# Large Numbers

First page . . . Back to page 10

### Footnotes

**1 :**
http://www.sizes.com/numbers/big_numName.htm
Anonymous author
at "sizes.com", Names of big numbers, 2004.

**2 :**
http://www.miakinen.net/vrac/nombres#lettres_zillions
Olivier
Miakinen, Écriture des nombres en français, (web page) 2003.

**3 :**
http://web.archive.org/web/20061021030550/http://www.io.com/~iareth/bignum.html
Gregg William Geist, Big Numbers (web page), 2006 (Latin number names,
some of the large examples like centumsedecillion)

**4 :**
http://www.miakinen.net/vrac/zillions
Olivier Miakinen, Les
zillions selon Conway, Wechsler... et Miakinen (web page), 2003.

**5 :**
http://www.graner.net/nicolas/nombres/wechsler.txt
Allan
Wechsler, "Re: Number names" (newsgroup message), 2000.

**6 :**
The Oxford English Dictionary (Second Edition), 1989, entry for
million (vol. IX, pp. 784-785), sense 1. a. (a)

**7 :**
Conway and Guy, The Book of Numbers. See bibliography entry
[44] below.

**8 :**
http://yudkowsky.net/singularity.html
Eliezer Yudkowsky, Staring
into the Singularity, web page (1996-2001).

**9 :**
Olivier Miakinen, personal communication, Sep 2004.

**10 :**
http://www.polytope.net/hedrondude/illion.htm
Jonathan Bowers
(AKA "hedrondude"), -Illion Numbers. Extensive list of his invented
large number names, in numerical order, and most ending in "-illion".

**11 :**
Stephen Hawking, God Created the Integers (an anthology of
translated works of great mathematicians throughout history), pp.
971-1039 (Georg Cantor)

**12 :**
http://www.toothycat.net/wiki/wiki.pl?CategoryMaths/BigNumbers
Douglas Reay, commenting on discussion of formal theory of
computation, toothycat.net wiki (created by Sergei and Morag Lewis),
CategoryMaths, BigNumbers.

**13 :**
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
(formerly at www.math.ohio-state.edu/~friedman/manuscripts.html)
Papers by Harvey M. Friedman. In the "lecture notes" section
is Enormous Integers in Real Life, 2000, which summarises
several methods of producing large integers, related to combinatorics
and theory of computation.

**14 :**
Harvey Friedman, Long Finite Sequences, 1998. Available at
the above website^{13}.

**15 :**
http://math.eretrandre.org/tetrationforum/showthread.php?tid=184
Henryk Trappman and Andrew Robbins, Tetration FAQ (online document)

Note: A previous version was here

**16 :**
Martin Gardner, The Colossal Book of Mathematics: Classic
Puzzles, Paradoxes, and Problems, W. W. Norton (2001), ISBN
0393020231.

The "Graham-Gardner number": pp. 448-450; also appeared in Scientific American in November 1977.

Most Gardner material has been published multiple times, so you might find it in one or another of his earlier books.

**17 :**
Knuth, Donald E., Coping With Finiteness, Science vol. 194 n.
4271 (Dec 1976), pp. 1235-1242.

**18 :**
http://www.stars21.com/translator/english_to_latin.html
InterTran English-Latin Translator, via Stars21.

**19 :**
Wikipedia, Names of large numbers, encyclopedia article
(accessed January 2010)

**20 :**
http://www.numericana.com/answer/units.htm#prefix
G{'e}rard P.
Michon's Numericana, Final Answers — Measurements and Units. (Has
lots of details about real and bogus SI prefixes) (formerly at
http://home.att.net/~numericana/answer/units.htm)

**21 :**
billion in the literal sense appears first in 1690, when it
unambiguously meant 10^{12}. The same applies to the first literal
usage of trillion (10^{18}) and all of the higher names in the table
— so these citation dates merely indicate how old the words are. See
my discussion
History of Short vs. Long Scale
for more details. All first citations in OED [43] are either
from Jeake in 1674 or Locke in 1690.

**22 :**
OED [43] does not cite billion in the superlative
sense, but milliard was used in the superlative sense as far back as
1823.

**23 :**
OED [43] cites one usage of sextillion in the
superlative sense by Walt Whitman in 1881; earlier editions of
Whitman's poetry collection show that he used this word and several
others as early as 1855, and trillions in 1847. Whitman also used
decillion as a superlative, and he used millions in the
superlative sense more than all the other -illions combined.
H. P. Lovecraft used vigintillion in the
superlative sense in 1926 and 1928, and used no other -illion words
above billion; his usage of million and billion was literal in
almost all cases.

