alright guys, here’s another old article from my large number site. it’s about andre joyce’s googo- and googolple- system, and it was in fact the very first article on my site. for a while it lurked around as an unfinished 3-part page, but i have now finished a new article on that system. for history’s sake i will archive the old unfinished googo- and googolple- article on this blog.
PART 1
Googologist Andre Joyce has made some odd observations about the googol and googolplex, and devised a googo- and googolple- number system. This page will showcase the two.
Googo-
Googo-x is (2x)^x, where x is written in Roman numerals.
If you don’t remember, they’re:
I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1,000
Since L is 50, googol would evaluate to 100^50 = 10^100, which is correct.
Googolple-x needs some modification though, because Joyce somehow fucked up the definition (in fact, most of his googology is broken and needs fixes to be able to work right). The correct definition would be x^x^x^2, where x is in roman numerals. Googolplex here would be 10^10^10^2 = 10^10^100, which is once again correct.
Let’s examine googo- first:
Googo-1 = googoi (goo-goy) = 2^1 = 2, which is trivial even in everyday life.
Googo-2 is differently named. One would think it’s googoii, but that can’t be pronounced any differently from googoi. So Joyce allows us to use ij in place of ii Therefore it’s googoij (goo-goyj), or 4^2 = 16. In day to day life it’s a sizable number (for example, having 16 kids is outrageous), but easy for us to comprehend, and trivial on the larger scale.
Since Joyce says we should use “ox” in place of “iii”, googo-3 should be called googoox, but that’s too odd, and confusable with “x” (10). I instead suggest “iji” instead, so googo-3 is googoiji (goo-goy-jee). Googoiji evaluates to 6^3 = 216, which is sizable in the everyday world and generally impressive, but still palpable to the mind.
Googo-4 is called “googoiv” (goo-goyv). It evaluates to 8^4 = 4,096, around the limits of what our minds can visualize.
Googo-5 is called googov (goo-gawv). It’s equal to 10^5, or exactly 100,000 – called lakh in the Indian number system.
Googo-6 is called googovi (goo-gaw-vye) = 12^6 = 2,985,984, or about three million. It’s approximately the population of Missisippi.
Googo-7 = googovij (goo-gawv-ij) = 14^7 = 105,413,504, a bit less than the population of Mexico.
Googo-8 = googoviji (goo-guh-vee-jee) = 16^8 = 4,294,967,296, around the population of Asia
Googo-9 = googoix (goo-goyx) = 18^9 = 189,359,290,368. Comparable to the number of stars in the Milky Way Galaxy.
Googo-10 = googox (goo-gawx) = 10,240,000,000,000, or 10 trillion. Comparable to the number of human cells in your body, and slightly more than the googologically important megafuga-three (7,625,597,484,987).
Googo-11 = googoxi (goo-gawx-ee) = 584,318,301,411,328, or 584 trillion. This many miles is around 10 light years, or the distance to the star Sirius.
Googo-12 = googoxij (goo-gawx-ij) = 36,520,347,436,056,576, or 36 quadrillion. This many miles is around 6,000 light years, or the distance to the Perseus Spiral Arm in the Milky Way Galaxy.
Googo-13 = googoxiji (goo-gawx-ee-jee) = 2,481,152,873,203,736,576, or 2.5 quintillion. This many miles is around 425,000 light years, or four times the diameter of the Milky Way.
Googo-14 = googoxiv (goo-gawx-iv) = 182,059,119,829,942,534,144, or 182 quintillion. This many miles is around 31 million light years, or the distance to the Sombrero Galaxy.
Googo-15 is a bit trickier, as googoxv is hard to pronounce. Fortunately, Joyce recently addd “vy” as in “ivy” in place of v. “googoxvy“, therefore, is pronounced “googox-vee”. It’s approximately 1.435*10^22, or 14 sextillion. At this scale it’s harder to come up with good and precise examples for the magnitude of these numbers.
Googo-16 = googoxvi (goo-gawx-vye) ~ 1.209*10^24, or 1.2 septillion.
Googo-17 = googoxvij (goo-gawx-vij) ~ 1.084*10^26, or 108 septillion.
Googo-18 = googoxviji (goo-gawx-vee-jee) ~ 1.031*10^28. This number is bigger than the diameter of the observable universe in meters!
Googo-19 = googoxix (goo-gawx-ix) ~ 1.037*10^30.
Googo-20 = googoxex (Joyce allows us to use “ex” instead of “x” to make it pronounceable) ~ 1.100*10^32.
We can continue withgoogoxexi, googoxexij, googoxexiji, googoxexiv, googoxexvy, googoxexvi, googoxexvij, googoxexviji, and googoxexix, but by now we can go at a faster rate and take jumps of tens.
Googo-30 = googoxexex ~ 2.210*10^53. That’s roughly the amount of atoms in Jupiter.
Googo-40 = googoxel ~ 1.329*10^76. Googoxeliji is the first googo- number more than the number of atoms in the observable universe, but still less that the number of small volumes in large spaces (like Planck volumes in the universe), and much less than many combinatorical numbers such as the odds of a monkey typing out Hamlet on its first try.
Googo-50 = the famed googol = 10^100. It’s often regarded as impractically large, but that isn’t true – it’s still less than, say, the number of Planck volumes in the universe. It’s also not too much less than (for instance) the number of ways to arrange a 6x6x6 Rubik’s cube (that’s around 10^116), so if you happen to get your hands on one, you have a real example of a number bigger than a googol RIGHT ON YOUR FINGERTIPS!
Googo-60 = googolex, a number Joyce writes on his website. It’s around 10^124.
Googo-70 = googolexex ~ 10^150. That’s almost as much as 4^^3 = 4^4^4 ~ 10^153.
Googo-80 = googolexexex ~ 10^176.
Googo-90 = googoxcy, since “cy” (as in Nancy) is offered as an alternative to c. In pronunciation, it could be confused with googoxi, so I’ll make it pronounced “goo-gawks-kee” instead. This number is around 10^203, or slightly more than a gargoogol (googol^2 = 10^200). In reality it’s a thousand times larger, but you get the point. It’s more than the number of Planck volumes in the observable universe!
Now for googo-s of 3-digit numbers:
Googo-100 = googoc (goo-gawk) ~ 10^230. This one seems to be brought up particularly often, probably due to its simple name. It’s equal to bingol (2^100) times gargoogol (10^200).
With that we can give a whole table of the first 100 googo- numbers:
1 2
2 16
3 216
4 4096
5 100,000
6 2.98598E+6
7 1.05413E+8
8 4.29497E+9
9 1.98359E+11
10 1.024E+13
11 5.84318E+14
12 3.65203E+16
13 2.48115E+18
14 1.82059E+20
15 1.43489E+22
16 1.20892E+24
17 1.08428E+26
18 1.03144E+28
19 1.03726E+30
20 1.09951E+32
21 1.22528E+34
22 1.43205E+36
23 1.75158E+38
24 2.23763E+40
25 2.98023E+42
26 4.13129E+44
27 5.95157E+46
28 8.89741E+48
29 1.37851E+51
30 2.21074E+53
31 3.66557E+55
32 6.27709E+57
33 1.10905E+60
34 2.01977E+62
35 3.78818E+64
36 7.31086E+66
37 1.45068E+69
38 2.95744E+71
39 6.18997E+73
40 1.32923E+76
41 2.92662E+78
42 6.6028E+80
43 1.52557E+83
44 3.60776E+85
45 8.72796E+87
46 2.15893E+90
47 5.45767E+92
48 1.40935E+95
49 3.716E+97
50 1.E+100
51 2.74542E+102
52 7.68659E+104
53 2.19387E+107
54 6.38091E+109
55 1.89059E+112
56 5.70439E+114
57 1.75217E+117
58 5.47726E+119
59 1.74196E+122
60 5.63475E+124
61 1.85332E+127
62 6.19647E+129
63 2.10545E+132
64 7.26839E+134
65 2.54869E+137
66 9.07568E+139
67 3.28111E+142
68 1.20405E+145
69 4.48391E+147
70 1.69419E+150
71 6.4934E+152
72 2.52406E+155
73 9.94852E+157
74 3.97527E+160
75 1.61007E+163
76 6.60865E+165
77 2.7485E+168
78 1.15803E+171
79 4.94208E+173
80 2.13599E+176
81 9.34794E+178
82 4.14186E+181
83 1.85768E+184
84 8.43295E+186
85 3.87399E+189
86 1.80072E+192
87 8.46807E+194
88 4.02823E+197
89 1.93812E+200
90 9.43029E+202
91 4.63976E+205
92 2.30802E+208
93 1.16066E+211
94 5.89981E+213
95 3.03104E+216
96 1.57369E+219
97 8.25596E+221
98 4.37617E+224
99 2.34344E+227
100 1.26765E+230
In exponential functions, the ratio between f(x) and f(x-1) is constant. But the ratio between googo-x and googo-(x-1) increases, albeit slowly. Therefore googo-x grows slightly faster than exponentiation.
