2.02. Knuth's Up-Arrows and the Hyper-Operators

(back to 2.01)

Introduction

We've finished with the "basic numbers" so to speak. We've gone through various -illion systems, the magnitude of various numbers, and also the googol family. We've also familiarized ourselves with the concept of recursion. So where do we go now? A good starting point is Knuth's up-arrow notation, which is a very well-known notation in googology. Bowers' and Bird's arrays, Conway's chain arrows, Hollom's hyperfactorials, Joyce's g function, and many of Aarex's notations are all based on up arrows, and so is the definition of Graham's number. Up-arrow notation is also an intuitive way to extend upon the ordinary tools for making large numbers. Therefore, I think up-arrows are a good starting point for discussing large number notations.

The Hyper-Operators

Up-arrow notation was invented by mathematician and computer scientist Donald Knuth in 1976, as a way to generalize expressing the hyper-operators. What are the hyper-operators? They extend upon the familiar sequence of addition, multiplication, and exponentiation—we looked at each of those in an earlier page.

To understand what that means, first notice that multiplication is repeated addition:

a*b = a+a+a ... +a+a
b

and exponentiation is repeated multiplication:

a^b = a*a*a ... *a*a
b

But what is a^a^a ... ^a^a?
b

That function (when solved right to left!) has a name: tetration. It's sometimes been denoted ^ba, especially before Knuth's arrows were devised. To better understand this operator, let's try a few examples. Note that here the expressions are solved right to left:

¹3 = 3
²3 = 3^3 = 27
³3 = 3^3^3 = 3^27 = 7,625,597,484,987, a number that shows up prevalently when working with threes in up-arrows.
⁴3 = 3^3^3^3 = 3^3^27 = 3^7,625,597,484,987 — this number has about 3.68 trillion digits.
⁵3 = 3^3^3^3^3 = 3^3^3^27 = 3^3^7,625,597,484,987 = 3^(3.68-trillion-digit number) — this number has about ⁴3 digits!

I'm assuming you get the pattern. But how would you denote repeated tetration (called pentation)? Or repeated pentation? Or repeated repeated repeated repeated pentation?!

Introducing the Up-Arrows

Although there have been various notations devised for tetration, pentation, and beyond, Knuth's up-arrows give us a simple and intuitive way to express them that can be extended as far as we wish. These operators can be defined with only three rules (note that x and y must be whole numbers):

Rule 1: x↑y = x^y

Rule 2: x↑^ay = x↑^a-1(x↑^a(y-1)) if y isn't 0

Rule 3: x↑^a0 = 1

(↑^a means ↑↑↑...↑↑↑ with a ↑s)

From here on out, ↑ will be instead denoted with the common ASCII substitution ^, which also nicely matches with a^b for exponentiation. If you don't get all this, here's a way to imagine it:

x tetrated to y = x^^y = x^x^x ... ^x^x^x with y x's

x pentated to y = x^^^y = x^^x^^x ... ^^x^^x^^x with y x's

x hexated to y = x^^^^y = x^^^x^^^x ... ^^^x^^^x^^^x with y x's

etc.

Now that we know how up-arrows are defined, let's look at the hyper-operators and numbers they produce.

Tetration

Properties and Notation of Tetration

Tetration, as we saw earlier, is the hyper-operator that comes after exponentiation. Unlike exponentiation, tetration is not present in mainstream mathematics, although it still has had its behavior quite extensively studied. The name "tetration" was coined by Reuben Goodstein from "tetra-" (Greek prefix for four) plus "iteration", meaning that it's the fourth operation in the sequence starting with addition, multiplication and exponentiation. Tetration is iterated exponentiation, so a tetrated to b would be a^a^{^}^{a^}^.^{..^a} with b a's. Expressions of the form a^a^{^}^{a^}^.^{..^a} are known as power towers, so a tetrated to b is a power tower of b a's.

Tetration has even more ways to denote than exponentiation. As we saw earlier, of the more common ways is ^ba, in analogy to a^b for exponentiation. That notation is most often used when there's no need for higher hyper-operators. The most common way, however, is as a↑↑b, or as a^^b when typing since ^ is easier to type than ↑.

Robert Munafo uses a^④b to denote tetration. Why is that? He recognizes that there are actually two ways to make a hyper-operator after exponentiation. The stronger way is as a^(a^(a^(a^(...^a)...)))) with b a's, and the weaker way is as ((...(a^a)^a)^a...)^a with b a's. The weaker way (Munafo calls it hyper₄) is denoted a_④b (we'll see how it compares against the hyper-operators in the next page) and is not as mathematically interesting as the stronger way (Munafo calls it hyper4), which is is denoted a^④b. In this article we'll focus on the stronger hyper-operators, and in the next we'll focus on the weaker hyper-operators.

Other ways to denote tetration include:

- a***b, on analogy with a*b for multiplication and a**b for exponentiation in some programming languages. The Big Psi web-book on large numbers uses this notation.

- a{2}b in Bowers' operator notation (we'll examine it in section 3)

- Harvey Friedman has used x^[y]

- a->b->2 in Conway chain arrows, and {a,b,2} in Bowers' arrays

Tetration lacks many of the symmetric properties of exponentiation. Like exponentiation it is neither commutative nor associative, but it also doesn't have many laws for simplifying expressions like exponentiation does. One law it does have (which holds for all hyper-operators) is that x^{x^^n} = x^^(n+1), and a law that can be used to upper-bound tetration expressions (shown by Sbiis Saibian as part of his Knuth Arrow Theorem) is that (x^^a)^^b will always be less than x^^(a+b). This actually generalizes to any hyper-operator more powerful than exponentiation, and that law (called the Knuth Arrow Theorem) is useful for upper-bounding and estimating expressions with Knuth's arrows.

Googology Wiki, on its article on tetration, mentions an interesting identity for tetration:

(x^^a)^{(x^^b)} = (x^^(b+1))^{(x^^(a-1))}

That identity is quite easy to see, since (x^^a)^{(x^^b)} = (x^{(x^^(a-1))})^{(x^^b)} = x^{x^^(a-1)*x^^b} = x^{x^^b*x^^(a-1)} = (x^{x^^b})^{x^^(a-1)} = (x^^(b+1))^{(x^^(a-1))}.