**24 :**
duodeviginti, undeviginti : These are two of the more
notable discrepancies with actual Latin number-names; see
the discussion here.

**25 :**
Typically the terms of such a competition require that each
number be finite, "well-defined", and "computable"; this last
requirement keeps the discussion within the realm of things that can
be proven. (Without it, the busy beaver functions prevail,
but it becomes nearly impossible to resolve the question of whose
function is larger).

An example of such a discussion is the long-running
xkcd forum discussion thread
"My number is bigger!".
This thread was begun on the 7^{th} of July, 2007, and remained
continually active for nearly three years (last checked May 2010). The
initial message began the competition with 9000; the
first respondent offered 3.250792...×10^{548}; several
class 2 replies brought it up to 3.454307...×10^{1661};
then it jumped to 10^{1010},
10^{101010}, 10↑↑512, and
10↑↑↑3=10↑↑(10↑↑10). All of this was within the first 24
hours. Up-arrow notation was no longer of any use by
the third day of the discussion, and the participants then began
defining recursive functions and discussing proofs. It continued along
those lines for the following three years.

**26 :**
See "Completeness of the S-K basis" on the Wikipedia page for
Combinatory logic,

### Bibliography

[27] Edward Brooks, The Philosophy of Arithmetic, 1904. Cited by [34].

[28] Edward Kasner and James Newman, Mathematics and the Imagination, Penguin, 1940

[29] Reuben Goodstein "On the restricted ordinal theorem", Journal of Symbolic Logic, 9 (2): 33–41, 1944.

[30] George Gamow, One, Two, Three... Infinity: Facts and Speculations of Science, Viking, 1947 (reprinted in paperback by Dover, 1988).

This was an early source for me and unfortunately gave me the
impression that the contimuum hypothesis had been proven.
This figure implies that the ℵ_{n} series of infinities
is the complete set of infinities:

*Gamow p. 23., implying CH*

If these are really "the first three infinite numbers", then there can
be nothing between ℵ_{0} and ℵ_{1}, and
that's CH.

[31] E. L. Kaufman, et. al., 1949: The discrimination of visual number. The American Journal of Psychology 62(4) (Oct. 1949), pp. 498-525.

[32] George Miller, The magical number seven plus or minus two: some limits on our capacity for processing information. The Psychological Review 63 (1956), pp. 81-97

[33] Philip J. Davis, The Lore of Large Numbers, New York: Random House, 1961

Much discussion of number writing systems, methods of arithmetic and estimation, names for large powers of 10, and so on. Covers many other topics, including: the method of finite differences; linear algebra and finite-element analysis; figurate sequences, prime numbers; large and small quantities encountered in science; SI prefixes and unit conversion.

This is one of the first books I found on the topic. Bits of it (such as the discussion of how many objects one can see at one time with one's eyes) are seen on this web page.

[34] Dmitri Borgmann, Naming the numbers. Word Ways: the Journal of Recreational Linguistics 1 (1), pp. 28-31, 1968. Cover and contents are here and article is here.

[35] Howard DeLong, A profile of mathematical logic (Addison-Wesley 1970, also Dover 2004) p. 192.

[36] R.L. Graham and B.L. Rothschild, Ramsey's Theorem for n-Parameter Sets. Transactions of the American Mathematical Society 159 (1971), 257-292. (Another PDF is here).

[37] Douglas Hofstadter, Gödel, Escher Bach: An Eternal Golden Braid, Vintage, 1979, ISBN 978-0394745022

[38] Philip Davis, and Reuben Hersh, The Mathematical Experience, Birkhaeuser, 1981.

infinities: pages 223-225

[39] Donald E. Knuth, Supernatural numbers. Appears as pp. 310-325 in The Mathematical Gardner, ed. David A. Klarner (1981).

[40] Douglas R. Hofstadter, On Number Numbness, Mathematical Recreations column, Scientific American, May 1982.

[41] Dewdney, A.K., The Busy Beaver, in Mathematical Recreations column, Scientific American, April 1985, p. 30.

[42] Douglas R. Hofstadter, Metamagical Themas, book collecting several articles from the Scientific American column of the same name, BasicBooks (1985), ISBN 0-465-04540-5.

[43] The Compact Oxford English Dictionary (Second Edition), 1991. This is the version that has 21473 pages photographically reduced into a single book of about 2400 pages.

