# What comes after tetration ? And after ? And after ? etc.

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

We can put those this way:

• 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 ?

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Take a look at en.wikipedia.org/wiki/Knuth's_up-arrow_notation. – 5xum Mar 28 '14 at 13:24
Thanks, but where is the digest for non-mathematician in there? That's some part of the question. – ogregoire Mar 28 '14 at 13:28
– geodude Mar 28 '14 at 13:39
Well, the 'digest' would be that there is no name for the processes, but if you want to describe them you can with the up arrows. Is also fairly easy to see that $2\uparrow^n2=2\uparrow^{n-1}2=...=4$ for any $n$. (Here, $\uparrow^n$ means $n$ consequtive up arrows). – 5xum Mar 28 '14 at 13:40
There is a name for this operation sequence. The sequence is called the hyperoperation sequence, and it can be represented in a variety of forms. – GregRos Mar 28 '14 at 20:32

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.

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It seems that you forgot the hexadecation ! – Claude Leibovici Mar 28 '14 at 13:41
This answers all my questions. Concisely. I have yet to properly understand the implications of the arrow notation, but thanks for clarifying this already :) – ogregoire Mar 28 '14 at 13:46
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 $a+b$. – Henning Makholm Mar 28 '14 at 19:52

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 $3\uparrow\uparrow\uparrow4 = 3\uparrow^34$.
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Clear explanation of Knuth's notation, thanks! – ogregoire Mar 28 '14 at 14:38
I've always struggled understanding the number of arrows' meaning. I've read Wikipedia and many other sites many times and still never fully grasped it. Thank you so much! – Cole Johnson Mar 28 '14 at 16:17

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

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Thanks for the proof! Very much appreciated :) – ogregoire Mar 28 '14 at 13:45
the symbols didn't get displayed properly – Jason S Mar 28 '14 at 18:44
I've never heard of tetration before so this may be an ignorant question. If exponentiation is 2^2, isn't 2 tetration 2 = 2^(2^2) = 2^4 = 16? And 2 pentation 2 = 2^(2^(2^2)) = 2^(2^4) = 2^16 = 65,536? – CramerTV Mar 28 '14 at 22:33
@CramerTV No. It means to take 2 copies of 2 and exponentiate, so it's 2^2=4. – Kundor Mar 28 '14 at 22:43