"Thy knowledge is become wonderful to me: it is high and I cannot reach it." (Psalm 138:6)
GOOGOLOGYtranslated and edited by Michael Joseph Halm
copyright 2006 Hierogamous Enterprises
In the beginning was the word and the word was "googol". That was the name given by Milton Sirotta, the nephew of mathematician Edward Kasner sometime before 1940 to the number represented by one followed by one hundred zeros. In John Horton Conway and Richard K. Guy's numbering system (a modified version of Nicolas Chuquet's) it is called ten duotrigintillion. In Pelletier's it is called ten sedecillion. In Donald Knuth's it is called one myriad undecyllion. In the Pelletier-Knuth system it would be called one million sextilliard. It the unambiguous compromise system using multiple millions it is the awkward ten thousand million million million million million million million million million million million million million million million million million or ten thousand sixteen-millions, which could be confused with a mere ten thousand sixteen million, 10016000000.
It can be represented mathematically in various ways, for example, in Ackermann’s Generalized Exponential notation, g(a, b, c) = g(a - 1, g(a, b - 1, c), c),
g(0, a, b) = a + b,
g(1, a, b) = ab:
g(2, 100, 10) = ten-to-the-hundredth =
g(2, 2, g(3, 2, 10)) = to-the-second-ten squared =
g(2, 50, g(1, 2, 50)) = twice-fifty-to-the-fiftieth.
Soon after googol came the term "googolplex" for the antilogarithm of the googol, ten-to-the-googolth = g(2, g(2, 100, 10), 2) = g(2, 2, g(3, 3, 10)). "Plexing" [from "plus exponent"] like this is a handy device for naming numbers both larger than a googol or a googolplex, from eleventyplex = g(2, 110, 10) = to millilliardplex = g(2, 6003, g(3, 2, 10)).
From the analogy gross:great gross::googol:?, André Joyce concluded that "great g(2, a, b)" meant g(2, a + 1, b), as in gross = 144 = g(2, 2, 12), great gross = g(2, 3, 12); so that the first Arabian number (i. e., with 1001 digits, from 1001 Arabian Nights) is the great googol = g(2, 1000, 10), and great googolplex = g(2, 1000, g(3, 2,10)). Then from another analogy this time from genealogy's great great grandfather = two-greats grandfather, he got to z-greats googol = g(2, n + 2, g(2, 100, 10)), and z-greats googolplex = g(2, n + 2, g(2, 100, g(3, 2, 10))). He also defined 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, 12), and since the Poulter's gross = g(2, 2, 14) = 196, then the Poulter's great gross = g(2, 3, 14) = 2744 and the n-greats Poulter's gross = g(2, n + 2, 14). The gross numbers beyond a googol start with 93-greats gross, 90-greats Baker’s gross, 88-great Poulter’s gross.
great googol = g(2, 3/2, g(2, 100, 10)) = g(2, 1000, 10)
googolteen = g(2, 100, 10) + 10
googolty = g(2, 101, 10)
googolilliard = g(2, 6g(2, 2, 100, 10) + 3, 10)
googolylliard = g(2, 16g(2, 100, 10) + 8, 10)
googolplex = g(2, g(2, 100, 10), 10) = g(2, g(2, g(2, 2, 10), 10), 10) = g(3, 2, 2, 10)
eleventyplex = g(2, 110, 10)
Doug of the Googolplex Project (at procrastinators.org) estimated printing out a googolplex would take another 3.125g(2, 85, 10) years to complete at the then current rate of 783400 zeroes/sec. Frank Pilhofer recalculated taking into account the growth in printer speed and figured a mere 566 years. Paul Durish sped up the process considerably by not waiting for a faster technology but by changing to a more convenient base and got: googolplex = 10 (base googolplex) in a few seconds.
André Joyce noticed that a googol could also be expressed as f (50) = g(2, 50, g(1, 50, 2)). The discovery that the prefixes googo- and the googolple- indicated the operations operating on Roman numeral, n, googo(n) = g(2, n, 2n) and googolple(n) = g(2, g(2, n, 2n), n), was one of the early developments in a more logical and laconic system of large number nomenclature now called googology (not to be confused, though it often is, with googlology, the art and science of googling or using the google.com search engine).
Many other Roman numerals can therefore also be used as an operand, besides the original -x and -l such as: -i = 1, -ij = 2, -iv = 4, -v = 5, -vi = 6, -vij = 7, -ix = 9, -xi = 11, -xij = 12, -xiv = 14, -xvi = 16, -xvij = 17, -xix = 19, -xxi = 21, -xxij = 22, -xxiv = 24, etc.