[44] John Horton Conway and Richard Guy, The Book of Numbers, Springer-Verlag, New York, 1996. ISBN 038797993X.

Page numbers for specific topics:

pp. 13-15 (-illion number-names)

pp. 59-61 (Knuth up-arrow notation)

p. 60 (Ackermann numbers)

p. 61 (Conway chained-arrow notation)

p. 61 (Skewes's number)

pp. 61-62 (the "Graham-Conway number")

pp. 266-276 (Cantor ordinal infinities)

pp. 277-282 (cardinal infinities and the continuum)

[45] Crandall, The Challenge of Large Numbers, Scientific American February 1997, pages 74-79.

[46] Georges Ifrah, The Universal History of Numbers, ISBN 0-471-37568-3. (1999).

[47] Chris Bird, Proof that Bird's Linear Array Notation with 5 or more entries goes beyond Conway's Chained Arrow Notation, 2006. Available here (and formerly at uglypc.ggh.org.uk/chrisb/maths/superhugenumbers/array_notations.pdf)

[48] Harvey Friedman, n(3) < Graham's number < n(4) < TREE{3}, message to FOM (Foundations of Mathematics) mailing list.

[49] A. Nabutovsky and S. Weinberger, Betti numbers of finitely presented groups and very rapidly growing functions, 2007. Available here

[50] Chris Bird, personal communication, 2008.

[51] Donald E. Knuth, personal communication, 2010 Feb 26.

[52] N. Mohan Kumar, Construction of Number Systems (for Math 310 course at Washington University in St. Louis), 2011.

[53] John Baez, Google+ post, 2013 Jan 11 (See also this mathoverflow question)

[54] Sbiis Saibian, 3.2.10 Graham's Number, web article, 2022 January 24.

[55] Adam Goucher, The Ξ function (blog article), 2013 Jan 6.

[56] Michael Taylor, Numbers, 2014.

### Other Links

Aaronson, Scott, Who Can Name the Bigger Number?, essay about how to win the often-contemplated contest; covers many of the topics discussed here.

Bird, Chris, Array Notations for Super Huge Numbers, 2006. (An older version of his work, which includes much of the material found here).

----, Super Huge Numbers, 2012. There are several sections, with the simplest and slowest-growing functions first. The initial chapter "Linear Array Notation" is roughly comparable to Bowers arrays; the other chapters define higher and higher recursive functions.

Bowers, Jonathan, Big Number Central.

----, Exploding Array Function.

----, Infinity Scrapers.

Hudelson, Matt, Extremely Large Numbers

Knuth, Mathematics and Computer Science: Coping with Finiteness. Advances in our ability to compute are bringing us substantially closer to ultimate limitations., Science, 1976, pages 1235-1242

Kosara, Robert, The Ackermann Function

MacTutor history of Mathematics page on Chuquet

Matuszek, David, Ackermann's Function

McGough, Nancy, The Continuum Hypothesis (web pages)

Munafo, Robert, hypercalc (the Perl
calculator program that handles numbers up to 10^{④}10000000000)

Pilhofer, Frank, Googolplex and How to get a Googolplex

Rado, Tibor, On non-computable functions, Bell System Tech. Journal vol. 41 (1962), pages 877-884. (busy beaver function)

Rucker, Rudy, Infinity and the Mind, 1980. (ordinal infinities: the relevant chapter was reproduced here the last time I checked.)

Spencer, Large Numbers and Unprovable Theorems, American Mathematical Monthly, 1983, pages 669-675

Steinhaus, Hugo, Mathematical Snapshots (3^{rd} revised edition)
1983, pp. 28-29.

Stepney, Susan, Ackermann's function

----, Big Numbers

----, Graham's Number (referring to the more well-known version, i.e. the "Graham-Gardner number")

Teoh, H. S., The Exploding Tree Function, 2008.

Weisstein, Eric (ed.), Ackermann Function

----, Large Number

Wikipedia, Veblen function.

## Acknowledgments

To Morgan Owens (packrat at mznet gen nz) for news of the Knuth -yllion names and the Busy Beaver function

Unconfirmed SI prefixes: Sci.Math FAQ, Alex Lopez-Ortiz, ed. (formerly at http://www.cs.unb.ca/~alopez-o/math-faq/mathtext/node25.html)

Japanese readers should see: 巨大数論 (from @kyodaisuu on Twitter)

If you like this you might also enjoy my numbers page.

s.27