1.8. Extensions to the -illions II: Jonathan Bowers' -illions

Introduction

Perhaps the best known -illion system amongst the googology community is a system proposed by Jonathan Bowers, an important figure in googology. We'll learn more about him and his work later, but for now we'll only focus on his -illions.

Bowers presents his -illion scheme on this page on his website. His website has a variety of pages divided into topics. The two topics he is best known for are polytopes (multi-dimensional shapes) and large numbers. Bowers' work in polytopes involves working with figures in the familiar one, two, three dimenions, and also four or more dimensions, and classifying those figures. He has made some pretty stunning graphics with the polytopes; my favorites are his visualizations of polyterons (5-dimensional figures). Bowers' work in large numbers is two things: an expansive -illion scheme that solves many problems of other schemes, and a powerful array notation that set the foot of modern googology. Here we'll examine this -illion scheme of his, and we'll examine his array notation in detail starting in section 3.

Bowers' names

First off, here's all the numbers Bowers names on his website:

Million - 1,000,000

Billion

1,000,000,000

Trillion - 10^12

Quadrillion - 10^15

Quintillion - 10^18

Sextillion - 10^21

Septillion - 10^24

Octillion - 10^27

Nonillion - 10^30

Decillion - 10^33

Undecillion - 10^36

Doedecillion - 10^39

Tredecillion - 10^42

Quattuordecillion - 10^45

Quindecillion - 10^48

Sexdecillion - 10^51

Septendecillion - 10^54

Octodecillion - 10^57

Novemdecillion - 10^60

Vigintillion - 10^63

Trigintillion - 10^93

Googol - 10^100 - shown for comparison

Quadragintillion - 10^123

Quinquagintillion - 10^153

Sexagintillion - 10^183

Septuagintillion - 10^213

Octogintillion - 10^243

Nonagintillion - 10^273

Centillion - 10^303

Cenuntillion - 10^306

Duocentillion - 10^309

Centretillion - 10^312

Ducentillion - 10^603

Trecentillion - 10^903

Quadringentillion - 10^1203

Quingentillion - 10^1503

Sescentillion - 10^1803

Septingentillion - 10^2103

Octingentillion - 10^2403

Nongentillion - 10^2703

Millillion - 10^3003

*Platillion - 10^6000

Myrillion - 10^30003

Micrillion - 10^ 3000003

Nanillion - 10^ 3billion3

Picillion - 10^ 3trillion3

Femtillion - 10^ 3quadrillion3

Attillion - 10^ 3 quintillion3

Zeptillion - 10^ 3sextillion3

Yoctillion - 10^ 3septillion3

Xonillion - 10^ 3octillion3

Vecillion - 10^ 3nonillion3

Mecillion - 10^ 3decillion3

Duecillion - 10^ 3undecillion3

Trecillion - 10^ 3doedecillion3

Tetrecillion - 10^ 3tridecillion3

Pentecillion - 10^ 3quattuordecillion3

Hexecillion - 10^ 3quindecillion3

Heptecillion - 10^ 3sexdecillion3

Octecillion - 10^ 3septendecillion3

Ennecillion - 10^ 3octodecillion3

Icosillion - 10^ 3novemdecillion3

Triacontillion - 10^ (3x10^90+3)

Googolplex - 10^10^100 - shown for comparison

Tetracontillion - 10^ (3x10^120+3)

Pentacontillion - 10^ (3x10^150+3)

Hexacontillion - 10^ (3x10^180+3)

Heptacontillion - 10^ (3x10^210+3)

Octacontillion - 10^ (3x10^240+3)

Ennacontillion - 10^ (3x10^270+3)

Hectillion - 10^ (3x10^300+3)

Killillion - 10^ (3x10^3000+3)

Megillion - 10^ (3x10^3million +3)

Gigillion - 10^ (3x10^3billion +3)

Terillion - 10^ (3x10^3trillion +3)

Petillion - 10^ (3x10^3quadrillion +3)

Exillion - 10^ (3x10^3quintillion +3)

Zettillion - 10^ (3x10^3sextillion +3)

Yottillion - 10^ (3x10^3septillion +3)

Xennillion - 10^ (3x10^3octillion +3)

Dakillion - 10^ (3x10^3nonillion +3) - old name vekillion

Hendillion - 10^ (3x10^3decillion +3) - old name mekillion

Dokillion - 10^ (3x10^3undecillion +3) - old name duekillion

Tradakillion - 10^ (3x10^3doedecillion +3) - old name trekillion

Tedakillion - 10^ (3x10^3tridecillion +3) - old name tetrekillion

Pedakillion - 10^ (3x10^3quattuordecillion +3) - old name pentekillion

Exdakillion - 10^ (3x10^3quindecillion +3) - old name hexekillion

Zedakillion - 10^ (3x10^3sexdecillion +3) - old name heptekillion

Yodakillion - 10^ (3x10^3septendecillion +3) - old name octekillion

Nedakillion - 10^ (3x10^3octodecillion +3) - old name ennekillion

Ikillion - 10^ (3x10^(3x10^60) +3) - old name twentillion

Ikenillion - 10^ (3x10^(3x10^63) +3)

