Talk:Large numbers

message: Sorry, I wasn't thinking for a second about the relative errors of really large numbers. Hopefully the current explanation explains it well. Revolver 05:23, 13 May 2005 (UTC)
thread_title:  Sorry 

message: How many possible characters are there to be chosen from? one human language? all human languages? ascii? ansi? the entire argument is invalid if we don't know how many individual characters there are to choose from. not trying to be mean, just a friendly request for clarification :)
Robby Shadowrunner340 02:02, 15 September 2006 (UTC)
replies: text: It's explained in the article: we're assuming each character is one byte, so 2^8 = 256 possible characters. —Lowellian (reply) 19:14, 8 April 2009 (UTC)
thread_title: About The Universe Computer Analogy...

message: :The heading "Uncomputably large numbers" is misleading, as no integer is uncomputably large (every value of a busy beaver function is an integer). It's a busy beaver function that's noncomputable, not the values in its range. Also, although Rado's sequences might have been the first ones, it's very easy to produce noncomputable functions that grow faster than every computable function. --r.e.s. 04:14, 14 October 2006 (UTC)
replies: text: Thanks for moving my comment (above) from the article page to here, where it belongs.  Sorry for the mix-up.--r.e.s. 19:16, 14 October 2006 (UTC)
thread_title:  From article page 

message: Many sections of this article, e.g. Systematically creating ever faster increasing sequences; Standardized system of writing very large numbers, have no citations.  Although they are very interesting, it appears they are original research which is not allowed by Wikipedia policy. True? Mytg8 20:44, 30 October 2007 (UTC)
thread_title:  Citations? 