There could also be the less proper, but more pronounceable, ones, like -xxxxxix = -il = 49, -lxxxxxix = -ic = 99, clxxxxix = -cic = 199, -cxxxxix = -cil = 149, -ccclxxxxix = -cid = 399, etc.
Joyce multilingually also used the Mayan ox = 3 instead of the Roman iii or iij to fill in the gaps: -ox = 3, -vox = 8, -xox = 13, -xvox = 18, etc.
Thus names can be formed like:
googoc = g(2, 100, 200) > g(2, 230, 10)
googoci = g(2, 101, 202) > 6g(2, 232, 10)
and the Italian-like googocci = g(2, 201, 402) > 2g(2, 523, 10)
the adjective-like googoccic = g(2, 299, 598) > g(2, 830, 10)
googolmox = g(2, 1003, 2006) > g(2, 3312, 10)
Many more Roman numerals can be made pronounceable using googology’s Principle of Equivalency which equates orthography and phonetics, x = "ex", l = "el", m = "em" [or alternatively from telegraphers’ jargon, "ump", from which comes the numbers "umpteen" = 1010, and "umpty" = myriad = 10000]
googolex = g(2, 60, 120) > 5g(2, 124, 10)
googoxem = g(2, 990, 1980) >5g(2, 3263, 10)
googomump = g(2, 2000, 4000) > g(2, 7204, 10)
By googology’s Principle of Recursivity the gaps left in the Roman numeral representations because of limitations of the Pronounceability Principle can be filled. Numbers can be constructed by adding together smaller repeating digital strings, multiples of Samuel Yates' repunits ["replicated units"] = R(z) = [g(2, z, 10)/9] = z "1"s.
quadrix = 4 "9"s = 9999 = 9[g(2, 5, 10)/9] = g(2, 5, 10) - 1
vigintiv = 20 "4"s = 44444444444444444444 = 4[g(2, 20, 10)/9] > 4g(2, 19, 10)
quadrixvigintiv = 999944444444444444444444 = (g(2, 5, 10) - 1)g(2, 19, 10) + 4[g(2, 20, 10)/9] > 9g(2, 23, 10)
duquadrixvigintiv = 999944444444444444444444999944444444444444444444 = ((g(2, 5, 10) - 1)g(2, 19, 10) + 4[g(2, 20, 10)/9])g(2, 24, 10) + (g(2, 5, 10) - 1)g(2, 19, 10) + 4[g(2, 20, 10)/9] > 9g(2, 47, 10)
centix = g(2, 101, 10) - 1 = 100 "9"s
Another interesting set of related numbers, whose patterns persists in their multiples, could be named using the astronomical suffix, -ile, for 1/n:
the sacred number of the Pythagoreans, [g(2, 6, 10)/7] = 142857 = integal megaseptile
Robert Ripley's persistent number, [g(2, 18, 10)/19] = 52631578947368424 = integtal exaundevigintile
and similar numbers, like [g(2, 16, 10)/17] = 588235294117647 = integral dekapetaseptemdecile
and [g(2, 22, 10)/23] = 434782608695652173913 = integral dekazettatrevigintile
[g(2, 28, 10)/29] = 344827586206896551724137931 = integral myriayottaundetrigintile,
[g(2, 46, 10)/47] = 212765957446808510638297872340425531914893617 = integral dekazettayottaseptemquadragintile
Noting that the representation for thousands above M in Roman numerals is with an added bar above the numeral, Joyce defined -bar as a multiplicative operator on the Roman numerals:
googolbar = g(2, 50000, 100000) = g(2, 250000, 10)
googocbar = g(2, 100000, 200000) > 5g(2, 530103, 10)
googodbar = g(2, 500000, 1000000) = g(2, 3000000, 10)
googombar = g(2, 1000000, 2000000) > 9g(2, 6301029, 10)
googomembar = g(2, 2000000, 4000000) > 9g(2, 13204119, 10)
Applying the -bar operator not only to the Roman numerals, but also to the Latin infixes and itself forms much abbreviated googologisms, if "barbaric" ones, Using Joyce’s More Generalized Exponential notation to include nestings and simultaneous operations --
g(a, b, c, d) = g(a - 1, b, c, g(a, b, c))
g(a, 1, b, c, d) = g(a - 1, 1, b, g(b, c, d), d)
g(a, b, c, d, e) = g(a - 1, b, c, g(c, d, e), e)
g(a, b, c, d, e, f) = g(a - 1, b, c, g(d, e, f), e, f)
g(a, 1, 1, b, c, d) = g(a - 1, 1, 1, g(b, c, d), c, d)
g((a, b), c, d) = g((a - 1, b - 1), g((a, b), c - 1, d) , d) = g(a -1, g(b -1, g(b, c - 1, d) - 1, d)
we have:
umpbarbar = g(1, g(2, 1000, 1000), 1000) = g(3, 2, 1000) + 3, 10) = g(2, 3003, 10)
umpbarbarbar = g(1, g(2, g(2, 1000, 1000), 1000), 1000) = g(2 3g(2, 3003, 10) + 3, 10)
Multiple plexing has however became indicated by other googologists with infixes like du- (from the Latin "duplex" meaning "twofold" or "double"):
googolduplex = g(2, g(2, g(2, g(2, 2, 10), 10), 10), 10) = g(4, 1, 2, 2, 10)
googoltriplex = g(5, 1, 2, 2, 10), etc. [NOTE: The -triplex suffix is pronounced "try-pleks" NOT "triple-X"!]