Icodillion - 10^ (3x10^(3x10^66) +3)

Ictrillion - 10^ (3x10^(3x10^69) +3) - old name triatwentillion

Icterillion - 10^ (3x10^(3x10^72) +3)

Icpetillion - 10^ (3x10^(3x10^75) +3)

Ikectillion - 10^ (3x10^(3x10^78) +3)

Iczetillion - 10^ (3x10^(3x10^81) +3)

Ikyotillion - 10^ (3x10^(3x10^84) +3)

Icxenillion - 10^ (3x10^(3x10^87) +3)

Trakillion - 10^ (3x10^(3x10^90) +3) - old name thirtillion

Googolduplex - 10^10^10^100 - shown for comparison

Tekillion - 10^ (3x10^(3x10^120) +3) - old name fortillion

Pekillion - 10^ (3x10^(3x10^150) +3) - old name fiftillion

Exakillion - 10^ (3x10^(3x10^180) +3) - old name sixtillion

Zakillion - 10^ (3x10^(3x10^210) +3) - old name seventillion

Yokillion - 10^ (3x10^(3x10^240) +3) - old name eightillion

Nekillion - 10^ (3x10^(3x10^270) +3) - old name nintillion

Hotillion - 10^ (3x10^(3x10^300) +3) - old name hundrillion

Botillion - 10^ (3x10^(3x10^600) +3)

Trotillion - 10^ (3x10^(3x10^900) +3)

Totillion - 10^ (3x10^(3x10^1200) +3)

Potillion - 10^ (3x10^(3x10^1500) +3)

Exotillion - 10^ (3x10^(3x10^1800) +3)

Zotillion - 10^ (3x10^(3x10^2100) +3)

Yootillion - 10^ (3x10^(3x10^2400) +3)

Notillion - 10^ (3x10^(3x10^2700) +3)

Kalillion - 10^ (3x10^(3x10^3000) +3) - old name thousillion

Dalillion - 10^ (3x10^(3x10^6000) +3)

Tralillion - 10^ (3x10^(3x10^9000) +3)

Talillion - 10^ (3x10^(3x10^12,000) +3)

Palillion - 10^ (3x10^(3x10^15,000) +3)

Exalillion - 10^ (3x10^(3x10^18,000) +3)

Zalillion - 10^ (3x10^(3x10^21,000) +3)

Yalillion - 10^ (3x10^(3x10^24,000) +3)

Nalillion - 10^ (3x10^(3x10^27,000) +3)

Dakalillion - 10^ (3*10^ (3*10^30,000) +3) - also called *manillion

Hotalillion - 10^ (3*10^ (3*10^300,000) +3) - also called *lakhillion

Mejillion - 10^ (3x10^(3x10^3,000,000) +3)

Dakejillion - 10^ (3*10^ (3*10^30,000,000) +3) - also called *crorillion

Hotejillion - 10^ (3*10^ (3*10^300,000,000) +3) - also called *awkillion

Googoltriplex - 10^10^10^10^100 - shown for comparison

Gijillion - 10^ (3x10^(3x10^3,000,000,000) +3)

*Bentrizillion - 10^ (6*10^ (6*10^ (6*10^6billion)))

Astillion - 10^ (3x10^(3x10^3trillion) +3)

Lunillion - 10^ (3x10^(3x10^3quadrillion) +3)

Fermillion - 10^ (3x10^(3x10^3quintillion) +3)

Jovillion - 10^ (3x10^(3x10^3sextillion) +3)

Solillion - 10^ (3x10^(3x10^3septillion) +3)

Betillion - 10^ (3x10^(3x10^3octillion) +3)

Glocillion - 10^ (3x10^(3x10^3nonillion) +3)

Gaxillion - 10^ (3x10^(3x10^3decillion) +3)

Supillion - 10^ (3x10^(3x10^3undecillion) +3)

Versillion - 10^ (3x10^(3x10^3dodecillion) +3)

Multillion - 10^ (3x10^(3x10^3tredecillion) +3)

Googolquadraplex - 10^10^10^10^10^100 - shown for comparison

Googolquinplex - 10^10^10^10^10^10^100 - shown for comparison

I colored the names like so:

Red names are the well-established number names in English.

Blue names are numbers that already have English names, but are modified in this system.

Green names are -illions whose names are already established in -illion systems other than Bowers'.

Purple names are main-sequence names Bowers himself established in his system.

Pink names are names Bowers himself established in his system that are not main-sequence.

Teal names are names that were mentioned to Bowers but he has no idea where they originated.

Yellow names are miscellaneous numbers that are not named in English but are well-established in the googology community.

Analyzing Bowers' names

As you can see, Bowers starts off simply listing the first twenty -illions in the English language. The only peculiarity is that he says "doedecillion" instead of "duodecillion".

Then Bowers lists the popular -illion names (which were clearly based upon Conway and Guy's -illions) between a vigintillion and a millillion, also listing the googol for comparison. However one peculiarity of his system is two names: the 101st and 103rd -illions are called cenuntillion and centretillion respectively. The reason why "centretillion" for the 103rd -illion was used is clear: otherwise the 103rd -illion would have to be called "trecentillion" and share its name with the 300th -illion. I'm not sure why Bowers chose "cenuntillion" for the 101st -illion though, as "uncentillion" seems ok. In any case, we can treat "cenuntillion" and "centretillion" as special cases as we'll see later.