Googo-101 = googoci (goo-gawk-ee) ~ 10^232
Googo-110 = googocex (goo-goh-sex) ~ 10^258
Googo-150 = googocel (goo-goh-sel) ~ 10^372
….
Googo-200 = googocc (goo-gawch) ~ 10^520
Googo-201 = googocci (goo-gawch-ee). Its name is reminiscent of Italian, and it’s about 10^523.
Googo-300 = googoccc. That’s pronounced goo-goshk. It’s about 10^833.
Googo-400 = googocdy (goo-gawk-dee) ~ 10^1161. That’s more than Joyce’s great googol (10^1000)! Joyce offers “dy” (as in lady) as an alternative to “d”.
Googo-500 = googod. That’s exactly 10^1500, and it has an awesome name, doesn’t it?
Googo-600 = googodcy (goo-god-see) ~ 10^1847
Googo-666 = googobeast. Just kidding. It’s called googodecelexvi (goo-goh-dek-uh-lex-vye) ~ 10^2081.
Googo-700 = googodccy (goo-god-chee) ~ 10^2202
Googo-800 = googodcccy (goo-god-shkee) ~ 10^2563
Googo-900 = googocem (goo-goh-sem) ~ 10^2930
And finally:
Googo-1000 = googom ~ 10^3,301
and:
Googo-1337 = googoleet = 10^133,337. Just kidding. It’s actually googomcccexexexvij (goo-gawm-shkex-ex-ex-vij) ~ 10^4582.
Googo-2000 = googomem ~ 10^7204
Googo-3000 = googomemem ~ 10^11,334
We can go further only with a more confusing -bar system Joyce devises, but I won’t discuss that. Instead, in an upcoming article I’ll discus Joyce’s crazy extension, producing insane googologisms like geiggeim.
Googolple-
Remember that googolple-x is x^x^x^2, where x is in Roman numerals. From there we can devise some whole new numbers, far far larger than the googo- group.
Let’s start:
Googolple-1 = googolplei (goo-guhl-play, not to be confused with Google Play) = 1^1^2 = 1^1 = 1. Heh, pathetically trivial.
Googolple-2 = googolpleij (goo-guhl-playj) = 2^2^4 = 2^16 = 65,536. A good jump from before, but not quite … scary, to say the least. It’s between googoiv and googov, closer to googov.
Googolple-3 (googolpleiji [goo-guhl-play-jee]) is much larger. It’s equal to 3^3^9 = 3^19,683 =
15054164145220926243143298033398654543076143473537427823628361588037621543571301695623049264407339299686835128946425225443147826949787930561330768130940971342651498914747987903017936839366686
68122285885321242839049917911226083599955908180577131260402529543908255721569802733735838375801214858950208374821540309800787899591183588972935501900799964862481847477524438997394562350473644
23343402644491809238620323737211030410625065937801257455332335784042728176943554941190340878924155811595534441776658455610674141780279842593542755786271110153920935777032178422467577159938505
27863143440137825302954400096599965422035350068885003919811126843083535495234137589102455390614147745453121569003468531097431528705505762776585634579969292789571943150663200583121057376633971
55950814575556513728209760157092383202207851341895364601388219241291019094085464805222876015496530944536702393684636887824070251360043094725585077115276499291932021506184721825866662885353023
50121947129574003848397764517133687439155570784450614961385303575613672545295015461941410267436925527708675304434976978636357475563244598471100543462532174113098209437635895680706269332201309
99528465911723696379451234379509178882111255496057359862889587710804096797256376246508804835264315162962684856485371330030695400531740336618389165874799967876888281848913327759470167768279402
68227670787008122473227438465924988007834165046966129808212405101073107931452434998938884757603097100780693400465838004587643991517221807140739809642545486349996417593604558889951110081812729
64236496758611805752994091256345788319505005821968348818474679135992464027438049382453287803718357005538946466964392535328124851806786635309633204043458041190231101887532289483993379253245413
38866774767148916824570033330260935113206834204238946077142955428600706271826005072958108100916095852989600626381105886901811561810802820260654105888681595530534070524243859292040849898489759
57742588092960418645417139538474674716046764422278824663181859179485126366228226004376682061461574224794668088891760943111196444384545011301201605902825294925144437051101686433854042100735707
73180717542632593987768439892681504279529955980336528154808032987921930046000288456363234407837694895755866892539105215806311767596959264298587086225271519503981087785251935938103414167637704
38316694501847590075040210810716906685947380077182469744725823543447706588324768834763195562774985477798540217873298009604291129795418962227074102940207034948505153392457242087232988694897916
91862793888648534039423093969963180426640098614852078911399894210447463851579049729817718094811734766888142838089799443262986011308104807090334628668739758557563777773656514009181195918140539
06367489580734644665573635472026643705241514001892107417716476679602610068477743660703952193115271863792094390272269071105649712903756188007430371634791670954836379620178208621889170715533057
89955126588684827797726211120735968197477850571316770150287165878389839672342545545460165773088511587169977051585969513304977545948135123659981533754846412899129174484606253824123635632075542
51529660415067186590795883628190290742679023788227130256328861343661223764938604882647827086688997158553097664398056755908158836799421631641553437731062521084297834752274999830405319675220175
42913575958666910314274385704884134369821590072521927025261154356504567731871874815655259394180313386453818548990724430642292711400862527302677489677119581804986116070499932039146909826779483
15892617601093090131230276263165247404303103302830312349238931851942362810611095371728245035650497938755128164297995830988798545113694176618213622919275401110086443810298346987303592255976821
25598053265711642177090789521634707884339612776492195292345627422856642958733287401389676372766106937960073050219522331531296711575177063790024692247009382718269065923191078746793786488421113
88870411689752215816158097250146242508949390790357014488697012259354957348686900831082746435117785965062647883184263555516853250579736344248156027880208870506443865894707757792653477625986038
18475791916367797747391246095655602223800648036700788484402544656890674685922345400047918107273508338752592743072335213702513608098172675113664693724883548584507464227538749558370794862387028
82783041912295005257520168593640771376928283057475962262590235407451837572496591952540151469643891258305554077898701840591354243299595767072484544231446080372548854276128228661686214442124253
98114100332726614290241369682705362036869989302516352514876444687137480169890770582097154355517668512838410073545158750807961331953508628678112921092870442226630161324953159425661691063898565
54513738407490597282746783788129812979028109503241161061784577387960452863724350171347066771273284045313314470700496181528490374495712531002858110305051595394688910479451764010214199411507762
27070726503478608301848060879630242508714747051988948924438643066475221039036228700696823050705819313903685229891467141694208717757668109081437751670713437088231757600814931888193645203119660
82783347757550465775029509330216583855932612042243995254737104307599710317049446667891195628131718766271780007967513816371461621576211724953300368967613967531131798612756313114761640200224109
09897096047882659825435994118092499814957541952576888465220038655008884016681944324249355478975098739325343679508671952394895274568060812616701676943632799462273012221044093036491711931096707
95305419042758455956370136929991588347873436551031717288346472593184126811451335285232619415026231482733157865852304777200560717784533913447137923443291597744849489623876110627621601709355477
48003587025250643494560628609733512364554912338175278013957215960132077055964040436855747700475035989721572625042506506781751568259590390676696040486777935752801698638129934481379615318674945
31022322307558457523498158640533674141280895252001960337471335391147429195976673894626270274982809064126149798002611144117956042410908431484297293144189777073277773490492738066740664233210622
71202575841491499413844857238879529343168429045708304972544241165775135548874100071149860327599246015118832031545959631308402088913869429074533110070447157369812888530586061683277930009265590
39310111364541240125028344094645848443330095774012832934930390083677878882792442662255360692722175902251974455090104074624458040221965114664787905053443096653683423225747602736649082735329869
07239429047653543889522510964447987592981261101983587282354735843446222690200592988638558297753636554986441366606794955936586071815580897396383767600604085569719731573737409125886970183544850
35857040259593077929434206514998356736919278228358576787713830183148493547819329897495236875539995126635484535340043242852374530124804404470839344040057144673909558000526586419130504740032442
50930337745502541153588944206146195153471001028032810370179966060777340043184303673501358572952980715367093890586299813496871369671768970129687475538583843155711360664976538108442601926592955
31264454812441067392610313006281638288464133553640683734596126398182938134198336119384208860678303356828827368326257900841199908953148996330700884531858248914582382414327794347267353691404871
79015162036960769713882565226212063116929267313791250181117863133139961036249841585335851225354780328319115856806899249126309946669172121073950067827921672393353562458738876886256277266269800
90338922528182072746544055132320403552278402265750464963932754934197602248577455183320937197719474744902380550959723523292999316508756432230826796287450938281110163159333931248721297748087055
67286065277119175281497113116191399734630378918950419509118629966509123869503823181113293719579137656793995668211103411950951209945959841659011610246278075282594644961723663161143100131743272
92844581392929206690578459061904202026145658890975285698330349572543960219492942303951972966927969919531242393967216151541079060032561093865827495099304347761717005314830798152671008900002312
55348846775478426172672486030229928075384101305178749555385440519775761026538154474442642024921839398986039320604203530228489300855150872774019263237545927066668584541668663191720435630143953
32450824604323134628233189726259964817347699334724873094310181800468506104501596188346299057817355426208801711234312356953071644447461655788233775970095817597818456353908694666333903059449195
25753433790458465344715232382753401483963349947225710436171972490636778916156329920537172202699850030801801007533083423053436271396373649453262162193702377701412525836542640122855963226082885
32782374130967638664840256772747285396629699564479786202210968893484811408669320432788860164489126981426787703315324211766127065634230291759906139056289751186983843217665999151660734010896680
33866547334467420905300410309064838693758551618802141617224288315773463395943125049927438541252450205809247513836038150781515868192120377789747366699247103946722211891432646926889280180230599
07712423811011748456575642057560101330906137143133614027629998937565251511369043608901561084711135286813991196261256304807840842927033826124454050520121385079509260672867230052761865624665898
51740748488277973314447750368758527886697733525248563329059990781475893232868814770118996304617660797626984215320374729275395730214352147964248404198973377840192662536451626396849226092380575
21564433762094003846822425260538637704370058723047408010462311425378038791427283199211721973874526685271925015961226614367134563010518266085961390928949673283942571081582168645131291677580454
185880097800510818762686617859227
Wow! That’s impressive, but it’s still possible to write in full, even by hand – its expansion is above. This number is 9,392 digits long, placing it between googomem and googomemem, and around the breaking point of the googo- system without the bar stuff.