Joyce's barbarian numbers would therefore be rather umpdubar, umptribar, etc.
Young Kieran Cockburn named a googolplex googolplex or googolplex squared = g(2, 2, g(2, 100, 10) = g(2, 200, 10) a "gargoogol" . His father, Alistair Cockburn added fuga- = g(3, 2, z), gag- = g(z + 1, z, 2) and named Stephan Houban's megafuga- = g(4, 2, z), to get fugagoogol, gaggoogol, and megafugagoogol = g(4, 2, g(2, 100, 10)).
Tom Kreitzberg extrapolated that to mag-, which in Joyce's notation is g(z, z + 1, z, 2), while Joyce in turn extrapolated to -ag-, by analogy to -illion, and so to baggoogol = g(2, z, z + 1, z, 2), traggoogol = g(3, z, z + 1, z, 2), quadraggoogol = g(4, z, z + 1, z, 2), etc.
"Mega-" usually however refers to the Greek-based SI (metric) prefixes:
g(2, 6, 10) = mega- ["large"], g(2, 9, 10) = giga- ["giant"], g(2, 12, 10) = tera- {"monstrous"], g(2, 15, 10) = peta-, g(2, 18, 10) = exa-, g(2, 21, 10) = zetta-, g(2, 24, 10) = yotta-.
googolyottaplex = g(g(2, 24, 10) + 3, 1, 2, 2, 10) = g(2, 2, g(3, 24, 10) + 3, 10))
Stephan Houban estimated megafugafour = g(4, 2, 4) > g(2, 153, g(3, 2, 10)) and claimed "We can safely say that computing all the digits of megafugafour will never happen.", to which Sunir Shah responded with "Never happen? Nonsense! Here they are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. (Some assembly required.)" The same of course applies to nearly all larger numbers, since above the first pandigital number, 1023456789, the probability of a number not having at least one of each digit approaches zero
Joyce expanded his system by noting that not only did the o-count indicate the operation number of the generalized exponential, but the vowel sound indicated its English name. o = 1, oo =2, and so ee = 3, or = 4, ie = 5, i = 6, e = 7 and ei = 8. The so-called g-count could also be extrapolated to three by noting googol = g(2, 50, g(1, 50, 2)).