Another strange thing is that he names the twelfth -illion doedecillion but the 102nd -illion duocentillion. What's that all about? Why use "doe" to add two in one place and "duo" in another? If I were Bowers I would just use "duo" in all cases to be faithful to the dictionary -illions, but for the sake of faithfulness to Bowers' system we can say that usage of "doe" versus "duo" depends on the case.

Continuing, the first name Bowers himself came up with was millillion. He came up with the name like so:

decillion is the 10th -illion, deci- divides by 10

centillion is the 100th -illion, centi- divides by 100

therefore:

millillion is the 1000th -illion since milli- divides by 1000

Later, he realized that he wasn't the only one who came up with "millillion" for 10^3003. There's also Henkle's and Conway and Guy's systems that use "millillion" or a variant of that, which he found through reading Robert Munafo's large numbers site. But in any case, he continues with the peculiar name:

platillion = 10^6000

This is one of the names that was suggested to Bowers, but he has no idea where it came from. It would be the 1999th -illion in the short scale, so I don't consider it actually part of Bowers' -illion system. Since this is the 1000th -illion using the long scale instead of the short scale, I suppose whoever suggested "platillion" to Bowers just wanted to give an arbitrary name to the 1000th long scale -illion, although "millillion" would be the obvious Latin continuation.

Then he gives another unusual name:

myrillion = 10^30,003

This is a name for the 10,000th -illion based obviously on "myriad", an archaic word for ten thousand from ancient Greek "myria" meaning 10,000. I consider this another "off-the-sequence" name, and as we will see we can just name this something like "decimillillion" with no problems and forget about this odd name.

Going back to the main sequence Bowers continues the sequence idea:

micrillion = 10^3,000,003

nanillion = 10^3,000,000,003

plus picillion, femtillion, attillion, zeptillion, yoctillion

Yoctillion is 10^(3*10^24+3), and it's as far through the -illions the SI prefix idea can take us. But instead of stopping here, Bowers continues with his own naming system with xonillion, vecillion, mecillion, duecillion ... ennecillion. After "ennecillion" Bowers makes a switch to using the Greek prefixes. Interestingly, a lot of them, like "icosillion", happen to share their names with Rowlett's -illions. Along the way he puts a googolplex in his -illion list for comparison.

Starting with "hectillion" Bowers makes a switch to using the large SI prefixes, aka kilo, mega, giga, etc., giving us names like "megillion". After that he goes back to his own original system, based upon his system for naming polytopes of any number of dimensions (e.g. polyhoton for 101 dimensions). Along the way also lists some of his old names for -illions, which he changed clearly because some of them could easily be confused with other -illions, e.g. vecillion vs. vekillion, and the English-based names could get a little confusing to work with, e.g. one hundred twentillion. He also lists the googolduplex for comparison.

"Thousillion" is clearly the limit of Bowers' old system, and Bowers presents his new freshly made extension to the -illions after that point, along with some unusual names suggested to him, which are manillion, lakhillion, crorillion, and awkillion. Lakhillion and crorillion clearly come from "lakh" and "crore", the Indian words for 100,000 and 10,000,000 (noted 10,00,000 and 10,00,00,000 in India's system), but the origin of "manillion" and "awkillion" is unkown. The manillion family appears to be an extension of Bowers' old system which was limited at "thousillion".

Right before the name "gijillion", Bowers lists the number "googoltriplex" for comparison, although he puts it in the wrong spot on his list; the googoltriplex is larger than any of Bowers' -illions. Then Bowers lists another peculiar -illion, "bentrizillion" equal to 10^ (6*10^ (6*10^ (6*10^6billion))). The number would be the billionth -illionth -illionth -illion in the long scale. Because of this long scale usage, it is unclear whether billion refers to 109 (its short scale value) or 1012 (its long scale value), but Bowers clearly interpreted it as the former.

Starting with "astillion" Bowers then takes a switch to astronomy-based names, naming his -illions on increasingly large astronomical objects: astillion is from "asteroid", lunillion from "lunar" (moon), fermillion from "terra firma" (Earth), jovillion from "Jovian" (Jupiter), solillion from "solar" (sun), betilllion from Betelgeuse, glocillion from "globular cluster", gaxillion from "galaxy", supillion from "supercluster", versillion from "universe", and finally multillion from "multiverse". He then lists a googolquadraplex (also known as googolquadriplex) and googolquinplex (also known as googolquintiplex) for comparison, and ends his list of -illions there.

The First Thousand -illions

Now that we've familiarized ourselves with Bowers' names let's look at how exactly we'll name the first 1000 -illions. We'll use all the names Bowers himself gives, even weird things like "doedecillion".