How about the NEXT number?
Googolple-4 = googolpleiv (goo-guhl-playv) = 4^4^16 = 4^4,294,967,296.
We can approximate this in scientific notation as follows: = 10^log(4^4,294,967,296) = 10^(4,294,967,296*log(4)) ~ 10^(2,585,827,972.983642077204) = 10^2,585,827,972 * 10^0.983642077204 ~ 9.63035*10^2,585,827,972. We’ll need to use googo- on numbers in the billions to get to a number this big! It’s too large to directly store on this website, but it’s possible to put as a downloadable text file. Even then, it’s ridiculously big for a text file, and would take up a rather enormous portion of a hard drive.
From there on we can now that it’s around 2.6 billion digits long, making it comparable to Jonathan Bowers’ nanillion (10^3,000,000,003, the billionth -illion). It begins 9630350133920400475534…..
This number can still digitally be written out in full, but it would take up gigabytes of space. The last digits are harder to compute, but still possible. Here’s what we’ll do for the last two:
4^1 ends 4
4^2 ends 16
4^3 ends 64
We can continue, knowing that the 4*(the last two digits as a 2-digit number) will be the last two digits of (4*the whole number)
4^4 ends 56
4^5 ends 24
4^6 ends 96
4^7 ends 84
4^8 ends 36
4^9 ends 44
4^10 ends 76
4^11 ends 04
Then it starts over, and cycles at a sequence of 10 terms. From here on out we can determine that the last 2 digits of 4^x = f(x mod 10), where f(n) is the last 2 digits of 4^n, which can be looked up above.
4,294,967,296 mod 10 is equal to 6, so we can say this number ends …..96. More digits are possible, but it’s a little trickier from here on. I might do it sometime, but for know, here’s the beginning and end of googolpleiv: 96303……(2.6 billion digits)…..96.
Up next is googolplev, which is equal to 5^5^25, which is around 5^(2.98*10^17). In terms of 10 it’s around 10^(2.08*10^17), or around E17#2 (that’s 10^10^17) in Hyper-E Notation. In Hyper-E, Ea#b = 10^10^10…..^10^b, with a 10s. The number is around 200 quadrillion digits long. It’s possible to compute both the leading and ending digits easily, but let’s do that later. Storing this number is possible on a supercomputer with around an exabyte (exabyte is 1 million terabytes) of space or around a million terabyte files, but it’s hard and time-consuming to do so.
Googolplevi is next in line – it’s equal to 6^6^36 ~ 10^(8.01*10^27) ~ E28#2 – that’s 8 octillion digits long! It can’t be stored on all the data in the world (somewhere in the zettabytes, a zettabyte is 10^21 (sextillion) bytes or a billion terabytes), but it probably will be possible in the future … but it’s safe to say that it’s quite a while before we are able write this monster in full. It would take up over a brontabyte of space to store in full.
The seventh member of the series is the googolplevij, equal to 7^7^49 ~ 10^(2.7*10^41) ~ E41#2. Writing this number is near impossible, but it’s hard to say whether it’s “possible” or “impossible”. It will require technology over a QUINTILLION (that’s a million million million, or a billion billion) TIMES more powerful than what we have today (keep in mind that a quintillion is HUGE), and let’s just say that.
Number the eighth is googolpleviji = 8^8^64 ~ 10^10^58 = E58#2. The number of digits is a billion times the number of atoms on Earth!!! It’s impossible to write out, as we’ll need a billion Earths at LEAST, but definitely more because most atoms on planets are in the inaccessible core. The number of atoms on Earth can be considered a limit of “writable”, but it’s still kind of arbitrary.
What’s next? The googolpleix (goo-guhl-playx) = 9^9^81 ~ 10^10^78 = E78#2 – the number of digits is approximately the number of atoms in the observable universe, and a hard limit of “writable” – even numbers at this scale are guaranteed to be unwritable.
And now for number ten – the legendary googolplex = 10^10^100. It’s a well-established statement that it’s impossible to write a googolplex, but unlike the notion that “there isn’t a googol of anything”, this one is true. But it can still represent real-world things – for instance Sbiis Saibian’s promaxima is around 10^10^343, and it’s an estimate of the number of possible histories of the universe, start to finish, assuming Planck units aren’t meaningful.
These first ten terms alone are a pretty interesting sequence. But what unfathomable stuff lies ahead?
Googolplexi is next in line. It’s equal to 11^11^121 ~ E126#2.
Now let’s skip to googolplexvy (goo-guhl-plex-vee) = 15^15^225 ~ E265#2 – the number of digits this has is more than the number of PLANCK VOLUMES in the observable universe!
Googolple-20 = googolplexex ~ E520#2. As you can see the growth rate of googolple- is faster than hyper-exponential! Hyper-exponential growth is when the number of digits grows exponentially. But it’s still much much slower than hyper-hyper-exponential, where the number of digits increases hyper-exponentially.
Googolple-30 = googolplexexex ~ E1329#2.
Googolple-40 = googolplexel ~ E2563#2
Googolple-50 = googolplel ~ E4247#2
Googolple-60 = googolplelex ~ E6401#2
Googolple-100 = googolplec = 10^(2*10^20,000) – even the exponent now is becoming very, very large.
Googolple-200 = googolplecc (goo-guhl-plech) ~ E92,041#2 ~ E5#3. This number has nearly agoogolgong (10^100,000) digits!!!
Googolple-500 = googolpled ~ E674,742#2, between E5#3 and E6#3, closer to the latter. Writing the number is impossible, and now even the exponent takes up megabytes of space.
