{ times}}$.

So $2+_{n+1}2=2+_n2=4$.

ajotatxe
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