Mathematics, psychology and sociology, philosophy.

Monday, May 3, 2010

An "Official" Nomenclature for Large Numbers?

2010 May 3rd

A former co-worker recently told me that his son has been learning (with his help) about very large numbers, including Graham's number, and asked me "if I know of any more 'official' nomenclature [for] numbers higher than centillion".

The higher the numbers go, the less official the names get. I have written much on this in the first section of my Large Numbers page.

Most folks who ask this question want to go more than just a little bit beyond centillion (10303 or 10600). Let's use 1012345 and 101027 as examples.

The only really official nomenclature is to say, for example, "ten to the power of ten to the power of twenty-seven".

I would give the prize for "second place" to Conway and Guy, The Book of Numbers (1996) pp. 13-15, who set out the system that I describe here. Under thier system, 1012345 is "one quadrilliquattuordecicentillion" and 101027 is "ten trestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestri-gintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestriginta-trecentillitrestrigintatrecentilliduotrigintatrecentillion".

I think the Knuth -yllion system would come in third; under his system, 1012345 is "ten myllion byllion tryllion decyllion undecyllion" and 101027 is "one quinvigintyllion septemvigintyllion octovigintyllion novemvigintyllion duotrigintyllion trestrigintyllion quattuortrigintyllion quintrigintyllion quinquadragintyllion quinquagintyllion duoquinquagintyllion tresquinquagintyllion quattuorquinquagintyllion quinquinquagintyllion sesquinquagintyllion septenquinquagintyllion octoquinquagintyllion unsexagintyllion quattuorsexagintyllion quinsexagintyllion sesexagintyllion septensexagintyllion unseptuagintyllion duoseptuagintyllion treseptuagintyllion quinseptuagintyllion octoseptuagintyllion novenseptuagintyllion unoctogintyllion duooctogintyllion tresoctogintyllion sexoctogintyllion septemoctogintyllion".

As you can see, systematic names for large numbers become unwieldy if you attempt to follow the classical system of giving names to each power of 10 (or powers of 1000 like Americans do today, or of a myriad as the Greeks and Chinese did, or of a million like Chuquet).

All of the other systems I have encountered are ad-hoc, unresearched and/or poorly thought out, imitations of the Chuquet names with clumsy or inconsistent decisions regarding how to proceed once the Latin ordinal number names run out. I describe some of these here.

The names googolplex for 1010100 and googolplexplex or googolduplex for 101010100 are fairly well-known. The number 1010101010000000 appeared in a 1994 journal article by Zarko Bizaca. Going beyond these, to numbers that are unwieldy to represent even as a succession of exponents:

Several academics (mostly mathematicians like Graham) have had to invent recursive function definitions to describe large finite numerical quantities, as part of a proof of some kind. As far as I have been able to tell, each such system is incompatible with every other such system.

Jonathan Bowers seems to have given more thought to this than anyone I have read about or been in contact with. His names (like exillion, tripent, baggol, trissol, dutridecal, goppatoth, golapulus, meameamealokkapoowa, and so on) are just convenient, arbitrary nicknames for various specific examples of his array notation and its multidimensional extensions. The array notation, in turn, is shorthand for a very complex set of recursively-defined functions.

Recursively-defined functions like those Bowers develops are extremely difficult to understand, and given two different recursive definitions, it can be even more difficult to prove which produces the more quickly-growing function. I am not sure how he developed his functions but I am reasonably confident that most of his claims about them are accurate. Checking his work is well beyond my patience, if not my ability. Bowers' keen abilities of comprehension are also evident in his descriptions of multi-dimensional geometric structures ("polychora", which are like polyhedra but with more dimensions).