Talk:Knuth's up-arrow notation

Untitled
This could do with conversion to use 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)

327=273
Why is $$3^{3^3}$$ = 327, not 273? Thanks, --Abdull 23:12, 30 September 2005 (UTC)


 * 273 = (33)3 = 39 &ne; 327. Writing the exponentials as $$3^{3^3}$$ = 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 &ne; 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 = $$a^{(a-1)^b}$$, rather than a^a^...^a^a (b a's). —Preceding unsigned comment added by 24.165.184.37 (talk) 23:12, 11 March 2008 (UTC)

Study (generalization to non-integer arguments)
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: $$a^{(n)}b|n\notin\mathbb{N}$$ 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)


 * This argument against noninteger arguments (sic/lol) seems fallacious. Usual exponentiation does not obey the rule (a^b)^c =?= a^(b^c) nor a^b =?= b^a, yet exponentiation is well defined for all positive real numbers (and more, specifically: complex numbers that are not negative reals).
 * So, these apparent contradictions based on the premise that we have to have rules which we can't and therefore don't expect. I think various (e.g., "log-linear") interpolations could be possible - e.g.:
 * between 2^^2 = 4 = 2^2, 2^^3 = 16 = 2^4, 2^^4 = 65536 = 2^16 and 2^^5 = 2^65536, it seems quite natural to have
 * 2^^2.5 = 2^3, 2^^3.5 = 2^8 = 2^2^3 and 2^^4.5 = 2^256 = 2^2^8; with obvious generalisation
 * 2^^(2+x) = 2^(2+2x), 2^^(3+x) = 2^2^(2+2x), 2^^(4+x) = 2^2^2^(2+2x), etc., when 0 <= x <= 1.
 * (I don't particularly like this idea, which came first to my mind. But the fractional derivatives are defined similarly, by separating the integer part and the fractional part. I guess that it should not be difficult to find a slightly "smoother" generalisation...) &mdash; MFH:Talk 22:31, 20 June 2024 (UTC)
 * PS: a different, probably better generalisation comes to mind: append a final exponent 2^x where x is the fractional part: for x=0 it's 1 and does not change the value, for x=1 it's 2 and gives the next term in the series. This will obviously work for any other base (e.g., 3, 4, 5, ...) instead of 2. (It will *not* yield 2^^2.5 = 8 but rather 2^^2.5 = 2^2^sqrt(2) ≈ 6.3429... &mdash; MFH:Talk 22:46, 20 June 2024 (UTC)

Formal definition
Okay, what's the formal definition of $$ a \uparrow^n b $$ when $$ n = 0 $$? It was originally written as 1 in the formal notation, but as $$ ab $$ in the tables below. Now the line has been removed altogether, meaning the function is currently undefined for $$ n = 0 $$. It should be either



a\uparrow^n b= \left\{ \begin{matrix} a^b, & n \isin \{ 0, 1 \}; \\ a\uparrow^{n-1}(a\uparrow^n(b-1)), & \mbox{otherwise} \end{matrix} \right. $$

or



a\uparrow^n b= \left\{ \begin{matrix} ab, & n = 0; \\ a^b, & n = 1; \\ a\uparrow^{n-1}(a\uparrow^n(b-1)), & \mbox{otherwise} \end{matrix} \right. $$.

cBuckley (Talk &#8226; Contribs) 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 &ge; 1. Dysprosia 13:40, 21 May 2006 (UTC)


 * Okay, my mistake. So what's with the first line in each of the tables? cBuckley (Talk &#8226; Contribs) 14:26, 21 May 2006 (UTC)

I see no reason why it isn't defined for n = 0. The definition $$a\uparrow^n b = a^{(n-2)} b$$ gives a perfect definition for all n &ge; 2; n $$\in\mathbb{Z}$$He Who I s 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 &ge; 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 $$a\uparrow^n b = a^{n-2} b$$ . But, as you said, anyone can resort to the standard notation if they have to, so there's no reason to argue this point.He Who Is 19:45, 23 May 2006 (UTC)