Googolple-1000 = googolplem = 1,000^1,000^1,000,000 = 10^(3*10^3,000,000),, between E6#3 and E7#3. Interestingly, this number (hyper-exponentially) is very close to Bowers’ megillion, equal to 10^(3*10^3,000,003+3)
Since the googolple- system is now near its breaking point, here’s a table showing the number of digits in each of the first 100 googolple- numbers:
1 1
2 5
3 9,392
4 2.58583E+9
5 2.08309E+17
6 8.02616E+27
7 2.17125E+41
8 5.66879E+57
9 1.8763E+77
10 1.E+100
11 1.06201E+126
12 2.72391E+155
13 2.01043E+188
14 5.01565E+224
15 4.90879E+264
16 2.16463E+308
17 4.89555E+355
18 6.41251E+406
19 5.45551E+461
20 3.35958E+520
21 1.65957E+583
22 7.25215E+649
23 3.0768E+720
24 1.38551E+795
25 7.21084E+873
26 4.70658E+956
27 4.16797E+1043
28 5.3999E+1134
29 1.10092E+1230
30 3.78904E+1329
31 2.35583E+1433
32 2.82571E+1541
33 6.96754E+1653
34 3.75607E+1770
35 4.69881E+1891
36 1.4457E+2017
37 1.15751E+2147
38 2.54746E+2281
39 1.62527E+2420
40 3.16712E+2563
41 1.98234E+2711
42 4.18861E+2863
43 3.13491E+3020
44 8.71242E+3181
45 9.41136E+3347
46 4.13369E+3518
47 7.71512E+3693
48 6.38841E+3873
49 2.44711E+4058
50 4.52028E+4247
51 4.19088E+4441
52 2.02981E+4640
53 5.33931E+4843
54 7.92156E+5051
55 6.88236E+5264
56 3.6319E+5482
57 1.20749E+5705
58 2.61835E+5932
59 3.8362E+6164
60 3.93012E+6401
61 2.91081E+6643
62 1.61056E+6890
63 6.88122E+7141
64 2.34325E+7398
65 6.56447E+7659
66 1.56052E+7926
67 3.24437E+8197
68 6.08464E+8473
69 1.05937E+8755
70 1.76313E+9041
71 2.88807E+9332
72 4.78992E+9628
73 8.27211E+9929
74 1.52903E+10236
75 3.10779E+10547
76 7.13901E+10863
77 1.9011E+11185
78 6.03099E+11511
79 2.33782E+11843
80 1.13599E+12180
81 7.09369E+12521
82 5.83438E+12868
83 6.48633E+13220
84 9.96906E+13577
85 2.17224E+13940
86 6.86866E+14307
87 3.22675E+14680
88 2.30471E+15058
89 2.55862E+15441
90 4.52104E+15829
91 1.2993E+16223
92 6.2089E+16621
93 5.04631E+17025
94 7.12066E+17434
95 1.78366E+17849
96 8.10317E+18268
97 6.80852E+18693
98 1.0816E+19124
99 3.31545E+19559
100 2.E+20000
The number of digits of the exponents increases at a rate we will call R. R grows, but slowly. It can be thought of as the difference between the number of digits of the exponent of googolple-x and the number of digits of the exponent of googolple-(x-1). In a hyper-exponential function (like 10^10^x), the ratio between the logarithm of f(x) and f(x-1) is constant, but for googolple-x it increases slowly. Therefore it grows slightly faster than hyper-exponential functions.
Conclusion
In conclusion, the googo- and googolple- systems are pretty cool, but they don’t match with the goal of modern googology, which is to invent, analze, and understand new notations instead of just devising whimsical naming systems. But then again, at the time of Joyce’s writing googology was at its early larval stages, so of course there was some odd experimenting. This may also tell us why Joyce’s notation is broken; he didn’t find googology to be serious business and just some playful experimenting, not bothering to check how big his numbers are. Then again, Bowers’ work (which existed at the time) already was like modern googology, except for many larger numbers having a definition left for googologists to fully formalize and understand themselves.
What’s next? Joyce continued his googo- system with a crazy vowel and letter repeating scheme – it’s actually kind of cool.
————————————————————————————————————
PART 2
Note: Andre Joyce changed his extended googo- system to make it much less powerful – however, I prefer the more powerful old one, which I will discuss here.
Remember the googo- system? Well, Joyce decided to continue it with a crazy scheme. Here it is:
A googol type number is written like so, according to Joyce: (modified from the g function to standard mathematical/googological forms)
Gabcd
Where:
a, b, c, d are each letters or combinations of letters
a and c are vowels determined by variables A and C, respectively:
| A | a |
| 0 | o (pronunciation depends on context) |
| 1 | oo (as in food) |
| 2 | ee (as in feet) |
| 3 | or (as in born) |
| 4 | ie (i as in fine) |
| 5 | i (as in kid) |
| 6 | e (as in leg) |
| 7 | ei (a as in game) |
(the pronunciations were by me)
b is a sequence of g’s, the number of g’s is a variable B minus 1
d is the variable D written in Roman numerals (modifications allowed for pronounceability are ij for ii, iji for iii, ex for x, el for l, ec for c if it isn’t preceded by another c, ed for d, and em for m)
Gabcd = (B{C}D){A}D
(where {x} is ^^^…^^^ with x ^s, and {0} is *)
First let’s check that our rules properly evaluate googol:
In googol, we need to find A, B, C, and D.
A = 1 (oo)
B = 2 (one g)
C = 0 (o)
D = 50 (L is 50 in Roman numerals)
Then plug in the values:
(B{C}D){A}D
(2{0}50){1}50
(2*50)^50
100^50
= 10^100, which is correct.
Now let’s work out some examples, to better comprehend this crazy system.
1. Examples with “o” as vowels
Numbers of this type are of the form gobod, where b and d are described above. Let’s work out some examples:
First, the smallest example, gogoi (goh-goy). First we determine A, B, C, and D:
A = 0 (o)
B = 2 (one g)
C = 0 (o)
D = 1 (i in roman numerals)
Then we plug the values in:
(B{C}D){A}D
(2{0}1){0}1
= 2*1*1
= 2. Trivial value.
Next, we try gogoij (goh-goyj):
A = 0
B = 2
C = 0
D = 2
(2{0}2){0}2
= 2*2*2
= 8. Somewhat bigger.
Third number in the sequence, gogoiji:
A = 0
B = 2
C = 0
D = 3
(2{0}3){0}3
= 2*3*3
= 18. Still bigger, but so far growing quite slowly.
Here we can notice some patterns: only D in the sequence is changing. We can devise a definition for the gogo- prefix:
gogo-x (where x is written in Roman numerals) = (2{0}x){0}x = 2*x*x = 2x2 (that’s two times the square of x)
With that, it’s trivial to continue with some values: (for pronunciation, refer to part 1 of googo- and googolple-)
| x | number name | value |
| 4 | gogoiv | 32 |
| 5 | gogov | 50 |
| 6 | gogovi | 72 |
| 7 | gogovij | 98 |
| 8 | gogoviji | 128 |
| 9 | gogoix | 162 |
| 10 | gogox | 200 |
| 11 | gogoxi | 242 |
| … | ||
| 15 | gogoxvy | 450 |
| 20 | gogoxex | 800 |
| 30 | gogoxexex | 1,800 |
| 40 | gogoxel | 3,200 |
| 50 | gogol* | 5,000 |
| 51 | gogoli | 5,202 |
| 52 | gogolij | 5,409 |
| … | ||
| 60 | gogolex | 7,200 |
| 70 | gogolexex | 9,800 |
| 80 | gogolexexex | 12,800 |
| 90 | gogoxcy (goh-gox-kee) | 16,200 |
| 100 | gogoc | 20,000 |
Hmmm. These first 100 aren’t that impressive, even by everyday standards. The gogo- function only has modest quadratic growth – it’s literally a quadratic function. But let’s continue.
*Not to be confused with googol, or Jonathan Bowers’ goggol (that’s 10{6}100). Though that could be a continuation of the giggol, gaggol, geegol, gigol, goggol, gagol series (giggol, gaggol, geegol, etc are 10{x}100 where x is 2, 3, 4, etc, so giggol is 10{2}100 = 10^^100, gaggol is 10^^^100, geegol is 10^^^^100, etc). None of these numbers are to be confused with any of my googo- numbers (like googoxex vs gogoxex).
| 101 | gogoci | 20,402 |
| 102 | gogocij | 20,808 |
| 110 | gogocex | 24,200 |
| 120 | gogocexex | 28,800 |
| 150 | gogocel | 51,200 |
| 200 | gogocc (goh-goch) | 80,000 |
| 300 | gogoccc (goh-goshk) | 180,000 |
| 400 | gogocdy | 320,000 |
| 500 | gogod | 500,000 |
| 600 | gogodec | 720,000 |
| 700 | gogodecc (goh-goh-dech) | 1,008,200 |
| 800 | gogodeccc (goh-goh-deshk) | 1,280,000 |
| 900 | gogocem | 1,620,000 |
| 1,000 | gogom | 2,000,000 |
| 1,100 | gogomec | 2,420,000 |
| 1,500 | gogomed | 4,500,000 |
| 2,000 | gogomem | 8,000,000 |
| 3,000 | gogomemem | 18,000,000 |
We can see that this system (not counting the -bar stuff) reaches its breaking point in the ten millions order of magnitude – the gogo- prefix is relatively weak, even compared to the humble googo- prefix, whose breaking point is over a millillion (10^3003, the thousandth -illion).
But we can go further with examples only using “o” as a vowel by repeating the “g”. Let’s try some examples, with two g’s:
First: goggoi (gog-oy). First we find A, B, C, and D:
A = 0 (o)
B = 3 (two g’s)
C = 0 (o)
D = 1 (i in roman numerals)
Then we calculate:
(B{C}D){A}D
(3{0}1){0}1
= 3*1*1
= 3. Once again trivial.
The second term, goggoij (gog-oyj) is similar, but d is 2 instead of 1. So we evaluate:
(3{0}2){0}2
= 3*2*2
= 12
Right here we can conclude that a formula for goggo-x is 3x^2 (that’s three times the square of x),where x is written in Roman numerals (same variations allowed as previous).