Now that we've familiarized ourselves with the properties of tetration, let's take a look at the numbers tetration produces.

Examples of Tetration

We'll start with cases with the base of two, since cases with the base of 1 always degenerate to 1. For example:

1^^3
= 1^(1^^2) (rule 2)
= 1^(1^(1^^1)) (rule 2)
= 1^(1^(1^(1^^0))) (rule 2)
= 1^(1^(1^1)) (rule 3)
= 1^(1^1) (rule 1)
= 1^1 (rule 1)
= 1 (rule 1)

All such expressions with tetration and higher operators will simply degenerate to power towers of 1, which always evaluate to 1. So let's go on to examples with the base of 2.

First we'll try 2^^1:

2^^1
= 2^(2^^0) (by rule 2)
= 2^1 (by rule 3)
= 2 (by rule 1)

Next let's try 2^^2:

2^^2
= 2^(2^^1)
= 2^2 as we saw earlier
= 4. The case of 4 is very prevalent with twos in the hyper-operators, as you will see.

2^^3:

2^^3

= 2^(2^^2)

= 2^4

= 16 — quite modest so far ...

2^^4:

2^^4

= 2^(2^^3)

= 2^16

= 65,536. This number crops up pretty often when working with twos in up-arrow notation.

2^^5:

2^^5

= 2^(2^^4)