I haven't come across a source with a definition for zero or less n arrows. There may be an inductive argument that $$a\uparrow^0 b = a$$ if b is non-zero. Unfortunately, this would need to be backed up by a citation. We need to avoid original research. --Ashawley (talk) 18:29, 28 March 2020 (UTC)

We currently have... - The general definition of the notation (by recursion) is as follows (for integer a, positive integer n and non-negative integer b):

a\uparrow^n b=  \begin{cases} a^b, & \text{if }n=1; \\ 1, & \text{if }b=0; \\ a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise } \end{cases} $$ -

I feel that this would look better as:

a\uparrow^n b=  \begin{cases} 1, & \text{if }b=0; \\ a^b, & \text{if }n=1; \\ a\uparrow^{n-1}(a\uparrow^{n}(b-1)), & \text{otherwise } \end{cases} $$ So that the three expressions $$ 1, a^b, a\uparrow^{n-1}(a\uparrow^{n}(b-1)) $$ are in order of 'increasing complexity'. Also, any tiny ambiguity about the value of $$ 0\uparrow 0 $$ would be immediately resolved. Clive tooth (talk) 17:14, 29 March 2020 (UTC)

Bogus table row?
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)


 * I agree. Seems like there's no way to have zero arrows... --Ashawley (talk) 22:06, 19 March 2019 (UTC)

How much is $$a\uparrow\uparrow 0$$? 0 or 1 or another number? -- Athanatophobos August 4th 2016


 * 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 $$a\uparrow\uparrow n=a\uparrow(a\uparrow\uparrow(n-1))$$. Kestrelsummer (talk) 23:10, 30 October 2016 (UTC)

OR for citation
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?
It seems to me that there is an error in the first table under table of values. First, we have $$2\uparrow\uparrow 6 = 2^{2^{65536}}\approx 10^{6.0 \times 10^{19,728}}$$

Then, in the very next table we have $$2\uparrow\uparrow 7 = 2^{2^{2^{65536}}}\approx 10^{10^{6.0 \times 10^{19,728}}}$$.

Assuming the first one is correct, shouldn't this be $$2\uparrow\uparrow 7 = 2^{2^{2^{65536}}}\approx 2^{10^{6.0 \times 10^{19,728}}}$$ 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
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 $$( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \uparrow\uparrow\uparrow, \ \dots)$$ 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. Nevertheless, I used mostly Knuth arrow notation in the examples. --r.e.s. (talk) 21:23, 14 August 2008 (UTC)

Googolplex
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. 91.105.25.16 (talk) 18:15, 14 January 2009 (UTC)

Incorrect Information
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)

Up-arrow notation defined on ordinals?
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. 75.22.201.232 (talk) 22:10, 24 April 2010 (UTC)

Graham's Number too large?
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?
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 $$3\uparrow\uparrow\uparrow\uparrow 3$$. Kestrelsummer (talk) 18:20, 30 October 2016 (UTC)

The definition of up-arrow notation
Why $$a\uparrow^n b$$ is $$H_{n+2}(a,b)$$ instead of $$H_n(a,b)$$ ? — Preceding unsigned comment added by 140.113.136.218 (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

Severe error right at the beginning
before even the table of contents, we see https://wikimedia.org/api/rest_v1/media/math/render/svg/7bde1c371dfcf9daeaabd7284d046e92bebae399 which is soon followed by https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e1cbace2adb782550d7a5c3fffbc4e5add62d THESE ARE INCONSISTENT. If the first is correct, then we need to replace an 'n' with 'n-1' in the second. If the second is correct, then an extra arrow is needed in the first. I have added an ERROR remark. DO NOT REMOVE THE ERROR MARKER until the error is fixed.Jamesdowallen (talk) 12:41, 2 November 2017 (UTC)

Second definition is correct (note that the recursive definition works the same as the Ackermann function's recursive case). I have fixed the first definition to match the second. 70.72.216.53 (talk) 22:37, 23 November 2017 (UTC)

