
Tetration - billpg
http://en.wikipedia.org/wiki/Tetration
======
akshaym
This is a fantastic article on "big numbers" (and discusses tetration):

<http://www.scottaaronson.com/writings/bignumbers.html>

(I think someone has posted it to HN before, which is why I am leaving it as a
comment)

~~~
warfangle
Definite upvote. That essay is insanely fascinating; I tend to re-read it
every few months.

~~~
michael_dorfman
I like the article, too, but I'm always disappointed when it doesn't cover
Knuth's up-arrow, which seems absurdly relevant.

~~~
ableal
Thanks. Obviously <http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation>

Shorter read: [http://www.daviddarling.info/encyclopedia/K/Knuths_up-
arrow_...](http://www.daviddarling.info/encyclopedia/K/Knuths_up-
arrow_notation.html) , which comes quickly to next step:

 _Three up-arrows together represent a still more vastly powerful operator,
equivalent to hyper5 or pentation, or a power tower of power towers_

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jacquesm
Addition, multiplication and exponentiation all have pretty clear day to day
applications.

Is there a laymans explanation for how you could apply this in an 'ordinary'
setting, in other words is this from my non-mathematical background a useful
construct or should I treat it as a curiosity?

I find it interesting that there is a simple order to the operations +, *, ^
and so on that apparently can be extended further than I was aware of, is
there a limit to that?

How would a fifth element in that series be applied?

Edit: as to the last, it seems the series builds on the previous operation in
the series to build the next, so possibly 'pentation' would be a series of
tetrations?

2+4 = 6

2x4 = 2+2+2+2 = 8 (edit2: using an 'x' here because hn eats asterisks thinking
you mean italics)

2^4 = 4x4x4x4 = 256

2^^4 = 2^(2^(2^2)) = 65536

2^^^4 = ...?

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jonp
This ([http://en.wikipedia.org/wiki/Knuths_up-
arrow_notation#Introd...](http://en.wikipedia.org/wiki/Knuths_up-
arrow_notation#Introduction)) describes how it can be extended without limit.
And there's are numerical examples at the end of that section demonstrating
how. And this (<http://en.wikipedia.org/wiki/Pentation>) focusses specifically
on the next step on "pentation".

I'm not aware of any "real-world" applications of tetration in the way that eg
exponentials can describe certain physical or economic processes. But it does
have some genuine uses in pure mathematics such as calculating the upper bound
of some numbers. It's not just making big numbers for the sake of it.

~~~
jacquesm
hehe, funny, I _just_ caught on to that (see edit above).

It's pretty logical really, but I can't seem to come up with what it should
work out to. But the notation, once you go from ^ for exponentiation to ^^ for
tetration immediately suggests ^^^ for the next step. I feel pretty dumb now
for not realizing right away that was what was intended. (edit: I suck at
math, in case that wasn't clear yet...).

Thanks!

------
jonp
Tetration is used to express Graham's number
(<http://en.wikipedia.org/wiki/Grahams_number>) which appeared in the Guinness
book of records as the largest number ever used in a serious mathematical
proof.

