Talk:Knuth's up-arrow notation

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This could do with conversion to use Wikipedia:TeX notation

I'll get right on it! Dysprosia

Can someone move this to "Knuth's up-arrow notation", please?

If it's gonna be moved, I'd vote for Knuth arrow. BTW I'd removed Ackermann function since Ackermann page no longer refers to it. Kwantus

use Knuth arrow for relatively small numbers LMAO! =) Kwantus

It was either that or relatively small large numbers! :) (Maybe smaller magnitude...) Dysprosia 05:49, 30 Aug 2003 (UTC)

I tried to set it so that 3(arrow arrow)5 was shown as also being equal to 3^(3^7625597484987), but couldn't get the notation right. Can someone fix that? DS 14:15, 7 May 2005 (UTC)

Done. - Gauge 23:03, 2 October 2005 (UTC)


Why is = 327, not 273? Thanks, --Abdull 23:12, 30 September 2005 (UTC)

273 = (33)3 = 39 ≠ 327. Writing the exponentials as = pow(3, pow(3, pow(3, 1))), we see that we evaluate these functions from the inside out, so pow(3, pow(3, pow(3, 1))) = pow(3, pow(3, 3)) = pow(3, 27) = 327, rather than 273. - Gauge 22:59, 2 October 2005 (UTC)
Very big difference... 327 = 7,625,597,480,000 ≠ 273 = 19,683 Frozen Mists 21:59, 28 June 2007 (UTC)

Left-associative exponents and higher operators ARE used in some applications (although I can't site one offhand), but if you define higher operators based on left-association, you don't get essentially new operators, since in that case a^^b would = , rather than a^a^...^a^a (b a's). —Preceding unsigned comment added by (talk) 23:12, 11 March 2008 (UTC)


As I said in the article on the closely related Hyper Operator, I think it would beinteresting to investigate the notation for a non-natural number of arrows, equivalent to: I've already defined over all integers n, but I think it would be interesting to expand this to all reals, or possibly even all complex numbers. He Who Is 01:02, 23 April 2006 (UTC)

Sounds like hard work! I hope you can do it. Kaimiddleton 07:54, 22 April 2006 (UTC)

It can't be done unfortunately, at least not with any degree of rigour or consistency, because unlike addition and multiplication exponentiation is *neither associative nor commutative; hence in general a^^(b^^c) =/= (a^^b)^^c and a^^b =/= b^^a. Hence you get anomalies such as 2^^(1/3) < 2^^(2/7) even though 1/3 > 2/7.

A few people have worked on the problem and found partial solutions, and these may be worth looking into, but a definitive answer is always likely to prove elusive.

  • I am indebted to Dr. Roger Wheeler of Leicester University who pointed out to me a few years back that this is the reason any such attempts will fail.

Meltingpot (talk) 16:00, 6 September 2008 (UTC)

Formal definition[edit]

Okay, what's the formal definition of when ? It was originally written as 1 in the formal notation, but as in the tables below. Now the line has been removed altogether, meaning the function is currently undefined for . It should be either



cBuckley (TalkContribs) 13:31, 21 May 2006 (UTC)

The article had b=0, not n = 0. As the article stands, the up-arrow is defined for n ≥ 1. Dysprosia 13:40, 21 May 2006 (UTC)
Okay, my mistake. So what's with the first line in each of the tables? cBuckley (TalkContribs) 14:26, 21 May 2006 (UTC)

I see no reason why it isn't defined for n = 0. The definition gives a perfect definition for all n &ge -2; n He Who Is 20:24, 21 May 2006 (UTC)

Well, what would be the point of that? The arrow notation should be used when it should be used, so there's no practical need to make such a definition when one can resort to conventional notation. Dysprosia 22:35, 21 May 2006 (UTC)

I don't mean to say that there is a practical need for it, I merely mean to say that because of that definition, the Knuth up-arrow notation is defined for all n greater than of equal to -2.He Who Is 22:31, 22 May 2006 (UTC)

I think you mean to say that the up-arrow is defined for all n ≥ 2: as it stands, the hyper operator isn't defined for negative n. Dysprosia 22:38, 22 May 2006 (UTC)

Actually I did mean negative two, because the standard notation is defined for all n greater than or equal to zero (or depending on your point of view, all integers), and a^n b = a^(n-2) b. But, as you said, anyone can resort to the standard notation if they have to, so there's no reson to argue this point.He Who Is 19:45, 23 May 2006 (UTC)

OR for citation[edit]

I understand that this is WP:OR, but it's the "proof" for the citation (or an estimate).

let's define T=SQRT(10). (T is approximately equal to 3.16, which is close enough to 3 for something like "moderately sized hard drives"). Using T as a base, 10 (in decimal) would be expressed as 100T. 10010 is expressed as 10000T. The amount of space on a hard drive (d) it would take to store a number (using ASCII) on a hard drive is one byte per digit. For the powers of 10, let N be the power. the desired number to store is 10N. Expressed in N, the number of bytes it takes to store a number is N+1. If Expressed as base T, it would be 2N+1. Let's Examine T7625597484987. (I know it isn't equal to 37625597484987, but comparatively, they're close). Such a beast would have 7625597484988 digits when expressed in Base T. As a result, it would have approximately 3812798742494 digits when expressed in base 10. assuming each digit takes one byte, the number can be stored in 3 812 798 742 494 bytes, or approximately 4 Terabytes (hard drive manufacturers terabyte). If using a binary storage mechanism (as might be used to most efficiently store any number, say as in a twos compliment storage), 3 decimal digits, can approximately be stored in 10 binary digits. at 8 bits to a byte, this would take approximately 1 588 666 142 705 bytes, or 1.5 terabytes. Moderately large hard drives these days are about 300gb, (large ones at 500gb), which is 12ish (ascii) or 5ish (binary) "moderately large hard drives"? McKay 06:38, 19 January 2007 (UTC)

Error in table?[edit]

It seems to me that there is an error in the first table under table of values. First, we have

Then, in the very next table we have .