2^^^3 vs 2^^^4
I think it would be better to have the opening example show direct correlations to different examples of Arrows. Instead of using 2^4 2^^4 and 2^^^3 is it better to use 2 xUparrow 3 for all three to properly illustrate the difference between the 3 examples and to show the scale at which additional ^'s increase the results? Apriestofgix (talk) 18:43, 28 November 2017 (UTC)

associativity
The article claims that Knuth's arrow notation be right-associative, but I can't find any associativity rule in the cited links; they always use parentheses. Isn't it probably like exponentiation, where the writer has to specify the association (e.g. with parentheses or by using superscript notation)? Edit: In the first source right-associativity was used, but I'll check the source's sources.Ninjamin (talk) 16:29, 3 April 2019 (UTC)
 * I found Conway using right-associativity for Knuth's arrow notation, but most sources use parentheses, so there's no reason to see a general convention here. Also the article claims that the notation be defined to be associative, but Knuth's paper makes no mention of that, thus I'm going to remove that passage.141.99.250.138 (talk) 09:30, 4 April 2019 (UTC)

Down-arrow notation
There are a few sources on the internet, including Googology Wiki, for down-arrows which are a kind of inverse of Knuth's up-arrow. I'm not qualified to add it myself, but I think down-arrows merit a section in this article. Glumblebee (talk) 06:25, 25 June 2019 (UTC)

Definition
The formal definition in the first line uses parenthesis to define a right-associative operator. The last line says: "the operator is SOMETIMES used right-associatively". This is confusing. For a left associative operator the definition would have to be



a\uparrow^n b=(a\uparrow^n (b-1))\uparrow^{n-1} a $$ 2A02:A210:2142:6C00:D5BA:D812:1452:F501 (talk) 09:23, 12 November 2019 (UTC)

Where does the definition for b = 0 come from?
See title. Knuth's original article never said anything about what happens when either term is zero, and I find the definition of 0 \uparrow^n 0 particularly suspect -- after all, isn't 0^0 indeterminate? Arcorann (talk) 13:33, 19 May 2020 (UTC)


 * I've also raised the same concern. It doesn't come from anywhere.  See discussion under  above. --Ashawley (talk) 19:37, 30 June 2020 (UTC)

English syntax question
Shouldn't this sentence in the Definition section :
 * Remark however that Knuth did not define the "nil-arrow" ($$\uparrow^0$$).

more properly be
 * Note, however, that Knuth did not define the "nil-arrow" ($$\uparrow^0$$).

I would make that change, but I'm unsure whether the first version, using "Remark" is correct English as well and I'm just not aware of it. Thanks! -- ArglebargleIV (talk) 19:02, 23 March 2021 (UTC)

2↑↑4 and 2↑↑↑4 are bad examples because it is a coincidence that 2↑↑↑4 happens to be equal to 2^2^...^2 with 2↑↑4 2's
The examples for tetration and pentation used at the top of the article (2↑↑4 and 2↑↑↑4) are bad because they make it easy for a layperson like me trying to learn what pentation is from the article to come to a false conclusion about what pentation is.

Specifically, I saw that 2↑↑4=65536 and 2↑↑↑4=2^2^...^2 with 65536 2's and made the false assumption that the general version of this is true, i.e. that a↑↑↑b=a^a^...^a with a↑↑b a's (but this is false).

I highly doubt I'm the only person who came to this false conclusion about what pentation is when reading this example at the top of the article. I suggest a different example be used instead that doesn't have this coincidental feature, perhaps a=2 and b=3.

I leave it to a math person to make that judgment call and change it. Thank you! 70.123.9.81 (talk) 04:53, 5 March 2022 (UTC)

what is said here, if anything?

 * Since $$6\uparrow\uparrow4$$ = $$6^{6^{6^{6}}}$$ = $$6^{6^{46,656}}$$, Thus the result comes out with $$\underbrace{6^{6^{.^{.^{.^{6}}}}}}_{4}$$

If this is anything other than tautology, it is poorly phrased. —Tamfang (talk) 01:54, 23 December 2022 (UTC)