= 2^65,536

= 2003529930406846464979072351560255750447825475569751419265016973710894059556311453089506130880933
34810103823434290726318182294938211881266886950636476154702916504187191635158796634721944293092798
20843091048559905701593189596395248633723672030029169695921561087649488892540908059114570376752085
00206671563702366126359747144807111774815880914135742720967190151836282560618091458852699826141425
03012339110827360384376787644904320596037912449090570756031403507616256247603186379312648470374378
29549756137709816046144133086921181024859591523801953310302921628001605686701056516467505680387415
29463842244845292537361442533614373729088303794601274724958414864915930647252015155693922628180691
65079638106413227530726714399815850881129262890113423778270556742108007006528396332215507783121428
85516755540733451072131124273995629827197691500548839052238043570458481979563931578535100189920000
24141963706813559840464039472194016069517690156119726982337890017641517190051133466306898140219383
48143542638730653955296969138802415816185956110064036211979610185953480278716720012260464249238511
13934004643516238675670787452594646709038865477434832178970127644555294090920219595857516229733335
76159552394885297579954028471943529913543763705986928913757153740001986394332464890052543106629669
16524341917469138963247656028941519977547770313806478134230959619096065459130089018888758808473362
59560654448885014473357060588170901621084997145295683440619796905654698136311620535793697914032363
28496233046421066136200220175787851857409162050489711781820400187282939943446186224328009837323764
93181478984811945271300744022076568091037620399920349202390662626449190916798546151577883906039772
07592793788522412943010174580868622633692847258514030396155585643303854506886522131148136384083847
78263790459607186876728509763471271988890680478243230394718650525660978150729861141430305816927924
97140916105941718535227588750447759221830115878070197553572224140001954810200566177358978149953232
52085897534635470077866904064290167638081617405504051176700936732028045493390279924918673065399316
40720492238474815280619166900933805732120816350707634351669869625020969023162859350071874190579161
24153689751480826190484794657173660100589247665544584083833479054414481768425532720731558634934760
51374197795251903650321980201087647383686825310251833775339088614261848003740080822381040764688784
71647552945326947661700424461063311238021134588694532200116564076327023074292426051582811070387018
34532456763562595143003203743274078087905628366340696503084422585596703927186946115851379338647569
97485686700798239606043934788508616492603049450617434123658283521448067266768418070837548622114082
36579802961200027441324438432402331257403545019352428776430880232850855886089962774458164680857875
11580701474376386797695504999164399828435729041537814343884730348426190338884149403136613985425763
55771053355802066221855770600825512888933322264362819848386132395706761914096385338323743437588308
59233722284644287996245605476932428998432652677378373173288063210753211238680604674708428051166488
70908477029120816110491255559832236624486855665140268464120969498259056551921618810434122683899628
30716548685255369148502995396755039549383718534059000961874894739928804324963731657538036735867101
75783994818471798498246948060532081996066183434012476096639519778021441199752546704080608499344178
25628509272652370989865153946219300460736450792621297591769829389236701517099209153156781443979124
84757062378046000099182933213068805700465914583872080880168874458355579262584651247630871485663135
28934166117490617526671492672176128330845273936469244582892571388877839056300482483799839692029222
21548614590237347822268252163995744080172714414617955922617508388902007416992623830028228624928418
26712434057514241885699942723316069987129868827718206172144531425749440150661394631691976291815065
79745526236191224848063890033669074365989226349564114665503062965960199720636202603521917776740668
77746354937531889958786628212546979710206574723272137291814466665942187200347450894283091153518927
11142871083761592223802766053278233516615551493693757784666701457179719012271178127804502400263847
58788339396817962950690798817121690686929538248529830023476068454114178139110648560236549754227497
23100761513187002405391051091381784372179142252858743209852495787803468370333781842144401713868812
42499844186181292711985333153825673218704215306311977485352146709553346263366108646673322924098798
49256691109516143618601548909740241913509623043612196128165950518666022030715613684732364660868905
01426391390651506390819937885231836505989729912540447944342516677429965981184923315155527288327402
83526884424087528112832899806259126736995462473415433335001472314306127503903073971352520693381738
43322950701049061867539433130784798015655130384758155685236218010419650255596181934986315913233036
09646190599023611268119602344184336333459492763194610171665291382371718239429921627253846177606569
45422978770713831988170369645886898118632109769003557358846244648357062914530527571012788720279653
64479724025405448132748391794128826423835171949197209797145936887537198729130831738033911016128547
41537737771595172808411162759718638492422280237344192546999198367219213128703558530796694271341639
10338827543186136434901009431974090473310144762998617254244233556122374357158259333828049862438924
98222780715951762757847109475119033482241412025182688713728193104253478196128440176479531505057110
72297431456991522345164312184865757578652819756484350895838472292353455946452121583165775147129870
82259092926556388366511206819438369041162526687100445602437042006637090019411855571604720446436969
32850060046928140507119069261393993902735534545567470314903886022024639948260501762431969305640666
36662609020704888743889890749815286544438186291738290105182086993638266186830391527326458128678280
66013375000965933646251460917231803129303478774212346791184547913111098977946482169225056293999567
93483801699157439700537542134485874586856047286751065423341893839099110586465595113646061055156838
54121745980180713316361257307961116834386376766730735458349478978831633012924080083635682593915711
31309780305164417166825183465736759341980849589479409832925000863897785634946932124734261030627137
45077286156922596628573857905533240641849018451328284632709269753830867308409142247659474439973348
13081098639941737978965701068702673416196719659159958853783482298827012560584236558953969030647496
55841479813109971575420432563957760704851008815782914082507777385597901291294073094627859445058594
12273194812753225152324801503466519048228961406646890305102510916237770448486230229488966711380555
60795662073244937337402783676730020301161522700892184351565212137921574820685935692079021450227713
30999877294595969528170445821819560809658117027980626698912050615607423256868422713062950098644218
53470810407128917646906550836129916694778023822502789667843489199409657361704586786242554006942516
69397929262471452494540885842272615375526007190433632919637577750217600519580069384763578958687848
95368721228985578068265181927036320994801558744555751753127364714212955364940843855866152080121150
79075068553344489258693283859653013272046970694571546959353658571788894862333292465202735853188533
37094845540333656535698817258252891805663548836374379334841184558016833182767683464629199560551347
00391478768086403226296166415606675081537106467231084619642475374905537448053182260027102164009805
84497526023035640038083472053149941172965736785066421400842696497103241919182121213206939769143923
36837470922826773870813223668008692470349158684099115309831541206356612318750430546753698323082796
64574176208065931772656858416818379661061449634325441117069417002226578173583512598210807691019610
52229263879745049019254311900620561906577452416191913187533984049343976823310298465893318373015809
59252282920682086223033258528011926649631444131644277300323779227471233069641714994553226103547514
56312906688543454268697884477429817774937101176146516241836166802548152963353084908499430067636548
06102940094693750609845588558043970485914449584445079978497045583550685408745163316464118083123079
70438984919050658758642581073842242059119194167418249045270028826398305795005734171148703118714283
41844991534567029152801044851451760553069714417613685823841027876593246626899784183196203122624211
77391477208004883578333569204533935953254564897028558589735505751235129536540502842081022785248776
60357424636667314868027948605244578267362623085297826505711462484659591421027812278894144816399497
38818846227682448516220518170767221698632657016543169197426512300417573299044735376725368457927543
65412826553581858046840069367718605020070547247548400805530424951854495267247261347318174742180078
57469346544713603697588411802940803961674694628854067917213860122541950381970453841726800639882065
63287928395827085109199588394482977756471520261328710895261634177071516428994879535648545535531487
54978134009964854498635824847690590033116961303766127923464323129706628411307427046202032013368350
38542536031363676357521260470742531120923340283748294945310472741896928727557202761527226828337674
13934256526532830684699975970977500055608899326850250492128840682741398816315404564903507758716800
74055685724021758685439053228133770707415830756269628316955687424060527726485853050611356384851965
91896864959633556821697543762143077866593473045016482243296489127070989807667662567151726906205881
55496663825738292741820822789606844882229833948166709840390242835143068137672534601260072692629694
68672750794346190439996618979611928750519442356402644303271737341591281496056168353988188569484045
34231142461355992527233006488162746672352375123431189344211888508507935816384899448754475633168921
38696755743027379537852625423290248810471819390372206668947022042588368958409399984535609488699468
33852579675161882159410981624918741813364726965123980677561947912557957446471427868624053750576104
20426714936608498023827468057598259133100691994190465190653117190892607794911921794640735512963386
