Googol and Googolplex
PREVIOUS << The -illion numbers
In this article, we will cover the googol and googolplex, numbers invented by Edward Kasner's nephew Milton Sirotta in 1920 or 1938 depending on account. We will also cover the "plex function", as well as googolisms that other people have made based on the googol.
Googol
The googol is equal to 1 followed by 100 zeroes, i. e:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
As there are about 1080 elementary particles in the universe, about 1020 universes would contain a googol particles, so a googol has little use in the real world (or the universe for that matter). A googol Planck times is about 1.24*1039 times the age of the universe (that is, about 1.71*1049 years) . However, the number of ways to arrange 70 objects is about 1.2 times a googol, so a googol still has some use in fields like combinatorics.
A googol is exactly equal to 10 duotrigintillion in the short-scale -illions (or 10 sexdecilliard in the long-scale -illions). It is the only member of the googol family that can be named compactly in terms of any -illions.
Sbiis Saibian calls this number guppyding, and Aarex Tiaokhiao calls this number 100-noogol, or unoohol.
The googol has become the namesake of many googolisms (even some of my googolisms which we will see later have names ending with -oogol), and other people have created names to extend from the googol.
SuperJedi224's extensions
SuperJedi224 invented names for the hundredth powers of other bases, taken from a combination of the name of the numeral system with that base and the word "googol", such as:
bingol = 2100 = 1,267,650,600,228,229,401,496,703,205,376
terngol = 3100 = 515,377,520,732,011,331,036,461,129,765,621,272,702,107,522,001
quaterngol = 4100 = 1,606,938,044,258,990,275,541,962,092,341,162,602,522,202,993,782,792,835,301,376
pentengol = 5100 = 7,888,609,052,210,118,054,117,285,652,827,862,296,732,064,351,090,230,047,702,789,306,640,625
vigesigol = 20100 = 12,676,506,002,282,294,014,967,032,053,760,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Googolplex
A googolplex is 1 followed by a googol zeroes, or 10^(10^100). The most popular description of the size of the googolplex is that it cannot be written out in the entire universe even if you could fit a digit onto a particle. According to popular belief, this is the largest named number, but that is not the case, as we will see later. Sirotta first defined the googolplex as "one followed by writing zeroes until you get tired", but Kasner was dissatisfied with this definition, and so it was changed to the current value.
In Conway's -illion system, the googolplex is ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintat
recentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrig
intatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitre
strigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion, and in Bowers' -illion system, it also has a long name:
ten tretriotriaconto-tretrigintitrecentiduetriaconto-tretrigintitrecentimetriaconto-tretrigintitrecentitriaconto-tretrigintitrecentienneicoso-tretrigintitrecentiocteicoso-
tretrigintitrecentihepteicoso-tretrigintitrecentihexeicoso-tretrigintitrecentipenteicoso-tretrigintitrecentitetreicoso-tretrigintitrecentitrioicoso-tretrigintitrecentidueicoso-
tretrigintitrecentiicoso-tretrigintitrecentienneco-tretrigintitrecentiocteco-tretrigintitrecentihepteco-tretrigintitrecentihexeco-tretrigintitrecentipenteco-tretrigintitrecentitetreco-
tretrigintitrecentitreco-tretrigintitrecentidueco-tretrigintitrecentimeco-tretrigintitrecentimeco-tretrigintitrecentiveco-tretrigintitrecentixono-tretrigintitrecentiyocto-
tretrigintitrecentizepto-tretrigintitrecentiatto-tretrigintitrecentifemto-tretrigintitrecentipico-tretrigintitrecentinano-tretrigintitrecentimicro-tretrigintitrecentimilli-
doetrigintitrecentillion
Even if it was theoretically possible to write out a googolplex, it would take a hopelessly long time. Even at the way-beyond-superhuman rate of writing one zero every Planck time, it would still take a googol Planck times, which is about 1.24*1039 times the age of the universe.
A googolplex is very closely approximated by 233219280948873623478703194294893901758648313930245806120547563958159347766086252158501397433593701551, which is even closer to a googolplex in terms of ratio than 1024 is to 1000. Below are the first and last 400 digits of that number.
1002377740682581912913566526097994175990673431775996790758530904061458854912979657661923919218083722075116663303514798428256906940459230653694207161561501065167215347716834383797109317658318590295000962783339671345042484587987133244310587682466205280302810257128582454751290722851255166705264080427326896689625558029517079323376748928668051830999751919844535589861403521790910366111492469710057417110..................
.................8312434949920254580899238515052785234631060920679480407872223823791378877268727012770712134396407629618458815194436912996688787217800089991571540116187463121778681837618814390807978229214912925871022802243021924256035135699323140383379957801315450157478097593403821128414417301931440472524420762464427216228792515063917286647248208979586539590334550153561062215599182339992927666157787971954826805248
The binary expansion of the googolplex begins with 11111111..., and also the ternary expansion of the googolplex begins with 22222..., because 320959032742893846042965675220214012506075180067979301169235453386341774775719406287167658023089812370 is even closer to the googolplex than the number just above.