message: :Moved from page (original research):
Let a strictly increasing integer sequence f_0(n) (n≥1) be given (written as a function for conveniently writing the functional powers), with f_0(1)>1. Then a sequence of sequences can be found by f_k(n)=f_{k-1}^n(1), from which we can select the "cross-sequence"  f_0(10), f_1(10),.. Another number instead of 10 could also be used; it has to be more than 1, and if f_0(1)=2 it has to be more than 2 to get an increasing sequence.
Together this is a process of creating a new sequence from a given one. This can be repeated (i.e., we can apply recursion), and again we can select from the matrix of numbers a single sequence, by taking the 10th element of each. We can apply the same whole process again and again.
Starting from f_0(n)=10^n=(10 \to n) this process corresponds to adding an element 10 in the Conway chain before the variable n, which is at the end of the chain:  we get f_k(n)=f_{k-1}^n(1)=10 \uparrow^{k+1} n=(10 \to n \to (k+1)), and the new sequence selected from the matrix is that of which the kth element is f_{k-1}(10)=f_k^{10}(1)=10 \uparrow^k 10=(10 \to 10 \to k)=(10 \to 10 \to k \to 1). (See also Knuth's up-arrow notation and Conway chained arrow notation). Alternatively, selecting the "cross-sequence"  f_0(2), f_1(2),.., we get f_{k-1}(2)=f_k^{2}(1)=10 \uparrow^k 2=(10 \to 2 \to k=10 \uparrow^{k-1} 10=(10 \to 10 \to (k-1)), so the sequence is only shifted one position.
Repeating this process we get (10 \to 10 \to k \to n) for successive values of n, and selecting k=10 we get a single sequence (10 \to 10 \to 10 \to n).
Repeating this whole process we get ever longer chains. Selecting n=10 we get the sequence of (10→10), (10→10→10), (10→10→10→10),... This can be used as starting sequence to apply the process again, etc. Even the value f_1(2) for this sequence is already a Conway chain of length 10 billion plus one.
Each sequence in this whole process can be identified by its order type in the process:
*(10→n→k) is the sequence with index n with order type k - 1
*(10→10→n→k) is the sequence with index n with order type ω + k - 1
*(10→10→10→n→k) is the sequence with index n with order type 2ω + k - 1
*(10→10), (10→10→10), (10→10→10→10),...  is the sequence with order type ω²
In a similar way this can be continued, and we get a set of sequences, well-ordered by the procedure of construction. 
*zero case: f_0(n)=10^n
*successor ordinal: f_{x+1}(n)=f_x^n(1)
*for limit ordinals: f_{a}(n)=f_{a_{n-1}}(10) for a suitable sequence of ordinals a_{n-1} tending to a.
Thus we have transfinite recursion as far as we define for each limit ordinal a suitable sequence of ordinals tending to it.
Consider the Cantor normal form \omega^{\beta_1} c_1 + \omega^{\beta_2}c_2 + \cdots + \omega^{\beta_k}c_k, where k is a natural number, c_1, c_2, \ldots, c_k are positive integers, and \beta_1 > \beta_2 > \ldots > \beta_k are ordinal numbers. The limit ordinals cover the case \beta_k>0. For order types less than ε0 (epsilon nought) \beta_k is less than the ordinal itself. The following rules provide for each limit ordinal a suitable sequence of ordinals: 
If a=(\omega\uparrow)^p (u+v) where p ≥ 0, u ≥ v, and v is a  limit ordinal, we take a_n=(\omega\uparrow)^p (u+v_n) for a suitable sequence of ordinals v_n tending to v.
For (\omega\uparrow)^p v where p ≥ 0, and v is a  limit ordinal, we take the sequence (\omega\uparrow)^pv_n for a suitable sequence of ordinals v_n tending to v.
For (\omega\uparrow)^{p+1} (a+1) where p ≥ 0 and a ≥ 0, we take the sequence (\omega\uparrow)^p (\omega^a n).
For example, this procedure defines for the order type \omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot5+578 a particular, very rapidly increasing, sequence of integers, and to specify a particular large integer, we can refer e.g. to the 37th element in this sequence. The sequence is defined in 578 steps from the sequence with order type \omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot5, which in turn is defined from the sequences  with order types \omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot4+\omega^{\omega^n}. The twelfth element in this sequence is defined from the sequences  with order types \omega^{\omega^{\omega6+42}\cdot1729+\omega^9+88}\cdot3+\omega^{\omega^\omega}\cdot4+\omega^{\omega^{11}n}, etc.
For ε0 we can take the sequence \omega\uparrow\uparrow n, and for ε0(k+1) the sequence ε0k + \omega\uparrow\uparrow n, etc. We cannot reach omega-one (ω1), the set of all countable ordinal numbers, and the smallest uncountable ordinal number: no sequence of ordinal numbers below ω1 has that ordinal as limit. It is also clear from the fact that we define sequences of which each element is defined in a finite number of steps, so making use of only a finite number of auxiliary sequences. Therefore for any of our sequences the set of auxiliary sequences is countable.
We have f_{a}(n)<f_{a+1}(n), except that we have "=" for n = 1. Note however that for a < b we do not always have f_{a}(n)<f_{b}(n). For example:
*f_{3}(n)<f_{\omega}(n) for n = 1, 3, 4, 5, 6, .., but we have "=" for n = 2.
*f_{4}(n)<f_{\omega}(n) for n = 1, 4, 5, 6, .., but we have ">" for n = 2, 3.
Tag for eventual archiving.  — Arthur Rubin  (talk) 03:53, 5 January 2016 (UTC)
thread_title: Systematically creating ever faster increasing sequences

message: I cannot deal with this article.  —Preceding unsigned comment added by 68.46.238.3 (talk) 21:59, 31 May 2008 (UTC)
thread_title:  This article 

message: Leviathan number is a long-standing stub on a number apparently invented by one author for the sole purpose of having a sexy example of a large number, and never used for any non-pedagogical purpose, but only as a curiosity or a guaranteed stumper for anyone who has not made a point to pay attention to useless information. Barring evidence that the British Pickover is an employee or owner of the US-based Mathematica, it does seem to have found slight recognition so i'm not inclined to AfD. But i think it would be more valuable as a section in the accompanying article (with a section lk from its current title) than standing alone. --Jerzy•t 22:30, 26 February 2009 (UTC)
thread_title:  Merge to accompanying article of Leviathan number  