We continue with a table, this time shorter than previous:
| x | number name | value |
| 3 | goggoiji | 27 |
| 4 | goggoiv | 48 |
| 5 | goggov | 75 |
| 10 | goggox | 300 |
| 20 | goggoxex | 1,200 |
| 50 | goggol* | 7,500 |
| 100 | goggoc | 30,000 |
| 200 | goggocc | 120,000 |
| 500 | goggod | 750,000 |
| 1,000 | goggom | 3,000,000 |
| 2,000 | goggomem | 12,000,000 |
| 3,000 | goggomemem | 27,000,000 |
*Goggol is also the name of a Bowersism equal to 10^^^^^^100 (that’s 6 ^s), which is FAR FAR FAR UNFATHOMABLY larger. I shall call this “goggol” the Joycian Goggol, as it’s created with a Joycian system. But this number is so small that it’s full decimal expansion (7,500) is easy to remember, so why bother naming it?
Well, this certainly grows faster than previous, but literally only 50% faster – not even twice as fast. Let’s try numbers with more g’s:
(gogg…ggo-x with y g’s in a row is equal to (y+1)*x^2)
gogggol (go-guh-guh-guhl) = 4*50^2 = 10,000
gogggom (go-guh-guh-gom) = 4*1000^2 = 4,000,000
gogggomemem (go-guh-guh-goh-mem-em) = 4*3000^2 = 36,000,000
goggggom (go-guh-guh-guh-gom) = 5*1000^2 = 5,000,000
gogggggom (go-guh-guh-guh-guh-gom) = 6*1000^2 = 6,000,000
This whole system with only o’s as vowels only reaches the ten millions order of magnitude here – with more g’s, the g’s get hard to keep track of, and are no longer worthy of continuing. Even then, with larger (20-100) amounts of g’s, the system barely reaches a billion, due to the innocent quadratic growth rates (with cubic growth rates overall). Overall, they can’t go further than my quadratic number group in my Pointless Gigantic List of Numbers. So now let’s explore the possibilities of adding the “oo” vowel – these numbers will probably reach exponential/hyper-exponential levels.
2. Examples with “oo” as vowels
Here we can reach new possibilities with “oo” – here we’ll try numbers that start with goo- instead of go-.
We’ll start with the googo- prefix, which we know is (2x)^x, and give an example:
googoi
A = 1 (oo)
B = 2 (one g)
C = 0 (o)
D = i (1 in roman numerals)
googoi = (2{0}1){1}1 = (2*1)^1 = 2.
And:
googo-x = (2{0}x){1}x = (2*x)^x = (2x)^x, where x is written in roman numerals
Here’s a table of some more numbers to recap from part 1:
| x | number name | value |
| 2 | googoij | 16 |
| 3 | googoiji | 216 |
| 4 | googoiv | 4,096 |
| 5 | googov | 100,000 |
| 6 | googovi | 2,985,984 |
| 7 | googovij | 105,413,504 |
Wow! The seventh member of the sequence already transcends ANY number we’ve covered so far. Let’s continue:
| 8 | googoviji | 4.3E9 |
| 9 | googoix | 1.9E11 |
| 10 | googox | 1.0E13 |
| 11 | googoxi | 5.8E14 |
| 12 | googoxij | 3.7E16 |
| 13 | googoxiji | 2.5E18 |
| 14 | googoxiv | 1.8E20 |
| 15 | googoxvy | 1.4E22 |
| … | ||
| 20 | googoxex | 1.1E32 |
| 30 | googoxexex | 2.2E53 |
| 40 | googoxel | 1.3E76 |
| 50 | googol | 1E100 (exactly equal) |
| 51 | googoli | 3E102 |
As you can see, the 51st entry even transcends the googol – oh wait, googol is part of the googo- naming system. Let’s finish up: (keep in mind that E(n) means 10^n, so E100 = 10^100 = googol)
| 60 | googolex (1) | E124 |
| 70 | googolexex | E150 |
| 80 | googolexexex | E176 |
| 90 | googoxcy | E203 |
| 100 | googoc | E230 |
| 101 | googoci | E232 |
| 110 | googocex | E258 |
| 150 | googocel | E372 |
| 200 | googocc | E520 |
| 201 | googocci | E523 |
| 299 | googoccexecix (googoccic) (2) | E830 |
| 300 | googoccc | E833 |
| 400 | googocdy | E1161 |
| 500 | googod | E1500 (exactly equal) |
| 600 | googodec | E1847 |
| 700 | googodecc | E2202 |
| 800 | googodeccc | E2563 |
| 900 | googocem | E2930 |
| 990 | googocemexec (googoxem) (2) | E3264 |
| 1000 | googom | E3301 |
| 1003 | googomiji (googolmox) (3) | E3312 |
| 2000 | googomem (googomump) (4) | E7204 |
| 3000 | googomemem | E11,334 (5) |
(1) Italic names are defined by Joyce himself.
(2) Joyce offers alternative number naming (example: ic here means 99, because c is 100 and i is 1, 100 – 1 = 99) to make numbers easier to pronounce (I don’t offer this, for consistency with Roman numerals’ sake).
(2) Joyce offers alternative number naming (example: ic here means 99, because c is 100 and i is 1, 100 – 1 = 99) to make numbers easier to pronounce (I don’t offer this, for consistency with Roman numerals’ sake).
(3) Joyce offers “ox” as an alternative to “iii” just as I offer the less confusing “iji”. By Joyce’s system googo-1003 should be googomox, but he makes a mistake and calls it googolmox.
(4) Joyce offers both “em” (which I use) and “ump” as alternatives to “m”.
(5) Joyce also defines a few -bar numbers, but like I said, they’re not of interest here.
So as you can see, the googo- system easily crushes all numbers with only “o” as vowels – but it’s only the start of more cool numbers. It’s breaking point is just over a great great googol (10^10,000).
Behold the googgo- prefix. Let’s calculate how much googgo-x evaluates to (we’ll leave D as the variable x):
A = 1 (oo)
B = 3 (two g’s)
C = 0 (o)
… and plug it in …
(B{C}D){A}D
(3{0}x){1}x
(3x)^x
… and now we’ll test some values.
| x | number name | value |
| 1 | googgoi (goo-guh-goy)* | (3*1)^1 = 3 |
| 2 | googgoij (goo-guh-goyj) | (3*2)^2 = 36 |
| 3 | googgoiji (goo-guh-goyj-ee) | (3*3)^3 = 729 |
* Sbiis Saibian wrote in his article on Joyce’s googology (which serves mostly to show how broken it is) that he doesn’t know how to distinguish googol, googgol, and googgool in speech, and doubts Joyce cares. Here is my answer – just pronounce each repeated g individually as “guh”.
The first three members of the googo- sequence (3, 36, 729) are already noticeably larger than their googo- counterparts (2, 16, 216). Let’s continue:
| 4 | googgoiv | 20,736 – that’s 5x larger than googoiv. |
| 5 | googgov | 759,375 – 7.5x larger than googov |
| 6 | googgovi | 34,012,224 – 11x larger than googovi |
| 7 | googgovij | 1.8E9 – 18x larger than googovij |
| 8 | googgoviji | 1.1E11 – 44x larger than googoviji |
| 9 | googgoix | 7.6E12 = 7,625,597,484,987 – that’s exactly equal to the important megafugathree, which is 3^3^3 = 3^27. Megafugathree shows up a lot when working with powers of 3. |
| 10 | googgox | 5.9E14 – that’s almost 600x larger than googox. |
As you can see, these numbers grow notably faster than googo-. Let’s continue:
| 11 | googgoxi | 5.1E16 |
| 12 | googgoxij | 4.7E18 |
| 13 | googgoxiji | 4.8E20 |
| 14 | googgoxiv | 5.3E22 |
| 15 | googgoxvy | 6.3E24 |
| … | ||
| 20 | googgoxex | 3.7E35 |
| 30 | googgoxexex | 4.2E58 |
| 40 | googgoxel | 1.5E83 |
| 47 | googgoxelvij | 3E101 – this is the first larger than a googol. |
| 50 | googgol (goo-guh-guhl) | 6E108 – defined by Joyce himself. |
| 60 | googgolex | 2E135 |
| 70 | googgolexex | 4E162 |
| 80 | googgolexexex | 3E190 |
| 90 | googgoxec | 7E218 |
| 100 | googgoc | 5E247 |
| ….. | ||
| 200 | googgocc | 4E555 |
| 300 | googgoccc | 2E866 |
| 500 | googgod | E1588 |
| 1000 | googgom | E3477 |
| 2000 | googgomem | E7556 |
| 3000 | googgomemem | E11,863 |
Hmm. Let’s compare this with googo-, with the largest example on both lists.
