"Thy knowledge is become wonderful to me: it is high and I cannot reach it." (Ps 138:6) GOOGOLOGY
translated by Razilee Purdue
C1978-2003 Hierogamous Enterprises
"Googol" is the name given by Dr. Edward Kasner's nine-year-old nephew, Milton Sirotta, to the number represented by one followed by one hundred zeros, aka ten duotrigintillion or ten sedecilliard [British]. It can also be represented as ten-to-the-hundredth, G(2, 100, 10), or ten-to-the-tenth-to-the-tenth-squared, G(2, 2, G(2, 10, G(2, 10, 10))), to-the-second-ten-squared, G(2, 2, G(3, 2, 10)).
Milton also coined the term "googolplex" for the antilogarithm of the googol, (10
10)100. "Plexing" like this is a handy device for naming numbers both larger than a googol or a googolplex, from eleventyplex = G(2, 110, 10) to noncentilliardplex = G(2, 1003, G(2, 10, 10)). From the analogy gross:great gross::googol:? Andre Joyce concluded that "great G(2, b, a)" means "G(2, b + 1, a)", as in gross = 144 = G(2, 2, 12); great gross = G(2, 3, 12); so that great googol = G(2, 3, G(2, 10, 10), and great googolplex = G(2, 2, G(2, 10, G(2, 10, 10))). Then from another analogy this time from genealogy's great great grandfather = two-greats grandfather, he got to n-greats googol = G(2, n + 2, G(2, 10, 10)), and n-greats googolplex = G(2, n + 2, G(2, 10, G(2, 10, 10))). He also concluded the n-greats gross = G(2, n + 2, 12), and since the Baker's gross = G(2, 2, 13) = 169, then the great Baker's gross = G(2, 3, 13) = 2197, the n-greats Baker's gross = G(2, n + 2, 13), and since the Poulter's gross = G(2, 2, 14) = 196, then the Poulter's great gross = G(2, 3, 14) = 2,744 and the n-greats Poulter's gross = G(2, n + 2, 14).
Larger and no less awkward numbers can be formed by adding additional -illiard- infixes to get: eleventyilliardplex = G(2, 663, 10) to great noncentilliardilliardplex = G(2, 5404, G(2, 10, 10)). This can rapidly become quite formidable.
Noting that googolplexplex = G(2, 2, G(3, 3, 10)) = googolple(xplex = googolplexple(x), it is easy to see that G(2, 10, 10) =G(3, 2, 10) = xplex. This is also consistent with Roman numerals where x = 10. The final l in googol would then logically stand for 50 and googo- indicate an operator like -ple-. The discovery that these functions operated on Roman numerals was the beginning ofthe development of a more logical and laconic system of large number nomenclature now called googology.
Googological operations can be defined for analogous prefixes of the form g-g-, based on the "o-count" in the googo-. Using the Roman numerals, r and s, as integer variables, we now have:
gogor = G(2, r, r)
gogoor = G(2, 2r, r)
googolpler = G(2, 100, G(2, 10, r))
googor = G(2, r, 2r)
googoor = G(2, 2r, 2r)
rples = G(2, r, s)
Many other Roman numerals can be substituted, besides x and l (both proper ones, such as iv, ix, and less proper, but pronounable ones, like il = 49, ic = 99, id = 499, im = 999, jiv = 3, jix= 8, jil = 48, jic = 98, jid = 498, jim = 998) as suffixes. Thus names can be formed from gogoci = G(2, 101, 101) = G(3, 2, 101) > 2.731861968G(2, 202, 10)
to googoomdix = G(2, 3108, 3108) = G(3, 2, 3108) > 6.203725265G(2, 10501, 10). The largest three-syllable number name would then be googoomplem = G(2, 1000, 1000) = G(3, 3, 1000).
By considering googolplexplex as googolpl(explex) = googolplex(plex) many more Roman numerals can be made pronouncable using the Principle of Equivalency: x = "ex", l = "el", m = "em" and the corollary equivalencies -- c = "cy" [as in Nancy], d = "dy" [as in Lady], v = "vy" [as in Ivy].
By the Principle of Recursivity the gaps left in the Roman numeral representations because of limitations of the Pronouncability can be filled. Numbers can be constructed by repeating smaller ones digitally, multiples of Samuel Yates' repunits [rep(licated )units] = [G(2, r, 10)/9]. Extrapolated further "repreps" [rep(licated) rep(etition)s] can also be defined as: [G(2, r, 10)/s], naming them with hyphenated googologisms.
The largest such two- and three-syllable googologisms would then be:
mem-ij = [G(2, 2000, 10)/2] = 5G(2, 1999, 10)
mem-ij-plem = G(2, 5G(2, 1999, 10), 1000) = 5G(2, 2002, 10).
An interesting subset of these repreps includes:
the sacred number of the Pythagoreans, [G(2, 6, 10)/7] = vi-vij = 142857;
Robert Ripley's persistent number, [G(2, 18, 10)/19] = jixex-xix = 52631578947368424 and similar numbers, like [G(2, 26, 10)/17] = exvi-exvij = 588235294117647
and [G(2, 22, 10)/23] = xexij-jivxex = 434782608695652173913
that maintain their digital sequence when multiplied by integers below s.
gogom = G(2, 1000, 1000) = G(2, 3000, 1000)
gogoom = G(2, 2000, 1000) = G(2, 6000, 10)
googom = G(2, 1000, 2000) = G(2, 3000, 20) > 1.1230231922G(2, 3906, 10)
googoom = G(3, 2, 2000) = G(2, 6000, 20) > 1.513470582G(2, 7806, 10)
In an analogy with the usual exponential (right superscription) and tetrational (left superscription) notation, Joyce spoonerized the syllables of these to get:
gomgo = G(3, 1000, 1000) = G(4, 2, 1000)
gomgoo = G(3, 1000, 2000)
goomgo = G(3, 2000, 1000)
goomgoo = G(3, 2000, 2000) = G(4, 2, 2000)
Folkman's number, G(2, 3, 3, G(2, 901, 2)), would, e. g., translate to pleplejivcymipleijij with the rhyme pattern aabc daee;
Graham's number, G(64, 3, 3, 3).
Noting that the representation for thousands above M in Roman numerals is with an added bar above the numeral, the suffix -bar can be defined as an operator on the Roman numerals with higher rank than the g-g- or -ple- operators:
embar = 1,000,000
goombargoo = G(3, 2, 2000000)
Applying the -bar operator not only to the Roman numerals, but also to itself forms much abbreviated googologisms, if "barbaric" ones. The suffix, -barbar, would mean, not -bar applied twice, but -bar applied 1000 times or multiplication by G(3, 2, 1000) = G(2, 3000, 10)21,000 with each additional -bar adding 1 to the first index:
embarbar = G(2, 1003, 1000) = G(2, 3009, 10)
embarbarbar = G(1, 1000, G(3, 3, 1000))
for more see
Let Us Remember Andre Joyce |
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