45230356733455880333131970803654571847915504326548995597058628882868666066180218822486021449999731
22164138170653480175510438406624412822803616648904257377640956326482825258407669045608439490325290
52633753231650908768133661424239830953080654966187938194912003391948949406513239881664208008839555
49422370967348400726427057011650890751961553701862647974563811878561754571134004738107627630149533
09735174180655479112660938034311378532532883533352024934365979129341284854970946826329075830193072
66533778255931433111096384805394085928398890779621047984791968687653998747709591278872747587443980
67798249682782722009264499445593804146087706419418104407582698056880389496546165879839046605876453
41810289907194293021774519976104495043196841503455514044820928933378657363052830619990077748726922
99860827905317169187657886090894181705799340489021844155979109267686279659758395248392673488363474
56516870161662406424242412289611180106156823425393921800524834547237792199112285959141918774917938
23340010078128326506710281781396029120914720100947878752551263372884222353869490067927664511634758
10119387531965724212147603828477477457170457861041738574791130190858387789015233434301300528279703
85803598151829296003056826120919509437373254541710563838870475289505639610298436413609356416325894
08137981511693338619797339821670761004607980096016024823096943043806956620123213650140549586250615
28258803302290838581247846931572032323360189946943764772672187937682643182838260356452069946863021
60488745284243635935586223335062359450028905585816112753417837504559361261308526408280512138731774
90200249552738734585956405160830583053770732533971552620444705429573538361113677523169972740292941
67420442324811387507563131907827218886405337469421384216992886294047963530515056078812636620649723
12575790195988730411956262273437289005165611110941117452779654827904712505819990774980638215593768
85546498822938985408291325129076478386322494781016753491693489288104203015610283386143827378160946
34133538357834076531432141715065587754782025245478065730134227747061674424196895261316427410469547
46214837562882997718041867850845469656191509086958742511844358373065909514609804512474094113738999
27822492983367796011015387096129749705566301637307202750734759922943792393824427421186158236161317
88639255309511718842129850830723825972914414225157940388301135908333165185823496722125962181250705
81137594955250227472746743698871319266707692991990844671612287388584575846227265733307537355728239
51616964175198675012681745429323738294143824814377139861906716657572945807804820559511881687188075
21297183263644215533678775127476694079011705750981957508456356521738954417987507452385445520013357
20333323798950743939053129182122552598337909094636302021853538488548250628977156169638607123827717
25621313460549401770413581731931763370136332252819127547191443450920711848838366818174263342949611
87009150304916533946476371776643912079834749462739782217150209067019030246976215127852195614207080
64616313732365178539762920920255002889620129701413796400380557349492690735351459612086747965477336
92958773628635660143767964038430796864138563447801328261284589184898528048048844180821639423974014
36290348166545811445436646003249061876303950235640204453074821024136689519664422133920075747912868
38051751506346625693919377402835120756662608298904918772878338521785227920457718469658552787904475
62192663992008409302075673925363735628390829817577902153202106409617373283598494066652141198183810
88451545977289516457213189779790749194101314836854463961690460703010759681893374121757598816512700
07612627891695104063158576375347874200702220510708912576123616580268068158584998526314658780866168
00733264676830206391697203064894405628195406190685242003053463156621891327309069687353181641094514
28803660599522024824888671155442910472192913424834643870536850864874909917881267056566538719104972
18200423714927401644609434598453925367061322106165330856620211889682340057526754861014769936887382
09584552211571923479686888160853631615862880150395949418529489227074410828207169303387818084936204
01825522227101098565344481720747075601924591559943107294957819787859057894005254012286751714251118
43564371840535630241812254732660933027103979680910649392727226830354104676325913552796838377050198
55234621222858410557119921731717969804339317707750755627056047831779844447637560254637033369247114
22081551997369137197516324130274871219986340454824852457011855334267526471597831073124566342980522
14554941562527240289153333543493412178620370072603152798707718724912344944771479095207347613854254
85311552773301030342476835865496093722324007154518129732692081058424090557725645803681462234493189
70813889714329983134761779967971245378231070373915147387869211918756670031932128189680332269659445
92862106074388274169194651622676325406650708810710303941788605648937698167341590259251946118236429
45652669372203155504700213598846292758012527715422016629954863130324912311029627923723899766416803
49714122652793190763632613681414551637665655983978848938173308266877990196288693229659737995193162
11872154552873941702436698855938887933167445333631195415184040882838151934212341228200309503133410
50704760159987985472529190665222479319715440331794836837373220821885773341623856441380700541913530
24594391350255453188645479625226025176292837433046510236105758351455073944333961021622967546141578
11271970017386114942795014112532806212547758105129720884652631580948066336876701473107335407177108
76615935856814098212967730759197382973441445256688770855324570888958320993823432102718224114763732
79135756861542125284965790333509315277692550584564401055219264450531207375628774499816364633283581
61403301758139673594273276904489203618803867549557518068900585329272014939235005258451467069826285
48257883267398735220457228239290207144822219885587102896991935873074277815159757620764023951243860
20203259659625021257834995771008562638611823381331850901468657706401067627861758377277289589274603
94039303372718738505369129571267150668966884938808851429436099620129667590792250822753138128498515
26902931700263136328942095797577959327635531162066753488651317323872438748063513314512644889967589
82881292548007642518658649024111112730135719718138160258317850693224400799865663537154408845486639
31817083957357807990597308390948818040609359591909074739609044101505163217496814121007657191774837
67355751000733616922386537429079457803200042337452807566153042929014495780629634138383551783599764
70885134900485697369796523869584599459559209070905895689145114141268450546211794502661175016692826
02509507707782119504326173832235624376017767993627960993689751913949650333585071554184364568526166
74243688920371037495328425927131610537834980740739158633817967658425258036737206469351248652238481
34166380806150570482905989069645193644001859712042572300731641000991698752426037736217776343062161
67448849308109299010095179745415642512048220867145868492551324442667771278637282113315362243010918
24391243380214046242223349153559516890816288487989988273630445372432174280215755777967021666317047
96972817248339284101564227450727177926939992974030807277039501358154514249404902653610582540937311
46531049433824843797186069372144446008267980024712294894057618538922034256083026970528766213773735
94394224114707074072902725461307358541745691419446487624357682397065703184168467540733466346293673
98362000404140071405427763248013274220268539369886978760700959004868465062677136307097982100655728
51013066010107806337433447730734786538817426812307437660666433127753564665786037151929227684404582
73283243808212841218776132042460464900801054731426749260826922155637405486241717031027919996942645
62095561981645454766204502241144940474934983220680719135276798674781345820385957041346617793722853
49400316315995440936840895725334387029867178297703733328068017646395020900239419314991150091052768
21119510999063166150311585582835582607179410052528583611369961303442790173811787412061288182062023
26384986151565645123004779296756361834576810504334176954306753804111392855379252924134733948105053
20257087281863072911589113359420147618726642915640363719276023062838406504254417423354645499870553
18726887926424102147363698625463747159744354943443899730051742525110877357886390946812096673428152
58591992485764048805507132981429935991146323991911395992675257635900744657281019180584180734222773
47213977232182317717169164001088261125490933611867805757223910181861685491085008852722743742120865
24852372456248697662245384819298671129452945515497030585919307198497105414181636968976131126744027
00964866754593456705993699546450055892162804797636568613331656390739570327203438917541526750091501
11988568727088481955316769316812728921430313768180164454773675183534978579242764633541624336011259
60252109501612264110346083465648235597934274056868849224458745493776752120324703803035491157544831
29527589193989368087632768543876955769488142284431199859570072752139317683783177033913042306095899
91373146845690104220951619670705064202567338734461156552761759927271518776600102389447605397895169
45708802728736225121076224091810066700883474737605156285533943565843756271241244457651663064085939
50794755092046393224520253546363444479175566172596218719927918657549085785295001284022903506151493
73101070094461510116137124237614267225417320559592027821293257259471464172249773213163818453265552
79604270541871496236585252458648933254145062642337885651464670604298564781968461593663288954299780
72254226479040061601975197500746054515006029180663827149701611098795133663377137843441619405312144
52918551801365755586676150193730296919320761200092550650815832755084993407687972523699870235679310
26804136745718956641431852679054717169962990363015545645090044802789055701968328313630718997699153
16667920895876857229060091547291963638167359667395997571032601557192023734858052112811745861006515
25988838431145118948805521291457756991465775300413847171245779650481758563950728953375397558220877
77506072339445587895905719156736, which has 19,729 digits