For the first nine -illions we'll specifically use unique names:

million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion

After that we'll use a construction system:

To name the nth -illion, 10 ≤ n ≤ 1000, and if the units place is not 2 and it's not true that the tens place is 0 AND the units place is 1 or 3, combine the roots in the table:

Units Tens Hundreds
1 Un Deci Centi
2 Doe / Duo Viginti Ducenti
3 Tre Triginta Trecenti
4 Quattuor Quadraginti Quadringenti
5 Quin Quinquaginti Quingenti
6 Sex Sexaginti Sescenti
7 Septen Septuaginti Septingenti
8 Octo Octoginti Octingenti
9 Novem Nonaginti Nongenti

based on the hundreds, tens, and units place. Units comes first, then tens, then hundreds. If any one place is 0, use no root. After that append "llion" to the name.

Special rules are:

If the units place of n is 2 and the tens place isn't 0, then use "doe" as the units root.

If the units place of n is 2 and the tens place is 0, then use "duo" as the units root.

If the tens place is 0 and the units place is 1 or 3, construct the name with the hundreds root (drop the t) plus "untillion" if the units place is 1, or "tretillion" if the units place is 3.

If n is 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9, use the following names respectively:

thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion

These rules are enough to give us the start of the table of -illion names:


n

103n+3

0

thousand

1

million

2

billion

3

trillion

4

quadrillion

5

quintillion

6

sextillion

7

septillion

8

octillion

9

nonillion

10

decillion

11

undecillion

12

doedecillion

13

tredecillion

14

quattuordecillion

15

quindecillion

16

sexdecillion

17

septendecillion

18

octodecillion


19

novemdecillion

20

vigintillion

21

unvigintillion

22

doevigintillion

23

trevigintillion

24

quattuorvigintillion

25

quinvigintillion

26

sexvigintillion

27

septenvigintillion

28

octovigintillion

29

novemvigintillion

30

trigintillion

31

untrigintillion

32

duotrigintillion

40

quadragintillion

50

quinquagintillion

60

sexagintillion

70

septuagintillion

80

octogintillion

90

nonagintillion


100

centillion

101

cenuntillion

102

duocentillion

103

centretillion

104

quattuorcentillion

105

quincentillion

110

decicentillion

111

undecicentillion

112

duodecicentillion

120

viginticentillion

130

triginticentillion

140

quadraginticentillion

150

quinquaginticentillion

160

sexaginticentillion

170

septuaginticentillion

180

octoginticentillion

190

nonaginticentillion

200

ducentillion

201

ducenuntillion

210

deciducentillion

211

undeciducentillion

220

vigintiducentillion

300

trecentillion

400

quadringentillion


500

quingentillion

600

sescentillion

700

septingentillion

800

octingentillion

900

nongentillion

1000

millillion


This is quite nice so far. But what comes after millillion and all that? We know the names of the millionth -illion and stuff, but how exactly will we continue? Welcome to the second tier where we find all that out.

The Second Tier

When we reach the second tier of Bowers' -illions, things get a bit tricky. What should we name the next -illion after millillion? Continuing with something like "unmillillion, duomillillion, tremillillion ... " sounds like a pretty natural idea until you consider confusing parts. For example, duomillillion sounds like it could just as well refer to the 2000th -illion, tremillillion may sound like the 3000th -illion, and so on. Sbiis Saibian, on his article on Bowers' -illions, has a better idea. He suggests the name "milli-untillion" for the 1001st -illion, followed by milli-deutillion, milli-tretillion, and after this point milli-x-illion is the 1000+x-th -illion; for example, "milli-centillion" is the 1100th -illion. Thus we obtain names such as:


1000

millillion

1001

milli-untillion

1002

milli-deutillion

1003

milli-tretillion

1004

milli-quadrillion

1005

milli-quintillion

...

...

1010

milli-decillion

...

...

1020

milli-vigintillion

...

...

1100

milli-centillion

...

...

1200

milli-ducentillion

etc.

etc.


I'm assuming you get the pattern. Why exactly are we taking this pattern? Think back to Henkle's -illions and what happens with names where you don't see the most significant part of the number name (for example "billion" in "one billion three hundred milllion twenty-two thousand five hundred fifty-three") until the end of the -illion name. By keeping the most significant part early on in the name, we've got much more convenient names.

After the whole milli-x-illion family, Sbiis Saibian suggests that we name the 2000th -illion "duomillillion". This allows us to create another block of 1000 -illion names:

2000

duomillillion

2001

duomilli-untillion

2002

duomilli-deutillion

2003

duomilli-tretillion

2004

duomilli-quadrillion

2005

duomilli-quintillion

...

...

2010

duomilli-decillion

...

...

2020

duomilli-vigintillion

...

...

2100

duomilli-centillion

...

...

2200

duomilli-ducentillion

etc.

etc.


Then after this point things get fairly routine. We can name the 3000th -illion "tremillillion", the 4000th -illion "quattuormillillion" ... the 10,000th -illion "decimillillion" ... the 100,000th -illion "centimillillion" ... etc.