Googgo-3000 is around E11,863, while googo-3000 is around E11,334. This makes the first 10^529 times larger than the second. Unfortunately, we’re at an exponential scale with these numbers, so we’re looking more at the difference between the exponents. Therefore, the difference is no longer very large at this scale.
So let’s continue with a smattering of the many possible triple-G numbers. First we define the googggo- prefix…
googggo-x = (4x)^x, where x is written in Roman numerals
… and then we define numbers …
| x | number name | value |
| 1 | googggoi (goo-guh-guh-goy) | 4 |
| 2 | googggoij | 64 |
| 3 | googggoiji | 1,728 |
| 4 | googggoiv | 65,536 |
| 5 | googggov | 32,000,000 |
| 6 | googggovi | 1.9E8 |
| 7 | googggovij | 1.3E10 |
| 8 | googggoviji | 1.1E12 |
| 9 | googggoix | 1.0E14 |
| 10 | googggox | 1.0E16 |
| 15 | googggoxvy | 4.7E26 |
| 20 | googggoxex | 1.2E38 |
| 25 | googggoxexv | E50 (exactly equal) |
| 30 | googggoxexex | 2.4E62 |
| 45 | googggoxelvy | 3E101 (first term more than a googol) |
| 50 | googggol | 1E115 |
| 75 | googggolexexv | 6E185 |
| 100 | googggoc | 2E260 |
| 200 | googggocc | 4E580 |
| 250 | googggoccel | E750 (exactly equal) |
| 500 | googggod | E1651 |
| 1000 | googggom | E3602 |
| 3000 | googggomemem | E12,238 |
We can see that this function (on a logarithmic scale) grows only slightly faster than the previous. Let’s give a smattering of the ones with quadruple g’s:
| number name | value |
| googgggoi | (5*1)^1 = 5 |
| googgggoij | (5*2)^2 = 100 |
| googgggox | (5*10)^10 ~ 9.8E16 |
| googgggoxex | E40 (exactly equal) |
| googgggol | 8E119 |
| googgggoc | 8E269 |
| googgggocc | E600 (exactly equal) |
| googgggom | E3699 |
| googgggomem | E8000 (exactly equal) |
| googgggomemem | E12,528 |
Like you probably expected, this one grows ever so slightly faster than the previous. Now let’s try a few with five or more g’s:
| googggggol | (6*50)^50 ~ 7E123 |
| googggggoc | 7E277 |
| googggggom | E2778 |
| googgggggom | E3845 |
| googggggggom | E3903 |
Since it’s hard to keep track of all the g’s now, we can safely say that numbers with “oo” then “o” as vowels are at a breaking point. These numbers reach their breaking point between a great great googol (10^10,000) and a googolgong (10^100,000), and have exponential growth rates overall. They can be upper-bounded by a googolgong.
So what’s next? Let’s try having “oo” as both the first and second vowel, and see what possibilities that opens.
First, as usual, let’s try the smallest example: googooi (goo-goo-ee).
A = 1 (oo)
B = 2 (one g)
C = 1 (oo)
D = i (1 in roman numerals)
(B{C}D){A}D
= (2{1}1){1}1
= (2^1)^1
= 2
The overall formula for googoo-x is (2^x)^x, which simplifies to 2^x^2 (note that this is 2^(x^2), NOT (2^x)^2). Let’s try some examples:
| x | number name | value |
| 2 | googooij | 2^2^2 = 2^4 = 16. Interestingly, this is exactly equal to googoij. |
| 3 | googooiji | 2^3^2 = 2^9 = 512 |
| 4 | googooiv | 2^4^2 = 2^16 = 65,536 |
| 5 | googoov | 33,554,432 |
| 6 | googoovi | 6.9E10 |
| 7 | googoovij | 5.6E15 |
| 8 | googooviji | 1.9E19 |
| 9 | googooix | 2.4E24 |
| 10 | googoox* | 1.3E30 |
| 11 | googooxi | 2.7E36 |
| 12 | googooxij | 2.2E43 |
| 13 | googooxiji | 7.5E50 |
| 14 | googooxiv | 1.0E59 |
| 15 | googooxvy | 5.4E67 |
| 16 | googooxvi | 1.2E77 |
| 17 | googooxvij | 9.9E86 |
| 18 | googooxviji | 3.4E97 |
| 19 | googooxix | 5E108 |
*This is why I don’t like using “ox” in place of “iii” – the third member of the googo- series (googo-3) would be called googoox with Joyce’s system, but this is already an interpretation of googoox. Joyce’s googology is ambiguity galore!
Impressive so far! The 19th term already passes the googol, while the previous sequences could only pass it after forty-something terms. The exponent itself is growing at a quadratic rate, making this sequence faster than exponential growth, but still much much slower than hyper-exponential, where the exponent itself increases hyper-exponentially. Let’s see what happens when we continue:
| 20 | googooxex | 3E120 |
| 30 | googooxexex | 9E270 |
| 40 | googooxel | 4E481 |
| 50 | googool | 4E752 – that’s more than googol to the seventh! |
| 60 | googoolex | E1084 – passed a great googol! |
| 70 | googoolexex | E1475 |
| 80 | googoolexexex | E1927 |
| 90 | googooxcy | E2438 |
| 100 | googooc | E3010 |
| ….. | ||
| 150 | googoocel | E6773 |
| 200 | googoocc | E12,041 |
| 300 | googooccc | E27,093 – larger than anything we’ve seen so far! |
| 400 | googocdy | E48,615 |
| 500 | googood | E75,258 – good number, isn’t it? |
| ….. | ||
| 1000 | googoom | E301,030 – bigger than a googolgong! |
| 2000 | googoomem | E1,204,199 – even passes a millionplex (also called four-ex-great googol) |
| 3000 | googoomemem | E2,709,270 |
This system goes well beyond anything we’ve previously seen! It passes the googolgong, and even the millionplex – the smallest major googologism above it is Jonathan Bowers’ micrillion (the millionth -illion, equal to 10^3,000,003), and it’s a fairly strong bound, on a logarithmic scale.
Now what will googgoo- (goo-guh-goo) bring? We can calculate the formula of googgoo-x as 3^x^2, and calculate from there:
| x | number name | value |
| 1 | googgooi (goo-guh-goo-ee) | 3 |
| 2 | googgooij | 81 |
| 3 | googgooiji | 19,683 – sizable difference so far (googooiji, for comparison, is 512) |
| 4 | googgooiv | 43,046,721 |
| 5 | googgoov | 8.5E11 |
| 6 | googgoovi | 1.5E17 |
| 7 | googgoovij | 2.4E23 |
| 8 | googgooviji | 3.4E30 |
| 9 | googgooix | 4.4E38 |
| 10 | googgoox | 5.2E47 |
| ….. | ||
| 15 | googgooxvy | 2E107 – first one bigger than googol |
| 20 | googgooxex | 7E190 |
| 30 | googgooxexex | 3E429 |
| 40 | googgooxel | 3E763 |
| 50 | googgool | E1193 – defined by Joyce himself |
| 60 | googgoolex | E1718 |
| 100 | googgooc | E4771 |
| 200 | googgoocc | E19,085 |
| 300 | googgooccc | E42,941 |
| 500 | googgood | E119,280 |
| 1000 | googgoom | E477,121 |
| 2000 | googgoomem | E1,908,485 |
| 3000 | googgoomemem | E4,294,091 |
These numbers are pretty cool – the exponents become over 1.5x as large as their googoo- counterparts. This passes a micrillion but is easily beaten by Bowers’ nanillion (10^3,000,000,003, the billionth -illion). Now let’s give a few numbers with triple g’s:
| x | number name | value |
| 1 | googggooi | (4^1)^1 = 4 |
| 2 | googggooij | (4^2)^2 = 256 |
| 3 | googggooiji | 262,144 |
| 4 | googggooiv | 4.3E9 – interestingly, exactly equal to googoviji from part 1 and earlier here |
| 5 | googggoov | 1.1E15 |
| 6 | googggoovi | 4.7E21 |
| ….. | ||
| 10 | googggoox | 1.6E60 |
| 13 | googggooxiji | E102 – first one larger than googol |
| 15 | googggooxvy | E135 |
| 20 | googggooxex | E241 |
| 50 | googggool | E1505 |
| 100 | googggooc | E6021 |
| 500 | googggood | E150,515 |
| 1000 | googggoom | E602,060 |
| 3000 | googggoomemem | E5,418,540 |
… and then with more g’s …
| googgggool | E1748 |
| googgggooc | E6990 |
| googgggoom | E698,970 |
| googggggoom | E778,151 |
| googgggggoom | E845,098 |
| googgggggoomemem | E7,605,882 |
Not that the googgggggoom (with six g’s in a row) is almost equal to googoom CUBED.
But all these numbers are still bounded by E10,000,000, and can also be upper-bounded by a billionplex (10^10^9).