Wait now. Where did that even come from?!

It came from the pattern that occurs with tetration and twos: we took 2 to the 1st power = 2, then 2 to the 2nd power = 4, then 2 to the 4th power = 16, then 2 to the 16th power = 65,536, and then it's only the continuation of the sequence that 2 to the 65,536th power is this huge number.

So it shouldn't come as much of a surprise that 2^^6, or 2 to the power of 2003529930...............5719156736, is a number with about the previous number of digits, so big that you couldn't come close to being able to write it out even if you could fit as many digits as there are Planck volumes in the observable universe into every Planck volume in the observable universe! Or should it? This kind of surprise is common among new googologists. Welcome to the counterintuitive land of googology.

What about cases involving 3? As we saw earlier:

3^^1 = 3 — trivial case here.

3^^2 = 3^3 = 27 — the number 27 shows up often when plugging 3's into googological functions as we will see

3^^3 = 3^27 = 7,625,597,484,987, a number that shows up prevalently when working with threes in up-arrows and other googological functions.

3^^4 = 3^7,625,597,484,987 — this number has about 3.68 trillion digits. Even though it can still be computed, I can't store the trillions of digits on my site — besides, scrolling through its digits, even on a separate page, is a pain in the ass.

3^^5 = 3^(3.68-trillion-digit number) — this number has about 3^^4 digits!

You get the pattern from here.

Now for cases involving 4:

4^^1 = 4 — we can see that n^^1 is always equal to n. To prove this consider this:

n^^1

= n^(n^^0) [rule 2 of up-arrow notation]

= n^1 [rule 3]

= n

That said, we can skip cases of the form n^^1 now. So we continue:

4^^2 = 256 — nothing too scary yet

4^^3 = 4^256 = 13,407,807,929,942,597,099,574,024,998,205,846,127,479,365,820,592,393,377,723,561,443,721,764,030,073,546,976,801,874,298,166,903,427,690,031,858,186,486,050,853,753,882,811,946,569,946,433,649,006,084,096 — that has 155 digits but pales in comparison to:

4^^4 = 4^(155-digit number) ~ 10^(8.02*10^153)

That's a pretty cool number that can be named megafuga-four or booga-four. It's already bigger than a googolplex. We'll learn more about names like megafuga-four and booga-four in not too long.

We can continue with 4^^5 (4 to the power of 4^^4), 4^^6 (4 to the power of 4^^5), etc.

Now let's look at base 5:

5^^2 = 3125 — nothing too big

5^^3 = 5^3125 = 19110125979454775203564045597039645991980810489900943371395127892465205302426158030120593865197398
50265586440155794462235359212788673806972288410146915986602087961896757195701839281660338047611225
97553362610100148265112341314776825241149309444717696528275628519673751439535754247909321920664188
30117871691225524210700507090646743828708514499502565861944615431835113798491336917799281274338404
31549236855526783596374102105331546031353725325748636909159778690328266459182983815230286936572873
69142264813129174376213632573032164528297948686257624536221801767322494056764281936007872071383707
23553054463561539464011853484937927195145945055082327492216058489129109451899599486861995431476669
38013037176163592594479746164220050885079469804487133205133160739134230540198872570038329801246050
19701346739717590902738949392381731578699684589979478106804282243609378394633526542281570430283244
23855150823164909672857121717081232327904818172683275101127467823174109858886837085220007117334922
53913322300756147180429007527677793352306200618286012455254243061006894805446584704820650982664319
36096038873625851074707434063628697657670269925864995355797631817390255089133122329474393034395616
13283340728316634982581452268620043077990846881038041873683248009038735962129196336025831207816736
73742533322879296907205490595621406888825991244581842379597863476484315673760923625090371511798941
42426227022006628648686786871018298087280256069310194928083082504419842479679205890881711232719230
14555829167467951974305480264046468540027339938607985944659615017525869658114475685100415686877309
03712482535343839285397598749458497050038225012489284001826590056251286187629938044407340142347062
05578530532503491818958970719930566218851296318750174353596028220103821161604854512103931331225633
22607664362366882968502088394961428304847391139916696226499485636852347128732947966808845094058939
51104650944137909502276545653133018670633521323028460519434381399810561400652595300731790772711065
78349417464268472095613464732774858423827489966875505250439421823219135722305406671537337424854364
56637820457016545932181540535483936142506644985854033074664685418901481343477146503150379541757786
22811776585876941680908203125

Pretty nice number. It has exactly 2185 digits. We can have 5^^4, 5^^5, etc. Now for base 6:

6^^2 = 46,656

6^^3 = a 36,306 digit number (2659119772.........7863878656). This time I'll be nice and not make you have to scroll through the tens of thousands of digits, but if you want to see them click here.

we can have 6^^4, 6^^5, 6^^6, etc., but now for base 7:

7^^2 = 823,543

7^^3 = a number with 695,975 digits (all the digits)

and base 8:

8^^2 = 16,777,216

8^^3 = a number with over 15 million digits (all the digits)

and base 9:

9^^2 = 387,420,489

9^^3 = a number with 369 million digits (all the digits) — this number is mentioned in the novel Ulysses by James Joyce. For more about that look here.

and finally base 10:

10^^2 = 10,000,000,000 or ten billion. Sbiis Saibian gives this number a name, dialogue.

10^^3 = a number Sbiis Saibian calls a trialogue, equal to 1 followed by ten billion zeros. The number appears in a School House Rock song My Hero, Zero, and I have a page about the number here (as a subpage of part 2 of my number list) which discusses the sequence 1^1^1 to 10^10^10 and the number's appearance in School House Rock.

10^^4 = 1 followed by "1 followed by 10 billion zeros" zeros, a number Sbiis Saibian calls a tetralogue.