After we reach a micrillion (the millionth -illion) we can do almost the same process from the 1,000,001th to 1,999,999th -illions, using micro- in place of milli- to get names like micro-untillion, micro-millillion, micro-duomilli-trigintillion, etc. After that we use duo-, tre-, quattour- on micrillion to get multiplier roots (for example the 13,000,000th -illion is tredecimicrillion)

Then of course we can do the same thing with the nanillion family (e.g. nano-untillion or duonanillion), picillion, femtillion, attillion, zeptillion, and yoctillion. With only a few new roots we can name exactly 1027-1 -illions. Besides the milestone numbers (millillion, micrillion, nanillion, etc.) we can of course name more complicated numbers like:

yocto-zepto-atto-femto-pico-nano-micro-milli-untillion = 103,003,003,003,003,003,003,003,006

or even more complex names such as:

centiyocto-zepto-treatto-doetrigintifemto-undecicentipico-novemsexagintinano-duomicro-quingentimilli-septenseptuagintiseptingentillion = 10300,003,009,096,333,207,007,502,334

and the largest we can name:

novemnonagintinongentiyocto-novemnonagintinongentizepto-novemnonagintinongentiatto-novemnonagintinongentifemto-novemnonagintinongentipico-novemnonagintinongentinano-novemnonagintinongentimicro-novemnonagintinongentimilli-novemnonagintinongentillion = 103,000,000,000,000,000,000,000,000,000 — this is exactly equal to 1 followed by 3 octillion zeros

After this point we're out of SI prefixes to use. Therefore Bowers creates a system that extends further than that. The next roots after yocto- he names are:

xono-, veco-, meco-, dueco-, treco-, tetreco-, penteco-, hexeco-, hepteco-, octeco-, enneco-, icoso-.

That's nice! We now have a full 20 separator roots to name a vigintillion minus one -illions total. But what comes after icoso-? Bowers doesn't specify, but Sbiis Saibian suggests that we could continue with:

meicoso-, dueicoso-, trioicoso-, tetreicoso-, penteicoso-, hexeicoso-, hepteicoso-, octeicoso-, enneicoso-

Note that "trioicoso-" is used in place of "treicoso-" because treicosillion itself by the rules already refers to 109*10^60+3 (the 3*1060th -illion).

Then we can do the same to triaconto-, tetraconto-, pentaconto- ... ennaconto-. After hecto- (the 100th root in the milli-, micro-, nano- ... series) we can apply our previous roots again. For example the next few roots after hecto- are:

mehecto-, duehecto-, triohecto- ...

and the series continues with:

tetrehecto-, pentehecto- ... vecehecto-, mecehecto-, duecehecto- ... icosehecto- ... triacontehecto- ... and so on.

Bowers doesn't specify what the 200th root of that series would be, but Sbiis Saibian suggests to use dohecto-. Likewise, for the 300th, 400th, 500th, etc., roots of that series, Sbiis Saibian suggests triahecto-, tetrahecto-, pentahecto-, hexahecto-, heptahecto-, octahecto- and ennahecto-.

 And that's it with the second tier! To review here's a table of all the roots:

Tier 1

n Ones (option 1) Ones (option 2) Ones (option 3) Tens Hundreds
1 m un unt deci centi
2 b duo / doe deut viginti ducenti
3 tr tre tret triginti trecenti
4 quadr quattuor quadr quadraginti quadringenti
5 quint quin quint quinquaginti quingenti
6 sext sex sext sexaginti sescenti
7 sept septen sept septuaginti septingenti
8 oct octo oct octoginti octingenti
9 non novem non nonaginti nongenti

For ones, use the option 1 roots when you're naming a number below 1033, option 3 roots for the ones place where the 10s and 100s place are 0, and option 2 elsewhere.

Tier 2

n Ones (option 1) Ones (option 2) Tens Hundreds
1 milli me vec(o/e) hect(o/e)
2 micro due icos(o/e) dohect(o/e)
3 nano trio triacont(o/e) triahect(o/e)
4 pico tetre tetracont(o/e) tetrahect(o/e)
5 femto pente pentacont(o/e) pentahect(o/e)
6 atto hexe hexacont(o/e) hexahect(o/e)
7 zepto hepte heptacont(o/e) heptahect(o/e)
8 yocto octe octacont(o/e) octahect(o/e)
9 xono enne ennacont(o/e) ennahect(o/e)
10 veco vece

11 meco mece

12 dueco duece

13 treco trece

14 tetreco tetrece

15 penteco pentece

16 hexeco hexece

17 hepteco heptece

18 octeco octece

19 enneco ennece

Once again, the option 1 (-o) roots are used when it's not combining with another tier-2 root, and the option 2 (-e) roots are used when combining.

In addition here's a list of some ofthe -illions we can name using the first two tiers:


n

103n+3

103*103n+3

0

thousand

(million)