googgol = g(2, 50, g(1, 50, 3)) = g(2, 50, 150) = > g(2, 108, 10)
googgool = g(2, 50, g(2, 50, 3)) > g(2, 1192, 10)
geegeel = g(3, 50, g(3, 50, 2))
geeggeel = g(3, 50, g(3, 50, 3))
geiggeil = g(8, 50, g(8, 50, 3))
geiggeim = g(8, 1000, g(8, 1000, 3))
Jonathan Bowers came up with the names:
gaggol = g(3, 10000, 10)
gygol = g(3, 10000000, 10)
gagol = g(3, 1000000000, 10)
boogol = g(11, 100, 10) and
tritri = g(4, 3, 3) = g(5, 2, 3)
tridecal = g(11, 10, 10) = g(12, 2, 10)
tetratri = g(2, 1, 1, 4, 3, 3) = g(g(3, 3, 3), 3, 3) = g(g(2, 27, 3), 3, 3)
pentatri = g(3, 1, 1, 4, 3, 3) = g(g(g(3, 3, 3), 3, 3), 3, 3) = g(g(g(2, 27, 3), 3, 3), 3, 3)
grand tridecal = g(3, 1, 1, 11, 10, 10)
dimentri = g(g(2, 3, 3), 1, 1, 4, 3, 3)
dulatri = g(g(2, g(2, 2, 3), 3), 1, 1, 4, 3, 3)
trimentri = g(g(2, g(2, 3, 3), 3), 1, 1, 4, 3, 3)
decaltrix = g(g(11, 10, 10), 1, 1, 11, 10, 10)
googolplux = g(g(2, 100, 10), g(2, 100, 10), 1, 1, 2, 100, 10) = g(g(3, 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
Which extrapolates, since c = g(2, 2, x) and m = g(2, 3, x) to:
googolpluc = g(g(g(2, 2, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
googolplum = g(g(g(3, 3, 3), 2, g(2, 100, 10)), 1, 1, 2, 100, 10)
Beyond Cockburn's mag- is naturally bag-, trag- quadrag-, and other formations from the -ag- infix:
baggoogol = g(2, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
traggoogol = g(3, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
quadraggoogol = g(4, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
centaggoogol = g(100, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
dentaggoogol = g(500, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
mentaggoogol = g(1000, 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
bentaggoogol = g(g(1, 2, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
trentaggoogol = g(g(1, 3, 1000), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
yottaggoogol = g(g(2, 24, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
The higher Greek prefixes are: hecto- meaning hundred, kilo- meaning thousand, and myria- meaning ten thousand. These have been officially extrapolated in the international metric system to include the even higher prefixes:
mega- "large" or ten-to-the-sixth, giga- "giant" or ten-to-the-ninth, tera- "monstrous", [or from the contraction te(t)ra-] or ten-to-the-twelfth, peta- [pe(n)ta-] or ten-to-the-fifteenth, exa- [(h)exa-] or ten-to-the-eighteenth, zetta- or ten-to-the-twenty-first, and yotta- or ten-to-the-twenty-fourth
Since zetta- (z + (s)etta) g(2, 21, 10) and yotta- (y + otto) g(2, 24, 10) obviously seem formed from the reverse alphabet (as Aronson and Blowers noticed) and Italian numbers (nove, dieci, undici, dodica, tredici, quattordici, quindici, sedici, diciasette, diciotto, diciannove, venti, etc.) Joyce extrapolated getting xova- [x + nove] g(2, 27, 10), weica- [w + dieci] g(2, 30, 10). To keep them the traditional bisyllabic he used the just the initial syllables:
vunda- [v + undici] g(2, 33, 10), uda- [u + dodici] g(2, 36, 10), treda- [t + tredici] g(2, 39, 10) and satta- [s + quattordici], g(2, 42, 10), rinda- [r + quindici] g(2, 45, 10), qeda- [q + sedici] g(2, 48, 10), pica- [p + diciasette] g(2, 51, 10), oca- [o + diciotto] g(2, 54, 10), nica- [r + diciannove] g(2, 57, 10), menta- [m + venti] g(2, 60, 10).
Beyond menta- -(e)nta- is used until sara- [s + quaranta] g(2, 120, 10), -(a)ra- until inqua- [i + cinquanta] g(2, 150, 10), -(i)nqua- until yessa- [y + sessanta] g(2, 180, 10), -(e)ssa- until otta- [o + settanta] g(2, 210, 10), (e)tta- to fetta- [f + settanta] g(2, 237, 10). To prevent confusion with eta- or yotta- higher infixes would have increase to trisyllablic, (o)ttanta- until uvanta- [u + novanta] g(2, 270, 10), (o)vanta- to lovanta- [l + novanta] g(2, 297, 10). This is the limit of this system since to apply it any higher we would have both kenta- [k + venti] g(2, 66, 10) and [k + cento] g(2, 300, 10).
So we reach lovantaggoogol, g(g(2, 297, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)) and beyond that:
myriaplexaggoogol, .g(g(2, 10000, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100))
megaplexaggoogol, g(g(2, 1000000, 10), 1, 1, g(2, 50, 100), g(2, 50, 100), g(2, 50, 100), g(2, 50, 100)), etc.
These very, very, very small numbers once again confirm the Frivolous Theorem: "Almost all natural number are very, very, very large."
for more see
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