10^^5 = 1 followed by "1 followed by "1 followed by 10 billion zeros" zeros" zeros, a number Sbiis Saibian calls a pentalogue. Once again I'm assuming you get the pattern. Just continue with the Greek prefixes for numbers (same ones used to name polygons), and remember to use ennea- for nine and not nona-. :)

And now for two impressive power towers of tens:

10^^10 = 10^10^10^10^10^10^10^10^10^10 (a power tower of 10 tens) is a number Jonathan Bowers calls decker, one of my own favorite power tower numbers. Sbiis Saibian gives this number an alternate name, dekalogue.

10^^100 (a power tower of 100 tens) is a number Jonathan Bowers calls giggol similarly to how 10^100 = googol. Similarly he defines giggolplex to be 10^^10^^100 (remember to solve right to left), a power tower of a giggol tens.

I think you get the idea. As powerful an operator as tetration is, when looking at pentational numbers the tetrational numbers will utterly vanish in the dust!

Pentation

Properties and Notation of Pentation

We've seen and are now familiar with the power of tetration. But what about repeated tetration, known as pentation? Let's first learn about the history of pentation.

Pentation is less known than its little brother tetration, and as we will see it produces far larger values than tetration does. The name was coined by Goodstein from penta- (meaning five) + iteration, indicating that it's the fifth hyper-operator. It does not have a popular notation like ^ba for tetration, though in 2004 Mark Cutler proposed a^b̄ for a pentated to b (a bar above the b). Sbiis Saibian instead proposes _b<--a for pentation, and he uses that with the hyper-operators when writing them by hand. The most common notation for pentation is by far a^^^b (a pentated to b), though it can also be notated a{3}b in Bowers' operator notation, a->b->3 in Conway chain arrows, and {a,b,3} in Bowers' arrays. Sunir Shah continued a^b for exponents and ^ba for tetration with a(*)b for pentation.

Robert Munafo uses a^⑤b for pentation (the hyper5 operator, a^④(a^④(a^④(...a^④a)...))), and additionally a_⑤b for the weaker hyper₅ operator (a.k.a. weak pentation), equal to (...(((a_④a)_④a)_④a...)_④a with b a's. We'll examine this operator, and see how it compares to the normal hyper-operators in the next article. As for the hyper5 operator, a^^^b is a^^a^^a^^a.....^^a with b a's (remember that the expressions are always solved right to left. In terms of power towers this can be visualized as:

a^{a^}^{a^}^.^{.^.a} } a^{a^}^a^{^}^.^{..^.a} } a^{a^}^{a^}^.^{..^a} ... ... ... } a^{a^}^{a^}^.^{..^a} } a with b levels, including a

where the number of a's in each power tower is represented by the number to the right. This means that with each step in pentation we're feeding in the previous step into the height of the power tower. Tim Urban, on his blog post on large numbers titled "From 1,000,000 to Graham's Number", calls this "a power tower feeding frenzy".

Pentation has a few properties similar to those of tetration that can be used to simplify or estimate expressions: a^^(a^^^b) = a^^^(b+1), and (a^^^b)^^^c < a^^^(b+c) (as a consequence of the Knuth Arrow Theorem). But other than that, it has even fewer nice properties than tetration has.

As you can see pentation already looks scary, and we haven't examined any of the numbers it produces yet! So now it's time to look at the power of pentation.

Examples of Pentation

What horrifying numbers will pentation bring? Let's jump right in:

2^^^1 = 2^^(2^^0) = 2^^1 = 2 — once again a trivial case. We can skip cases of the form n^^^1 from here on out, as it's clear that n^^^1 will always evaluate to n (in fact this is true for all the hyper-operators).

Now let's try 2^^^2:

2^^^2
= 2^^(2^^^1)
= 2^^2
= 4 — trivial again

2^^^3 now:

2^^^3 = 2^^2^^2 = 2^^4 = 2^2^2^2 = 2^2^4 = 2^16 = 65,536 — we encounter 65,536 again. That's a good jump, but we've already examined numbers way bigger than this.

What about 2^^^4?

2^^^4 = 2^^2^^2^^2 = 2^^2^^4 = 2^^65,536

= 2^2^2^2^2^2^2^2^.......................^2^2^2^2^2 with 65,536 2's — YIKES!!! This is so much bigger than ANYTHING previously seen with the only exception being the giggolplex!

Once again, where did that even come from?!? Once again, that jump seemed completely out of nowhere ... until you observe the pattern:

2^^^1 is 2 — meh
2^^^2 is power tower of 2 2's = 4 — meh
2^^^3 is a power tower of 4 2's = 65,536 — hmm
2^^^4 is a power tower of 65,536 2's = 2^2^2^2^2^2^2^2^.......................^2^2^2^2^2 with 65,536 2's — AAAAAAHHHHHHHHHHHHHH!!!!!!!!

Once again, it's only natural now that 2^^^4, a power tower of 65,536 2's, is so utterly unfathomable that it's waaaaaaaay beyond what we can appreciate.

The number 2^^^4, the power tower of 65,536 twos, is a number that never fails to impress me. Even as someone experienced in googology, I find that 2^^^4 is a great example of how horrifying these very large numbers are, and how erratic and jumpy the growth rates appear to be. Once again, welcome to the world of googology.

What about 2^^^5?! Continuing the sequence, this number is a power tower of 2^^^4 twos—in other words, a power tower of "a power tower of 65,536 twos" twos! This time, it's not only impossible to appreciate the number, but it's also impossible to even appreciate the power tower used to represent this number!

Continuing, 2^^^6 is a power tower of 2^^^5 twos, 2^^^7 is a power tower of 2^^^6 twos, etc. I think you get the pattern: pentating a number iterates the height of a power tower, feeding in the previous number into the height of the power tower of the next number.

Now let's try cases involving 3's:

3^^^2 = 3^^(3^^^1) = 3^^3 = 3^3^3 = 3^27 = 7,625,597,484,987 — we encounter this important power of three again.

3^^^3 = 3^^(3^^^2) = 3^^7,625,597,484,987 = a power tower of 7.6 trillion 3's! Jonathan Bowers calls this number tritri.