1

million

millillion

2

billion

micrillion

3

trillion

nanillion

4

quadrillion

picillion

5

quintillion

femtillion

6

sextillion

attillion

7

septillion

zeptillion

8

octillion

yoctillion

9

nonillion

xonillion

10

decillion

vecillion

11

undecillion

mecillion

12

doedecillion

duecillion

13

tredecillion

trecillion

14

quattuordecillion

tetrecillion

15

quindecillion

pentecillion

16

sexdecillion

hexecillion

17

septendecillion

heptecillion

18

octodecillion

octecillion


19

novemdecillion

ennecillion

20

vigintillion

icosillion

21

unvigintillion

meicosillion

22

doevigintillion

dueicosillion

23

trevigintillion

trioicosillion

24

quattuorvigintillion

tetreicosillion

25

quinvigintillion

penteicosillion

26

sexvigintillion

hexeicosillion

27

septenvigintillion

hepteicosillion

28

octovigintillion

octeicosillion

29

novemvigintillion

enneicosillion

30

trigintillion

triacontillion

31

untrigintillion

metriacontillion

32

duotrigintillion

duetriacontillion

40

quadragintillion

tetracontillion

50

quinquagintillion

pentacontillion

60

sexagintillion

hexacontillion

70

septuagintillion

heptacontillion

80

octogintillion

octacontillion

90

nonagintillion

ennacontillion


100

centillion

hectillion

101

cenuntillion

mehectillion

102

duocentillion

duehectillion

103

centretillion

triohectillion

104

quattuorcentillion

tetrehectillion

105

quincentillion

pentehectillion

110

decicentillion

vecehectillion

111

undecicentillion

mecehectillion

112

duodecicentillion

duecehectillion

120

viginticentillion

icosehectillion

130

triginticentillion

triacontehectillion

140

quadraginticentillion

tetracontehectillion

150

quinquaginticentillion

pentacontehectillion

160

sexaginticentillion

hexacontehectillion

170

septuaginticentillion

heptacontehectillion

180

octoginticentillion

octacontehectillion

190

nonaginticentillion

ennacontehectillion

200

ducentillion

dohectillion

201

ducenuntillion

medohectillion

210

deciducentillion

vecedohectillion

211

undeciducentillion

mecedohectillion

220

vigintiducentillion

icosedohectillion

300

trecentillion

triahectillion

400

quadringentillion

tetrahectillion


500

quingentillion

pentahectillion

600

sescentillion

hexahectillion

700

septingentillion

heptahectillion

800

octingentillion

octahectillion

900

nongentillion

ennahectillion


1000

millillion

1001

milli-untillion

1002

milli-deutillion

1003

milli-tretillion

1004

milli-quadrillion

1005

milli-quintillion

1010

milli-decillion

1020

milli-vigintillion

1100

milli-centillion

1200

milli-ducentillion

2000

duomillillion

3000

tremillillion

4000

quattuormillillion

5000

quinmillillion

6000

sexmillillion

7000

septenmillillion

8000

octomillillion

9000

novemmillillion

10,000

decimillillion

11,000

undecimillillion


12,000

duodecimillillion

20,000

vigintimillillion

30,000

trigintimillillion

40,000

quadragintimillillion

100,000

centimillillion

200,000

ducentimillillion

1,000,000

micrillion

1,000,001

micro-untillion

1,001,000

micro-millillion

2,000,000

duomicrillion

10,000,000

decimicrillion

100,000,000

centimicrillion

1,000,000,000

nanillion


So there you have it, the second tier. Hopefully none of this was too confusing or hard to wrap your head around. With that said, on to the third tier!

The Third Tier

We now reach the third tier of Bowers' unusual naming system. We begin of course with a name for the 1000th tier 2 separator (in the series milli-, micro-, nano- ... ): killo-. But after this point things get tricky. To distinguish between numbers like "killo-millillion" (that would be the 103000+1000th -illion), Sbiis Saibian suggests that we name the 1001st tier 2 root killamilli-. Note the usage of the vowel "a" here in place of "o". If this is confusing, simply consider that "o" is specifically used to indicate the end of the tier 2 root.

After killamilli-, we can of course combine "killa" with any of the previous 999 tier 2 roots: killamicro-, killaveco-, killatrioicosedohecto-, etc. But what would we call the 2000th tier 2 root? To match up with usage of terms like
"duomillillion", we'll name the 2000th tier 2 root "duekillo-". We can of course append "duekilla" with any of the first 999 tier 2 roots. After this point the 3000th, 4000th, etc., tier 2 roots are triokillo- (not trekillo!), tetrekillo-, pentekillo- ... we can continue with things like vecekillo, ennecekillo, triahectekillo, etc. With that the 999,999th tier 2 root is simply enneennaconteennahectakillaenneennaconteennahecto-. Pretty simple right?

We can go to even more drastic heights by adding the millionth, billionth, etc., up to septillionth roots, all based on the large SI prefixes: mego-, gigo-, tero-, peto-, exo-, zetto-, and yotto-. With those the largest tier 2 root we can name is:

enneennaconteennahectayottaenneennaconteennahectazettaenneennaconteennahectaexaenneennaconteennahectapetaenneennaconteennahectateraenneennaconteennahectagigaenneennaconteennahectamegaenneennaconteennahectakilloenneennacontexono-

Not bad so far! But a natural question is: How do we go further? At this point Bowers turns to something more unusual. He bases the names upon his polytope naming system. Now what exactly is that?

