| Large Numbers |
1 :
http://www.sizes.com/numbers/big_numName.htm
2 :
http://www.miakinen.net/vrac/nombres#lettres_zillions
3 :
http://www.io.com/~iareth/bignum.html
(Latin number names, some
of the large examples like centumsedecillion)
4 :
http://www.miakinen.net/vrac/zillions
: page by Olivier
Miakinen; and private communication.
5 :
http://www.graner.net/nicolas/nombres/wechsler.txt
: note from
Allan Wechsler
6 : (no web page)
Bulletino di Bibliographia e di Storia
delle Scienze matematiche e fisische. — Bologna volumes XIII, 1880,
ISSN 9012-9458.
7 : (no web page)
The Oxford English Dictionary (Second Edition),
1989, entry for million (vol. IX, pp. 784-785), sense 1. a. (a)
8 :
http://www.linguistlist.org/issues/7/7-451.html
9 : (no web page)
Le Nouveau Petit Robert (1993 edition), entry
for the word billion (page 223); entry for the word trillion (page
2312)
10 :
Conway, John Horton and Guy, Richard, The Book of Numbers,
New York: Springer-Verlag, New York, 1996. ISBN 038797993X.
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 (Graham's number)
pp. 266-276 (Cantor ordinal infinities)
pp. 277-282 (cardinal infinities and the continuum)
11 :
Hawking, Stephen, 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 :
http://www.math.ohio-state.edu/~friedman/
Web site of Harvey
M. Friedman. In the "preprints, drafts and abstracts" is a paper
Enormous Integers in Real Life, 2000, which summarizes several
methods of producing large integers, related to combinatorics and
theory of computation.
14 :
Harvey Friedman, Long Finite Sequences, 1998. Available at
the above website13.
15 :
http://math.eretrandre.org/tetrationforum/attachment.php?aid=189
Henryk Trappman and Andrew Robbins, Tetration FAQ (online document)
16 :
Martin Gardner, <:The Colossal Book of Mathematics: Classic
Puzzles, Paradoxes, and Problems, W. W. Norton (2001), ISBN
0393020231. Graham's number: pp. 448-450; also appeared in
Scientific American in 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://yudkowsky.net/singularity.html
Eliezer Yudkowsky, Staring
into the Singularity, web page (1996-2001).
Other References
Crandall, The Challenge of Large Numbers, Scientific American February 1997, pages 74-79
Davis, Philip and Hersh, Reuben. The Mathematical Experience, Birkhaeuser, 1981. pages 223-225 (infinities)
Davis, Philip J., The Lore of Large Numbers, New York: Random House, 1961
Dewdney, A.K., The Busy Beaver, in Mathematical Recreations column, Scientific American, April 1985, p. 30.
Gamow, George, 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 ℵn series of infinities was equivalent to a power-set series, and also to the continuum power-set series.
Hofstadter, Douglas, Gödel, Escher Bach: An Eternal Golden Braid
Hudelson, Matt, Extremely Large Numbers
Kasner, Edward and Newman, James, Mathematics and the Imagination, Penguin, 1940
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
Knuth, Supernatural Numbers, in The Mathmatical Gardener, D. A. Klarner, ed., 1981
Kosara, Robert, The Ackermann Function
MacTutor history of Mathematics page on Chuquet
Matuszek, David, Ackermann's Function
McGough, Nancy, The Continuum Hypothesis (web pages)
Miller, George: The Magical Number Seven Plus or Minus Two: Some Limits on Our Capacity for Processing Information (1956)
Munafo, Robert, hypercalc (the Perl calculator program that handles numbers up to 10④10000000000)
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 (3rd revised edition) 1983, pp. 28-29.
Stepney, Susan, Ackermann's function
Stepney, Susan, Big Numbers
Stepney, Susan, Graham's Number
Weisstein, Eric (ed.), Ackermann Function
Weisstein, Eric (ed.), Large Number
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.
Appendix: Notes on Certain Topics
List of Notes:
Personal History
Large numbers have interested me almost all my life. At age 5 100 was the biggest number I knew, by age 6 it was 1000000, at age 7 I asked my Mom what was after 1000 and a million and she told me about the (lesser) billion and trillion (1012); at age 8 I learned about vigintillion (1063) in a book from the school library. I loved vigintillion so much I wrote it in the sand in the schoolyard:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
prompting much harassment from the other children! I kept going: by age 10 I had invented higher dyadic operators and by age 13 I knew the Steinhaus-Moser notation. That was about as high as anyone had gone at the time, so I turned my attention to computers and began to write programs to manipulate large Class-2 numbers. My latest accomplishment in this area is hypercalc. It literally cannot overflow, except by dividing by zero.
If you like this you might also enjoy my numbers page.
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