Now 3^^^3 is a super-awesome number. It's commonly used when giving an example of an unfathomably large number, and it's my own favorite example of a number that's waaaay beyond what we can appreciate. You can imagine it like so:

3^3^3^3^3^3..................^3
7.6 trillion 3's

So how would you imagine this number? Imagine a power tower of those 7.6 trillion threes stretching all the way from Earth to the sun, with each three about 2 centimeters tall. Then, only the top five threes, only a few inches from the Sun, already make a number insanely bigger than a googolplex! Imagine the insanity with only the top meter of the tower, and that's less than a billionth of the way down the tower—with that in mind, you can hardly imagine how huge solving the entire tower down to the Earth would be!!!

What about 3^^^4? That number, by the sequence, is a power tower of 3^^^3 3's, or in other words, a power tower whose height is the number you get when evaluating a power tower of 7.6 trillion threes. Just imagine a power tower of such an unfathomable size, and imagine how insanely huge the number you get from evaluating it is.

This number can be imagined like so:

3^3^3^3^3^3..................^3
3^3^3^3^3^3..................^3 3's
7.6 trillion 3's

This time, just like with 2^^^5, the power tower is itself impossible to appreciate now!

Likewise, 3^^^5, a power tower of "a power tower of "a power tower of 7 trillion 3's" 3's" 3's, can be imagined like so:

3^3^3^3^3^3..................^3
3^3^3^3^3^3..................^3 3's
3^3^3^3^3^3..................^3 3's
7.6 trillion 3's

You get the pattern from here. How about cases with fours?

4^^^2 = 4^^4 = megafugafour ~ 10^(8*10^153).

But 4^^^3 = 4^^4^^4 = a power tower of 10^10^153 fours. It can be imagined like so:

4^4^4^4^4^4..................^4
10^10^153 4's

That's a power tower, of more than a googolplex fours! By now we've lost all hope for comprehending such gigantic values.

Likewise, 4^^^4 is a power tower of 4^^^3 fours, which can be imagined as:

4^4^4^4^4^4..................^4
4^4^4^4^4^4..................^4 4's
10^10^153 4's

etc.

Now let's try base 5:

5^^^2 = 5^^5 ~ 10^10^10^2184. That value is bigger than a googolduplex! It's actually impossible to even appreciate the gulf between this value and a googolduplex—remember how difficult it was to appreciate the jump from Skewes' number to Skewes' Approxima? Here, it's even worse: you would need to multiply a googolduplex by itself 10^10^2184 times to reach this number, and we have no hopes of comprehending 10^10^2184 as a value! But that's a kind of number we've seen when working with tetration!

5^^^3, on the other hand, is a power tower of 5^^5 5's, imaginable as:

5^5^5^5^5^5..................^5
10^10^10^2184 5's

That's a power tower of more than a googolduplex fives!

Now let's skip to 5^^^5, named "boogafive" with the booga- prefix (we'll examine it a few articles from now). It is imaginable as:

5^5^5^5^5^5..................^5
5^5^5^5^5^5..................^5 5's
5^5^5^5^5^5..................^5 5's
10^10^10^2184 5's

Note that this far far larger than boogafour, which is merely about 10^10^153.

I think you get the pattern from here on out. Let's skip to base 10:

10^^^2 = 10^^10, the decker which we saw earlier

10^^^3 = 10^^10^^10 = a power tower of a decker tens. Sbiis Saibian calls this number tria-taxis.

10^^^4 is a power tower of 10^^^3 tens. Sbiis Saibian calls this tetra-taxis; the names for higher numbers of the form 10^^^n continue with the Greek prefixes just like with the -logue numbers.

And now for two final numbers:

10^^^10 is a number Saibian calls deka-taxis. It can be imagined as:

And finally, 10^^^100 is a number Bowers calls gaggol, which can be imagined as:

That's an awesome number alright.

I TOLD you pentation will make tetration look adorable! But things get really crazy with the NEXT next operator, hexation.

Hexation

Properties and Notation of Hexation

Often, in an introduction to hyper-operators people will go no further in describing them than the sixth hyper-operator, known as hexation from hexa- (Greek prefix for six) + iteration. It is the largest hyper-operator Wikipedia mentions at all as of January 2016. Tetration has a long article on Wikipedia, pentation has a short article on Wikipedia, and hexation is mentioned in the articles on Knuth arrows and the hyper-operators. However, it does not mention any higher operators. Hexation is also the most powerful hyper-operator whose Googology Wiki article has examples of the operation.

Hexation is, of course, most often noted a^^^^b with Knuth's up-arrows. It's also denoted a{4}b with Bowers' operators, a->b->4 with Conway chain arrows, and {a,b,4} with Bowers' arrays. Sbiis Saibian proposes a_-->b for hexation similar to _b<--a for pentation. Sunir Shah uses a(**)b, and in Cutler's bar notation a hexated to b is ^b̄a. Robert Munafo uses a^⑥b for hexation (the hyper6 operator) and a_⑥b for the weaker hyper₆ operator (equal to (((....(a_⑤a)_⑤a)_⑤a).......)))) with b a's).

a hexated to b is equal to iterating the number of power-towers in pentation, aka a^^^a^^^a^^^a ... ^^^a with b a's. In terms of power-towers it can be visualized like so:

where there are b layers (including a), and the number of power towers in each layer is the leftmost tower in the previous layer. Tim Urban calls this a "power tower feeding frenzy psycho festival".

Notice the insane progression from each hyper-operator. We have the mundane addition and multiplication, then the more powerful exponentiation, and then the staggering power-tower tetration, then the crazy power-tower-iteration pentation, and now we have the utterly insane power-tower-super-iteration hexation. Now let's look at the numbers hexation produces.

Examples of Hexation

We now know the basics of hexation. Now it's time to find out how large the numbers hexation produces are. Let's go to some examples:

2^^^^1 = 2^^^(2^^^^0) = 2^^^1 = 2. We can skip cases of the form n^^^^1, as they clearly all evaluate to n.

2^^^^2 is, once again, 4

2^^^^3 is 2^^^2^^^2 = 2^^^4 = the power tower of 65,536 twos. A mind-boggling number, but one we've seen before.