First, consider that a polygon (means many knees) is a 2-dimensional figure made from connected line segments, and that a polyhedron (means many faces) is a 3-dimensional figure made from connected polygons. But what comes after that? As I said in the introduction to this page, Bowers has done a great deal of work in the field of figures in higher dimensions. A 4-dimensional figure is often known as a polychoron (according to Bowers [source] this name was coined by Norman Johnson), continuing the names "polygon" and "polyhedron" as "polychoron" means "many rooms". Although this is the agreed-upon name for 4-dimensional figures among the polytopist community, professional mathematicians will often not use such names and instead opt for the rather dull term "4-polytope". But note that polytopists will generally study higher dimensions much more passionately than professional mathematicians do—the same can in fact be said for large numbers, as we'll see in sections 2 and 3 when we compare large number notations professional mathematicians devised (mostly to give people a sense of how big infinity is) against the much more powerful systems people like Jonathan Bowers himself devised. Back to where we were, what words do polytopists like Bowers use to denote 5 or higher dimensional figures?

Bowers explains that he and George Olshevsky coined the names polytetron, polypenton, and polyhexon for 5, 6, and 7 dimensions, from the Greek prefixes for 4, 5, and 6. However, he says the names have a problem: he gives the example, "is a hecatonicosatetron a 120 sided polytetron (5-D) or a 100 sided icosatetron (25-D)?" To solve this problem he explains that Wendy Krieger came up with the names polyteron, polypeton, polyexon (Bowers changed the name to polyecton), polyzetton, and polyyotton for 5, 6, 7, 8, and 9 dimensions; the names of course come from the large scale SI prefixes tera- through yotta-. And then Bowers extended that system ... to a quattuordecillion dimensions! I'm not kidding, he has a naming system that goes all the way up to such an insane number of dimensions! And that is the base of the rest of his -illion system.

So applying the polytope roots to Bowers' -illions is quite simple. Consider the series mega-, giga-, tera- ... of tier-3 roots. The SI prefixes alone can take us to the 8th tier-3 root, but Bowers' system can take us way past that. First the 9th tier-3 root is xenna-, then we can continue with the names daka-, henda-, doka-, tradaka-, tedaka-, pedaka-, exdaka-, zedaka-, yodaka-, and nedaka-.

After this, the 20th tier 3 root is ika-, and after that we have ikena-, icoda-, ictra-, ictera-, icpeta, ikexa-, iczeta-, ikyota-, and icxena-. At this point there are a few things to note. First off, note with the 21st through 29th tier 3 roots we have the tens root (ic/ik here) before the ones root (ena-, oda-, tra-, etc.). This is in contrast with all the previous -illions; for example, in quattuortrigintillion, "quattuor" is the ones root and it comes before "triginti", the tens root. Also, there is the peculiar variation of using "ic" vs. "ik" in the names. This variance seems to be entirely ad-hoc, but we'll go with it for faithfulness's sake.

Now the 30th tier 3 root is traka-, but how do you go further than that? We can refer back to the polytope naming system and use the names, trakena-, tracoda-, tractra-, tractera-, tracpeta-, trakexa-, traczeta-, trakyota-, and tracxena-.

Just about the same can be done for the 40th to 49th tier 3 roots: just use the roots above but replace "tra" with "te". Then you can replace it with "pe" for the 50th to 59th roots, "exa" for the 60th to 69th, "za" 70th to 79th, "yo" 80th to 89th, and "ne" 90th through 99th.

After this point we use "hota" for the 100th root; you can append "ho" with "tena", "toda", "tra", "tera", "peta", "tecta", "zeta", "yota", "xena", "daka", and "tenda" for the 101st through 111th roots respectively. After that, you can append "ho" with the 12th through 99th tier 3 roots to get the 112th through 199th tier 3 roots, inserting a "t" after "ho" if the tier 3 root you append starts with a vowel. For example, hozacpeta- is the 175th tier 3 root, and hotictra- is the 123rd tier 3 root.

To make the 200th through 299th tier 3 roots, take one of the 100th through 199th tier 3 roots but replace "ho" with "bo". To get the 300th through 399th tier 3 roots replace "ho" with "tro". Replace it with "to" for the 400th to 499th tier 3 roots, "po" for 500th through 599th, "exo" 600th to 699th, "zo" 700th through 799th, "yoo" 800th to 899th, and "no" 900th to 999th. Then 999th tier 3 root, the largest nameable with tier 3 roots, is "nonecxena".

To recap, below is a table of all the tier 3 roots. It's bigger than the other tables because there is more variance in names and usage of the roots.

n

Ones (option 1)

Ones (option 2)

Ones (option 3)

Tens (option 1)

Tens (option 2)

Hundreds

0


ka

ta




1

killa

kena

tena

da

da

ho

2

mega

coda

toda

i

ti

bo

3

giga

ctra

tra

tra

tra

tro

4

tera

ctera

tera

te

te

to

5

peta

cpeta

peta

pe

pe

po

6

exa

kexa

tecta

exa

texa

exo

7

zetta

czeta

zeta

za

za

zo

8

yotta

kyota

yota

yo

yo

yoo

9

xenna

cxena

xena

ne

ne

no

10

daka


daka


11

henda


tenda


12

doka


doka


13

tradaka


tradaka


14

tedaka


tedaka


15

pedaka


pedaka


16

exdaka


exdaka


17

zedaka


zedaka


18

yodaka


yodaka


19

nedaka


nedaka



Note: For ones, use option 1 when the tens place is 0 or 1 and the hundreds place is 0. Use option 2 when the tens place is 2 through 9. Use option 3 when the tens place is 0 or 1 and the hundreds place is 2 through 9. For tens, use option 1 when the hundreds place is 0 and option 2 otherwise.