There’s still more though (though they won’t be nearly as big as the previous numbers). These unusual constructions will use “o” as a vowel and then “oo”.
First, an example:
gogool
A = 0 (o)
B = 2 (one g)
C = 1 (oo)
D = 50 (L = 50 in Roman numerals)
(B{C}D){A}D
= (2{1}50}{0}50
= (2^50)*50
~ 5.6E16, or 56 quadrillion. Quite modest compared to many of the previous numbers, which were millions of digits long, but still very large in everyday terms. It’s even far far less than a googol.
The gogoo- prefix can be defined as x*2^x, where x is written in Roman numerals. Let’s try some examples:
| x | number name | value |
| 1 | gogooi (goh-goo-ee) | 2 – note how the frst member of a sequence with one g in a row is 2, with two g’s in a row it’s 3, with three g’s it’s 4, etc. |
| 2 | gogooij | 8 |
| 3 | gogooiji | 24 |
| 4 | gogooiv | 64 – so far not very impressive. |
| 5 | gogoov | 160 |
| 6 | gogoovi | 384 |
| 7 | gogoovij | 896 |
| 8 | gogooviji | 2048 – also the name of an infamous game |
| 9 | gogooix | 4608 |
| 10 | gogoox | 10,240 |
| ….. | ||
| 15 | gogooxvy | 491,520 |
| 20 | gogooxex | 20,971,520 |
| 30 | gogooxexex | 3.2E10 |
| 40 | gogooxel | 4.4E13 |
| 50 | gogool | 5.6E16 |
| 75 | gogoolexexv | 2.8E24 |
| 100 | gogooc | 1.3E32 |
| 150 | gogoocel | 2.1E47 |
| 200 | gogoocc | 3.2E62 |
| 324 | gogoocccexexiv (goh-goosh-kex-ex-iv) | 1.1E100 – first to pass a googol |
| 500 | gogood | E153 |
| 1000 | gogoom | E304 |
| 2000 | gogoomem | E605 |
| 3000 | gogoomemem | E906 |
One thing to point out: gogoo-x is EXACTLY equal to f2(x) in the fast growing hierarchy. (if you don’t know, f0(x) = x+1, f1(x) = f0(f0(f0…..(f0(x))….)) nested x times = 2x, f2(x) = f1(f1(f1…..(f1(x))….)) = x*2^x, f3(x) = f2(f2(f2…..(f2(x))….)) nested x times, etc. It has been extended to far far bigger proportions involving bigger infinite ordinals, might discuss that in other pages.)
This system’s numbers start out rather unimpressive, but they do go faster: they eventually reach the astronomical range, and then something beyond that! It’s certainly far far faster than numbers with only “o” as vowels, but far weaker than googoo-, googgoo-, and the like and even slower than googo-. This one doesn’t make it to a great googol. Nevertheless, let’s try numbers with more g’s here.
First off:
goggoo-x (pronounced gog-oo-[x] where x depends on the number) = x*3^x
gogggoo-x (pronounced go-guh-guh-goo-[x] where x depends on the number) = x*4^x
goggggoo-x (pronounced go-guh-guh-guh-goo-[x] where x depends on the number) = x*5^x
etc.
Now for the double-g numbers:
| x | number name | value |
| 1 | goggooi | 3 |
| 2 | goggooij | 18 |
| 3 | goggooiji | 81 |
| 4 | goggooiv | 324 |
| 5 | goggoov | 1215 |
| 6 | goggoovi | 4374 – note that gogoovi is only 384 |
| 7 | goggoovij | 15,309 |
| 8 | goggooviji | 52,488 – not to be confused with 524,288, which is the 19th power of 2 |
| 9 | goggooix | 177,147 |
| 10 | goggoox | 590,490 |
| ….. | ||
| 15 | goggooxvy | 215,233,605 |
| 20 | goggooxex | 7.0E10 |
| 30 | goggooxexex | 6.1E15 |
| 40 | goggooxel | 4.9E20 |
| 50 | goggool (gog-ool) | 3.6E25 – compare to gogool, which is around 5.6E16 |
| 75 | goggoolexexv | 4.6E37 |
| 100 | goggooc | 5.2E49 |
| 200 | goggoocc | 5.3E97 |
| 205 | goggooccvy (go-gooch-vee) | 1.3E100 – first to pass a googol. This sequence passes a googol 36% earlier than the previous. |
| 300 | goggooccc | E146 |
| 500 | goggood | E241 |
| 1000 | goggoom | E480 |
| 2000 | goggoomem | E958 |
| 3000 | goggoomemem | E1435 |
This sequence passes the great googol, but not by too far and doesn’t reach a millillion.
Let’s just look at the 3000th members of the next four sequences:
| gogggoomemem | E1810 |
| goggggoomemem | E2100 |
| gogggggoomemem | E2338 |
| goggggggoomemem | E2538 |
None of them, not even the last, reach a millillion – it takes nine g’s in a row to make it to that number, and by then the g’s get hard to manage – let’s just say the go-x-oo-y family can be bounded by a millillion.
2.5. Hierarchy of numbers so far
Now’s a good time to put up a hierarchy of numbers so far.
There are two levels so far: those with o as the highest vowel and those with oo as the highest vowel. The higher “ee”, “or”, etc. levels will come later.
The first level has one family (the o-o family, like gogol and goggol), and the second has three (from lowest to highest, o-oo (like gogool), oo-o (like googol), and oo-oo (like googool)).
And each family is further divided into multiple series, depending on how much the g is repeated. For example, the oo-o family has the googo-, googgo-, googggo-, etc. series.
Each series is a group of numbers. For example, the googo- series starts with googoi, googoij, googoiji, etc. and also includes higher members like googox, googol, and googoc.
Notes:
1. cap is an upper bound for all numbers in a family/level.
2. First number in parentheses after a series name is the formula for numbers in the
3. Second number in parentheses after a series name is the value of the 50th member (for example, googol for the googo- series)
—————
So here’s a visualization:
o level – cap: E8
o-o family – cap: E8
gogo- series (2x^2 / 5000)
goggo- series (3x^2 / 7500)
gogggo- series (4x^2 / 10,000)
goggggo- series (5x^2 / 12,500)
etc.
oo level – cap: E7#2 = E10,000,000
o-oo family – cap: E3000
gogoo- series (x*2^x / 5.6E16)
goggoo- series (x*3^x / 3.5E25)
gogggoo- series (x*4^x / 6.3E31)
goggggoo- series (x*5^x / 4.4E36)
etc.
oo-o family – cap: E100,000
googo- series ((2x)^x / E100*)
googgo- series ((3x)^x / E108)
googggo- series ((4x)^x / E115)
googgggo- series ((5x)^x / E120)
etc.
oo-oo family – cap: E7#2 = E10,000,000
googo- series (2^x^2 / E753)
googgo- series (3^x^2 / E1193)
googggo- series (4^x^2 / E1505)
googgggo- series (5^x^2 / E1747)
etc.
* exact value
This should give us a quick overview of how powerful each system is, as well as its behavior.
So what’s next? The more powerful vowels of course. These will have chaotic growth rates. The tetrational ones (with ee) will be harder to work with, the pentational ones (with or) harder still, and hexation and beyond will be still even harder.
————————————————————————————————————
PART 3
3. Numbers with “ee” as vowels
In the googo- naming system, “ee” is the third vowel – it’s equivalent to tetration (a^^b).
Let’s try an example to see its power:
geegol (also the name of a Bowersism equal to 10^^^^100, but this is a different Joycian geegol)
A = 2 (ee)
B = 2 (one g)
C = 0 (o)
D = 50 (L)
(B{C}D){A}D
= (2{0}50){2}50
= (2*50)^^50
= 100^^50 = 100^100^100……^100 with 50 100’s = HUGE!
100^^50 is an EXTREMELY UNFATHOMABLE number – equal to 100^100^100^100…….^100 with 50 100’s. It’s larger than ANY number to appear in science at all – it’s unfathomably larger than a googolplex, -duplex, -triplex, -quadriplex, -quintiplex, and the decker (10^^10)! In terms of 10s, it can be approximated as 10^10^10^……..10^200 with 49 10’s, or just E200#49 – placing it just slightly larger than a penantalogue (10^^50), in tetrational terms.
But seriously, where the hell did that come from? We need to find out using examples…..lots of examples.
Let’s start small, and introduce the hierarchy of prefixes (families sorted from weakest to strongest):
ee level
o-ee family
gogee- series
goggee- series
etc.
oo-ee family
googee- series
googgee- series
etc.
etc.
ee-o family
geego- series
geeggo- series
etc.
ee-oo family
geegoo- series
geeggoo- series
etc.
ee-ee family
geegee- series
geeggee- series
etc.
…and then we’ll go through each one.