message: THe section on computational complexity (the part that talks about "what if the universe could be one big computer" seems to include some original research or conclusions drawn by inference from the cited paper.  Particularly where cryptology is concerned.  I would recommend using additonal sourcing for this section. HappyJake (talk) 18:33, 1 September 2009 (UTC)
replies: text: This section is much too long and to 95% out of subject, (doubtful) "explanations" on quantum computers etc belong elsewhere. It should be condensed to a dozen of lines giving the numbers and what they refer to, conveniently linked to the article featuring further reading. (as a side note, i don't know anybody using 40 character passwords exhausting the full range of 8 bits. 8 characters ranging from ascii 32 to 97 is much more realistic, and considering the number of printable characters in the latin-n char sets should put us definitely on the safe side) — MFH:Talk 15:04, 29 May 2011 (UTC)
thread_title:  Computational Complexity 

message: From the article:
Each sequence in this whole process can be identified by its order type in the process:
*(10→n→k) is the sequence with index n with order type k - 1
*(10→10→n→k) is the sequence with index n with order type ω + k - 1
*(10→10→10→n→k) is the sequence with index n with order type ω2 + k - 1
*(10→10), (10→10→10), (10→10→10→10),...  is the sequence with order type ω²
Don't all increasing sequences have order type ω? For example, all sets with order type k < ω have exactly k elements, but the sequence an = (10→n→k) is an infinite sequence.
What is the statement supposed to mean? Something about the complexity of the algorithm used to compute the numbers??? — sligocki (talk) 02:41, 19 August 2010 (UTC)
thread_title:  Order type of sequences 

message: The article seems to use two different conventions for the meaning of a "billion" in a single sentence: 13.7 billion years (4.3 × 10^17 seconds) old, ... and contains about 5 × 1022 stars, organized into around 125 billion (1.25 × 10^11) galaxies".   Tkuvho (talk) 19:51, 9 January 2011 (UTC)
replies: text: The same, 10^9.--Patrick (talk) 23:57, 9 January 2011 (UTC)
replies: text: OK, thanks.  I missed the "seconds".   Tkuvho (talk) 01:15, 10 January 2011 (UTC)
thread_title:  A Billion, Anyone? 

message: The statement "This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be 'relatively small' and therefore acceptable)" in the context of its previous statement is incorrect. The error in the exponent implies the error is already "relative". Thus, even though the number is large, the "relative error" is also very large. Subh83 (talk) 08:08, 29 January 2011 (UTC)
thread_title:  Accuracy section 

message: Regardless of whether Googology should exist and redirect here, it's a very rare term.  A more common use seems to be the study of Google.  I've nominated the redirect for deletion, but, even if the redirect should exist, the statement here requires a source, and there doesn't seem to be a source for the claim that it's used, just (rare) use.  — Arthur Rubin  (talk) 03:52, 5 January 2016 (UTC)
replies: text: Since it is a redirect to this page, it is a useless link and so I have removed it. --Bill Cherowitzo  (talk) 17:16, 29 May 2018 (UTC)
thread_title:  Googology 

message: Substituted at 02:17, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Text and references copied from Everett Dirksen to Large numbers.  See former article's history for a list of contributors.  7&6=thirteen (☎) 17:40, 1 February 2019 (UTC)
replies: text: Text and references copied from Carl Sagan to Large numbers.  See former article's history for a list of contributors.  7&6=thirteen (☎) 17:57, 1 February 2019 (UTC)
thread_title: Attribution

message: Please, could somebody add a brief explanation of what "relative error" means? --Backinstadiums (talk) 16:36, 21 March 2019 (UTC)
replies: text: I wikilinked the first occurrence of relative error. —[AlanM1(talk)]— 19:32, 21 March 2019 (UTC)
thread_title:  relative error  

message: This article is a little meandering, but seeing as "Googology" redirects here, perhaps that topic should be expanded to include discussions of limits of BB computability, nonstandard integers, perhaps the history of the field, etc. Would that be considered on topic?  — Preceding unsigned comment added by Kotlopou (talk • contribs) 21:37, 18 December 2020 (UTC)
thread_title:  Googology? 

message: In the introduction this apparent sentence "To indicate the correct integer of a count and to provide a valid context for the correct indication of a unit of volume that has been validated according to international principles." is not a sentence and I can't see how to fix it. It's not clear why a unit of volume is introduced here.  I hope someone else can fix the text. JRGp (talk) 10:13, 25 February 2023 (UTC)
thread_title:  Not a sentence 

message: 095 267 5888 197.156.107.184 (talk) 18:05, 9 February 2024 (UTC)
thread_title:  Abdukaarim Aliiyyii amd