2^^^^4, however, is equal to 2^^^2^^^^3 = 2^^^(2^^65,536) = :

2^2^2^2^2..................^2
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
: : : :
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
2^2^2^2^2..................^2 2's
65,536 2's

with 2^^65,536 minus 2 layers

That seems like a HUGE jump yet again, and it is indeed one, but now needless to say, it's all part of the pattern. Nonetheless this is a truly incredible number. Even the number of layers used in this visualization above is an unfathomable number!

How about 2^^^^5? That's:

with 2^^^^4 minus 2 layers

Notice that we're feeding the numbers into the number of layers now!

Let's move on to cases with 3's:

3^^^^2 = 3^^^3 = a power tower of 7.6 trillion 3's. Another cool number that we've seen before. The next number, however, is MUCH bigger:

3^^^^3 is equal to 3^^^(power tower of 7.6 trillion 3's). It's the first term in the sequence used to define Graham's number. That's not just a power tower of 3^^^3 threes—it's so much bigger than that. 3^^^^3 can be visualized as:

That's one of my own favorite large numbers. It's a huge number whose computation you can almost be able to imagine by iterating the height of power towers of 3's (starting with 3) 3^^^3 times. When you look at 3^3, 3^^3, 3^^^3, and 3^^^^3 as a family of large numbers, each has its own flavor. Sbiis Saibian describes this progression of numbers like so:

3^3 (palpable), 3^^3 (astronomical), 3^^^3 (unfathomable), 3^^^^3 (more unfathomable still?!?!)

While 3^^^3 is unfathomable, 3^^^^3 is a new kind of unfathomable, which we can call, for lack of a better term, "super unfathomable".

Likewise 3^^^^4 is like 3^^^^3, but with 3^^^^3 layers of power towers instead of just 3^^^3 layers. Unlike 3^^^^3, 3^^^^4 is defnitely past numbers whose computation you can really "see".

Of course, we can continue with cases involving fours. Let's just cut to the chase and describe 4^^^^4, a number Jonathan Bowers calls tritet. Here's a visualization of tritet:

The extra layer of iterated power towers is a big reason why this number is so much harder to understand than 3^^^^3, and also because the number in the righmost third of the picture, 4^^^4, is way harder to understand then 3^^^3. But this number pales in comparison to something like 10^^^^100, a number Bowers calls geegol to continue the googol, giggol, gaggol idea:

Are you dizzy yet?

Heptation, Octation, Enneation ...

After hexation comes heptation, octation, enneation, etc. Each name just uses the Greek prefix for what level of operation it is plus -ation, so heptation is iterated hexation, octation is iterated heptation, and so on.

I think that for heptation we can just give one example. 5^^^^^5 (five heptated to five) is a number Bowers calls tripent, which can be imagined like so:

Ok this is fucking huge. This right here is the power of just a few steps of recursion: this process can easily take you to some true eldritch abominations.

If you want to imagine octation, just imagine iterating the number of large horizontal layers in the image above. It's OK if that's hard for you to wrap your mind around. Then you can imagine enneation as iterating octation, decation as iterating enneation, etc., etc., etc., etc.

If these were huge, imagine how horrifying a tridecal, 10^^^^^^^^^^10 (that's ten ^s) would be!!! Visualizing a tridecal like the picture above would be difficult, since at that point the numbers exponentiated would require zooming in a lot to see. Or even worse, how about 10^^^^^^^^^^^^^^^^^^^^^^^^^^^10 (that's 27 ^s)?

Or 10^^^^^^... ... (100 ^s) ... ...^^^^^^10, which Bowers calls boogol?

Or 10^^^^^^... ... ... (7,625,597,484,987 ^s) ... ... ... ^^^^^^10?

Or 10^^^^^^... ... ... (3^^^3 ^s) ... ... ... ^^^^^^10?

Or 10^^^^^^... ... ... (5^^^^^5 ^s) ... ... ... ^^^^^^10?!

Or 10^^^^^^... ... ... (10^^^...(100 ^s)...^^^10 ^s) ... ... ... ^^^^^^10 which Bowers calls boogolplex?!?!

Before we get lost imagining iterating up-arrows, let's just say that there are in fact ways to express iterating these up-arrows in a whole new mind-blowing way! Three notable ways to do so are Conway's chain arrows, Bird's and Bowers' arrays (which are closely related), and Aarex's extended up-arrows. We'll get to those later, but for now I think these huge up-arrow numbers are satisfying to imagine. Let's wrap up with the sequence of Ackermann numbers, which is based on up-arrow notation.

Review: The Ackermann Numbers

I think a good way to wrap up this article is with the sequence of the Ackermann numbers. The Ackermann numbers can be easily defined with up-arrow notation, and they were described by Conway and Guy in their Book of Numbers. Sources vary on the origin (and definition) of Ackermann numbers, but my best bet is that they were either defined somewhere around 1928—the year the Ackermann function was first defined—or the name was coined by Conway and Guy.

They are defined like so:

the nth Ackermann number = n↑ⁿn

NOTE: Some sources say that the Ackermann numbers are defined as n n-ated to n, which creates the sequence 1+1, 2*2, 3^3, 4^^4, 5^^^5, etc. However this is errouneous as far as I am aware.

Let's start with visual representations of the first five Ackermann numbers:

First Ackermann number (1^1):

Second Ackermann number (2^^2):

Third Ackermann number (3^^^3):

Compared to the other crazy numbers we went through, 3^^^3 seems humble now, doesn't it?

Fourth Ackermann number (4^^^^4):

Fifth Ackermann number (5^^^^^5):

... and here's a bonus one ...

Sixth Ackermann number (6^^^^^^6)

(Click here for the full-size image.)

Notice the insane progression here. I'll leave you to imagine how big the 7th Ackermann number which Bowers calls "trisept" is.

Conclusion

Up-arrows are a notation that expands upon everyday math in an intuitive but mind-blowing way: it continues the familiar sequence of addition, multiplication, and exponentiation, and it doesn't take long at all to become scary. I think this notation makes a good first notation to discuss because it is a building block in many large number notations. Up next, I will go into more detail into the weak hyper-operators, an alternate version of the hyper-operators.

2.03. The Weak Hyper-Operators

Page updated

Google Sites

Report abuse