First the gogee- series ((2^^x)*x):
gogeei (goh-gee-ih) = (2^^1)*1 = 2*1 = 2 – note that the first member of any such series is ALWAYS degenerate, and equal to c in the formula.
gogeeij (goh-gee-ij) = (2^^2)*2 = (2^2)*2 = 4*2 = 8. Very pathetic so far.
gogeeiji (goh-gee-ij-ee) = (2^^3)*3 = (2^2^2)*3 = (2^4)*3 = 16*3 = 48
gogeeiv (goh-gee-iv) = (2^^4)*4 = (2^2^2^2)*4 = (2^16)*4 = 262,144 – certainly far larger, but certainly doesn’t strike fear into us.
gogeev = (2^^5)*5 ~ 1.0E19,729 – HOLY FUCKING SHIT THIS IS SO MUCH BIGGER THAN ANYTHING PREVIOUSLY ENCOUNTERED. But it’s still a real-world level number, and a number still less than the higher numbers in the oo level (like googoom). It’s also smaller than a googolgong, and smaller than fairly well-known numbers in mathematics, like the largest known primes (around 10^10^7). It’s also feasibly writable, even on this website without having to scroll too long. Nevertheless, this is still a very huge number, and let’s see how huge the next one is.
gogeevi = (2^^6)*6 ~ E19,728#2 – WHAT? THAT’S INSANE! This number is very unfathomably large – its number of digits is approximately the previous number. It even passes the googolplex, and any number we’ve encountered so far here (in parts 1 and 2) EXCEPT for the 100th and higher members of the googolple- series (googolplec and above). It’s completely impossible to write out, even if you put a digit in every Planck volume in the known universe (which is impossible in its own right). But this is still less than a googolduplex, a tetralogue (10^10^10^10), or even a googolplexigong (10^10^100,000, the gong version of a googolplex).
gogeevij = (2^^7)*7 ~ E19,728#3 – the exponent is approximately the previous number! This passes the googolduplex.
gogeeviji = (2^^8)*8 ~ E19,728#4 – passes the googoltriplex
gogeeviji = (2^^9)*9 ~ E19,728#5
gogeeix = (2^^10)*10 ~ E19,728#6
gogeex = (2^^11)*!1 ~ E19,728#7
………
Staggering, right? Yep – this easily crushes anything we’ve encountered so far. It’s also a great example of tetrational growth rate. What is tetrational growth? The growth rate of a function like this – the size of the power tower increases by one in each step, usually not until after a few steps. Here’s another few functions with tetrational growth:
The -logue series (L(x) = 10^^x): L(1) = 10, L(2) = 10^10 = dialogue, L(3) = 10^10^10 = E1#3 = trialogue, L(4) = 10^10^10^10 = E1#4 = tetralogue, L(5) = E1#5 = pentalogue, etc.
The megafuga- series (M(x) = x^^x): M(1) = 1, M(2) = 4, M(3) ~ E13, M(4) ~ E154#2, M(5) ~ E2184#3, M(6) ~ E36,305#4, M(7) ~ E695,974#5 ~ E6#6, M(8) ~ E7#7, M(9) ~ E9#8, M(10) = E10#9 = E1#10.
For a more complicated example, try the Catalan-Mersenne Sequence – it is the series 2, M2, MM2, MMM2, MMMM2, etc. (M[x] = (2^x)-1, so M1 = 1, M2 = 3, M3 = 7, M4 = 15, etc. Some of those numbers (3, 7, 31, 127…..) are prime and called Mersenne primes.)
Let’s try the first few terms:
CM(1) = 2, CM(2) = M2 = 2^2-1 = 4-1 = 3, CM(3) = MM2 = 2^M2-1 = 2^3-1 = 8-1 = 7, CM(4) = 2^M3-1 = 2^7-1 = 127, CM(5) = 2^M4-1 = 2^127-1 ~ 1.7E38, CM(6) ~ E38#2, CM(7) ~ E38#3, CM(8) ~ E38#4, etc.
The first five here are all known to be prime, but the sixth is too large for any known primality test, but is still conjectured to be prime.
What do these sequences have in common? The power tower height increases by around one in every step, after a few terms. The -logue series grows like this right away, megafuga- series grows like this from the third term, and the Catalan-Mersenne sequence from the fifth term. This is such a new type of growth rate – one that will be familiarized as we go through these later series.
Without further ado let’s finish the rest of the gogee- series:
| x | number name | value |
| 20 | gogeexex | E19,278#17 |
| 30 | gogeexexex | E19,278#27 |
| 40 | gogeexel | E19,278#37 |
| 50 | gogeel | E19,278#47 |
| ….. | ||
| 100 | gogeec | E19,278#97 |
| 200 | gogeecc | E19,278#197 |
| 500 | gogeed | E19,278#497 |
| 1000 | gogeem | E19,278#997 |
| 2000 | gogeemem | E19,278#1997 |
| 3000 | gogeememem | E19,278#2997 |
Pretty insane – this passes Bowers’ giggol (10^^100) and even my great giggol, analogous to the great googol, equal to 10^^1000. Its numbers EASILY dwarf anything from parts one and two, even the largest members of the googolple- series!
But this series is by necessity less than the goggee- (gog-ee) series (3^^x)*x…..but by how much? That’s what we’ll find out next.
First things first – goggeei (gog-ee-ih) = (3^^1)*1 = 3 – once again degenerate.
goggeeij = (3^^2)*2 = 3^3*2 = 27*2 = 54. A sizeable jump, but barely clears my secondary number range (7 up to about 50) in my number list.
goggeeiji = (3^^3)*3 = 3^3^3*3 = 3^27*3 = 3^27*3^1 = 3^(27+1) = 3^28 = 22,876,792,454,961. Exactly three times megafugathree.
goggeeiv = (3^^4)*4 ~ 10^(3.6*10^12), between E12#2 and E13#2 – this is a few trillion digits long, and when written out, would take terabytes of space.
goggeev = (3^^5)*4 ~ E13#3 – this is impossible to write.
goggeevi ~ E13#4
…….
goggeex ~ E13#8
goggeel ~ E13#48
goggeec ~ E13#98
goggeed ~ E13#498
goggeem ~ E13#998
goggeemem ~ E13#2998
The gogee- sequence is always around one power tower height behind the goggee- sequence – both grow at tetrational rates. Gogggeee-, goggggeee-, etc. behave similar, each ending slightly ahead of the previous. Their formulas are (4^^x)*4, (5^^x)*5, etc.
But what about the googee- family? Each series is by necessity faster than its gogee- counterpart, but by how much? We’ll find out here. First we make the formulas:
googee-x = (2^^x)^x
googgee-x = (3^^x)^x
googggee-x = (4^^x)^x
googgggee-x = (5^^x)^x
etc.
Then we calculate the values:
googeei = 2
googeeij = (2^^2)^2 = 4^2 = 16
googeeiji = (2^^3)^3 = 16^2 = 4096
googeeiv ~ 1.84E19 – so far satisfyingly faster than gogee-, and decently astronomical compared to gogeeiv, which is only in the tertiary range
googeev ~ E98,641 – has about twice as many digits as gogeev.
googeevi ~ E19,729#2 – what? That’s almost the same approximation given for gogeevi. This is because powers only multiply the exponent, which loses its effectiveness once the exponent becomes exponential in size.
googeevij ~ E19,728#3 – ditto.
googeeviji ~ E19,728#4
googeeix ~ E19,728#5
googeex ~ E19,728#6
…..
and the next googee- numbers are indistinguishable at a tetrational scale from their gogee- counterparts. Ditto for the googgee- series – see for yourself.
So the oo-ee family and the o-ee family, really, are on par with growth rates.
But what about some different families? First let’s try the ee-o family:
We’ll start off with what geego-x means: it means (2x)^^x, where x is in Roman numerals.
In any case, it’s trivial to continue:
geegoi = 2^^1 = 2
geegoij = 4^^2 = 256
geegoiji = 6^^3 ~ 10^(2.65*10^36,304) – a big jump, but still writable in full (see full decimal expansion).
geegoiv = 8^^4 ~ 10^10^15,131,337 – another huge jump, and impossible to write in full
geegov = 10^^5 = 10^10^10^10^10 – exactly equal to a pentalogue
geegovi = 12^^6 ~ E13#5
geegovij = 14^^7 ~ E16#6
geegoviji = 16^^8 ~ E19#7
geegoix = 18^^9 ~ E23#8
geegox = 20^^10 ~ E26#9
…..
geegoxex = 40^^20 ~ E64#19
geegol = 100^^50 ~ E200#49
geegoc = 200^^100 ~ E460#99
As you can see, this sequence grows faster than the previous ones…..but it’s only a faster version of tetrational growth. Nevertheless, it’s still a little faster than typical tetrational growth, similar in rate to the megafuga- prefix.
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this is the end of the googo- and googolple- article series. as i said it’s mostly archived for history’s sake – the new content is currently under construction and will be much better.
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