Also, below are the first and last digits of googolth powers of numbers other than 10 (the other base values of "1 followed by a googol zeroes"):
2^(10^100) = 255,178,906,420,018,795,763,280,064,062,091,846,617,185,......893,380,022,607,743,740,081,787,109,376 (3010299956639811952137388947244930267681898814621085413104274611271081892744245094869272521181861721 digits)
3^(10^100) = 1,272,468,170,422,665,199,830,086,974,301,075,003,897,298,......,000,000,000,000,000,000,000,000,000,001 (about 4.7712125471966*1099 digits)
4^(10^100) = 65,116,274,281,716,709,781,038,272,232,384,998,074,......,893,380,022,607,743,740,081,787,109,376 (about 6.0205999132796*1099 digits)
5^(10^100) = 39,188,192,081,755,467,490,251,570,000,237,353,675,......,106,619,977,392,256,259,918,212,890,625 (about 6.98970004336019*1099 digits)
6^(10^100) = 32470703618273781188494504454525046905520.........893380022607743740081787109376 (about 7.7815125038364363250876679797960833596831875x10^99 digits)
710^100 = 25956318868975222842893418491449895473404.........000000000000000000000000000001 (about 8.450980400142568307122162585926361934835724x10^99 digits
810^100 = 16,616,299,661,354,464,912,242,942,831,220,064,518,418.........893380022607743740081787109376 (about 9.0308998699194358564121668417347908030252238x10^99 digits
910^100 = 1,619,175,244,738,804,926,879,596,037,406,859,232,556,.........000,000,000,000,000,000,000,000,000,001 (about 9.5424250943932487459005580651023061840025773x10^99 digits)
The last digits of a^(10^b) converge (the googolth powers are the case where b=100). If a is even, the last digits of a^10^b converge to a leftward-infinite string of digits ending in ...9103890995893380022607743740081787109376, which is congruent to 1 mod 5^d and 0 mod 2^d. If a ends in a 5, the last digits of a^10^b converge to the ten's complement of the first sequence, with the final ...624 incremented to ...625: ...0896109004106619977392256259918212890625. If a ends in a 0, then the googolth power will end in a googol zeros. Otherwise, the googolth power ends in a sequence of 100 (more if the original number is a fourth root of unity mod 25) zeroes followed by a 1. Additionally, in a^10^b where a is coprime to 10, the digits just before the zeroes will also converge to a sequence:
3: ...30099616688311672716601826712349001982311284857649343931485069328008084427865522
7: ...395512806
9: ...60199233376623345433203653424698003964622569715298687862970138656016168855731044
11: ...8446
13: ...594
19: ...19508198378
Other base variants of the googolplex have also been named, such as the googolplexibit, which is equal to 22^100, or 21,267,650,600,228,229,401,496,703,205,376. Its decimal expansion begins 228,536,769,422,951,370,300,290,046,652,... and ends ...799,355,055,482,488,460,983,335,387,136. The ternary-googolplex is equal to 33^100, or 3515,377,520,732,011,331,036,461,129,765,621,272,702,107,522,001 and its decimal expansion starts 85,984,205,504,223,545,203,614,351,895,... and ends ...,404,622,494,102,822,640,358,094,520,003. The leading digits of these numbers are computed by multiplying the logarithm of the base by the exponent and extracting the fractional part, and the terminating digits are computed using modular exponentiation.
Factors of numbers near googolplex
Factors of numbers just over googolplex have been found. This may seem unbelievable, but it can be done by using tricks involving the modulo function. The smallest factor of 1010^100 + 1 is 316,912,650,057,057,350,374,175,801,344,000,001, or 2104*56 + 1.
googolplex + 1 = 316912650057057350374175801344000001 x 155409907106060194289411023528840396801 x 1467607904432329964944690923937202176001 x 11438565962996913740067907829760000000001 x 495176015714152109959649689600000000000001 x 7399415202816574979127045311692800000000001 x 9823157208340761024963422324575436800000001 x 153011449181500580269468675253...004467201
googolplex + 2 = 2 x 3 x 166666666666666666666666666666...666666666666666666666666666667 (10100 digits in the final factor)
googolplex + 3 = 7 x 1429 x 999700089973008097570728781365...8916325102469259222233330001 (10100 - 4 digits in the final factor)
googolplex + 4 = 22 x 1129 x 1181 x 18917 x 54917 x 14950993 x 2936848501 x 63822551333 x
googolplex + 5 = 3 x 5 x 29 x 666666667 x 344827...2069
The "plex function" can be defined as plex(n) = 10^n. Therefore, a googol is equal to plex(100), and can also be named hundredplex if n-plex is equal to plex(n). And, there are even further members of the googol series: the googolduplex, which can be called googolplexian or googolplexplex (1 followed by a googolplex zeroes), and then the googoltriplex which can be called googolplexianite, or the unwieldy googolplexplexplex, which is 1 followed by a googolduplex zeroes.
The 2-argument plex function can be defined like so:
plex(a, b) = 10^(10^(10^(...(10^(10^a))...))) with b 10s
So a googolplex can also be represented as plex(2, 3) or plex(100, 2), and the googolduplex can be represented as plex(2, 4) or plex(100, 3). In general, googol-n-plex is equal to plex(100, n+1).
NEXT >> The Factorials