Assuming the first one is correct, shouldn't this be since we are simply looking at 2 raised to the power in the previous column? I'm not sure which of these is incorrect, but I thought I would just bring this to the attention of someone who is more certain. GreekHouse 02:11, 12 October 2007 (UTC)

Compared to the effect of the rounding error in the 6.0, replacing 2 by 10 at the bottom has a negligible effect: a factor a little more than 3 at the second level, i.e. a term 0.5 at the third level.--Patrick 06:15, 12 October 2007 (UTC)

Small issue on convention[edit]

This article starts with the example of multiplication. Although I think that the example of 3x2 is the correct one, it is not consistent with the general example ab (a copies of b instead of b copies of a). However the general trend in this article is of course b copies of a. So should we make the first example ba or change the explicit example to two copies of three, which is counter-intuitive? Paul (talk) 08:15, 25 January 2008 (UTC)

→==Numeration systems based on the hierarchy (+, *, ^, ^^, ^^^, ...)==

Knuth arrows denote the 3rd and higher operations in the hierarchy which was introduced by Goodstein[1947] (he used other notation, but apparently also introduced the terms "tetration", "pentation", and so on). I wrote a description of Goodstein's base-b level-k system of uniquely representing any nonnegative integer (it uses digits 0,...,b-1 and the first k operators in the hierarchy), but I wasn't quite sure of the best WP article in which to enter it. Maybe it would be appropriate here (?), but because the family of hyper operations is precisely this hierarchy, I posted it over there, at Hyper_operator#Numeration_systems_based_on_hyper_operations. Nevertheless, I used mostly Knuth arrow notation in the examples.
--r.e.s. (talk) 21:23, 14 August 2008 (UTC)


The article claims "Note: (10\uparrow)^k denotes a functional power of the function f(n) = 10^n (the function also expressed by the suffix -plex as in googolplex)." Now, sorry to be picky, but this needs a citation. I know a googol is 10^100 and a googolplex is 10^10^100, but this alone does not imply that "-plex" generally means 10^n. And there are no other number names ending -plex to use as data points. The words "googol" and "googolplex" were introduced by Kasner as described in the googol article, and should be regarded as indivisible morphemes unless either (1) Kasner originally and explicitly intended "-plex" to be a generalisable suffix with this meaning, or (2) it has since come into widespread use with this meaning. Is either of those the case? I think not, but I'll put the question here in case anyone can answer more definitely. (talk) 18:15, 14 January 2009 (UTC)

Incorrect Information[edit]

The base-2 representations at the end of the article seem to be wrong. Earlier in the same section it says that a base-b representation should only contain digits 0,1,...,b-1 and so a base-2 representation should only contain the digits 0 and 1 and yet all the representations contain a 2 digit. I do not understand the representations enough to fix them, but I was able to understand enough to see this was wrong. Could someone more knowledgeable fix this apparent inconsistency? --SurrealWarrior (talk) 17:01, 22 August 2009 (UTC)

I think it's a matter of terminology, in the sense that although the "complete hereditary representation" of an integer in base b does involve b, only {0,1,...,b-1} are called "digits". To some extent this can be compared to calling the expression 2*102 + 6*101 + 6*100 (which contains "10") the base-10 representation of the number usually written as 266. Probably this ought to be clarified in the article.
r.e.s. (talk) 13:27, 3 September 2011 (UTC)

Bogus table row?[edit]

The first row in "Tables of Values" is for m = 0, but the definition above says that that number must be greater than or equal to one. This seems inconsistent.

Offby1 (talk) 00:05, 29 March 2010 (UTC)

Up-arrow notation defined on ordinals?[edit]

The ε₀ page uses Knuth's up-arrow notation extended to ordinals, but the definition for ordinals appears nowhere on this page, and perhaps it should. (talk) 22:10, 24 April 2010 (UTC)

Graham's Number too large?[edit]

The article currently reads:

"Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n-arrow operator is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.

Some numbers are so large that even that notation is not sufficient. Graham's number is an example."

To my non-mathematician eye, this means hyper operators describe bigger numbers than Knuth's notation does, and Graham's number is even beyond the conventional scope of hyper operators. This seems to contradict the Graham's number page, which states, "it can be easily described by recursive formulas using Knuth's up-arrow notation or the equivalent, as was done by Graham."

Maybe this makes sense at some level, but to a non-mathematician like me this appears contradictory, and at the very least isn't clear enough.--Atkinson (talk) 04:40, 27 July 2012 (UTC)

How do you say it?[edit]

What's the verbal way to express up-arrow notation? K.Bog 20:08, 22 April 2014 (UTC)

There is a video of Ron Graham explaining Graham's (ie his) number, in which he pronounces it several different ways. But they are all quite close to things like "three four arrows three" for . Kestrelsummer (talk) 18:20, 30 October 2016 (UTC)

The definition of up-arrow notation[edit]

Why is instead of  ? — Preceding unsigned comment added by (talk) 08:12, 13 May 2014 (UTC)

       Here's a number for you:

10^10^10^100 ^(10^10^10^100) 10^10^10^100

         10^10^10^100 layers

How much is [edit]

Athanatophobos August 4th 2016

0 or 1 or another number?

Well as usual it is up to us to decide what we want it to mean, but the sensible choice would be 1 to preserve the equation . Kestrelsummer (talk) 23:10, 30 October 2016 (UTC)