In addition here's a table of names we can make with everything we have so far:


n

103n+3

103*103n+3

103*103*103n+3

0

thousand

(million)

(millillion)

1

million

millillion

killillion

2

billion

micrillion

megillion

3

trillion

nanillion

gigillion

4

quadrillion

picillion

terillion

5

quintillion

femtillion

petillion

6

sextillion

attillion

exillion

7

septillion

zeptillion

zettillion

8

octillion

yoctillion

yottillion

9

nonillion

xonillion

xennillion

10

decillion

vecillion

dakillion

11

undecillion

mecillion

hendillion

12

doedecillion

duecillion

dokillion

13

tredecillion

trecillion

tradakillion

14

quattuordecillion

tetrecillion

tedakillion

15

quindecillion

pentecillion

pedakillion

16

sexdecillion

hexecillion

exdakillion

17

septendecillion

heptecillion

zedakillion

18

octodecillion

octecillion

yodakillion


19

novemdecillion

ennecillion

nedakillion

20

vigintillion

icosillion

icillion

21

unvigintillion

meicosillion

ikenillion

22

doevigintillion

dueicosillion

icdillion

23

trevigintillion

trioicosillion

ictrillion

24

quattuorvigintillion

tetreicosillion

icterillion

25

quinvigintillion

penteicosillion

icpetillion

26

sexvigintillion

hexeicosillion

ikectillion

27

septenvigintillion

hepteicosillion

iczetillion

28

octovigintillion

octeicosillion

ikyotillion

29

novemvigintillion

enneicosillion

icxenillion

30

trigintillion

triacontillion

trakillion

31

untrigintillion

metriacontillion

trakenillion

32

duotrigintillion

duetriacontillion

tracdillion

40

quadragintillion

tetracontillion

tekillion

50

quinquagintillion

pentacontillion

pekillion

60

sexagintillion

hexacontillion

exakillion

70

septuagintillion

heptacontillion

zakillion

80

octogintillion

octacontillion

yokillion

90

nonagintillion

ennacontillion

nekillion


100

centillion

hectillion

hotillion

101

cenuntillion

mehectillion

hotenillion

102

duocentillion

duehectillion

hotodillion

103

centretillion

triohectillion

hotrillion

104

quattuorcentillion

tetrehectillion

hoterillion

105

quincentillion

pentehectillion

hopetillion

110

decicentillion

vecehectillion

hodakillion

111

undecicentillion

mecehectillion

hotendillion

112

duodecicentillion

duecehectillion

hodokillion

120

viginticentillion

icosehectillion

hotikillion

130

triginticentillion

triacontehectillion

hotrakillion

140

quadraginticentillion

tetracontehectillion

hotekillion

150

quinquaginticentillion

pentacontehectillion

hopekillion

160

sexaginticentillion

hexacontehectillion

hotexakillion

170

septuaginticentillion

heptacontehectillion

hozakillion

180

octoginticentillion

octacontehectillion

hoyokillion

190

nonaginticentillion

ennacontehectillion

honekillion

200

ducentillion

dohectillion

botillion

201

ducenuntillion

medohectillion

botenillion

210

deciducentillion

vecedohectillion

bodakillion

211

undeciducentillion

mecedohectillion

botendillion

220

vigintiducentillion

icosedohectillion

boticillion

300

trecentillion

triahectillion

trotillion

400

quadringentillion

tetrahectillion

totillion


500

quingentillion

pentahectillion

potillion

600

sescentillion

hexahectillion

exotillion

700

septingentillion

heptahectillion

zotillion

800

octingentillion

octahectillion

yootillion

900

nongentillion

ennahectillion

notillion


1000

millillion

killillion

1001

milli-untillion

killamillillion

1002

milli-deutillion

killamicrillion

1003

milli-tretillion

killananillion

1004

milli-quadrillion

killapicillion

1005

milli-quintillion

killafemtillion

1010

milli-decillion

killavecillion

1020

milli-vigintillion

killaicosillion

1100

milli-centillion

killahectillion

1200

milli-ducentillion

killadohectillion

2000

duomillillion

micrekillillion

3000

tremillillion

nanekillillion

4000

quattuormillillion

picekillillion

5000

quinmillillion

femtekillillion

6000

sexmillillion

attekillillion

7000

septenmillillion

zeptekillillion

8000

octomillillion

yoctekillillion

9000

novemmillillion

xonekillillion

10,000

decimillillion

vecekillillion

11,000

undecimillillion

mecekillillion


12,000

duodecimillillion

duecekillillion

20,000

vigintimillillion

icosekillillion

30,000

trigintimillillion

triacontekillillion

40,000

quadragintimillillion

tetracontekillillion

100,000

centimillillion

hectekillillion

200,000

ducentimillillion

dohectekillillion

1,000,000

micrillion

megillion

1,000,001

micro-untillion

megamillillion

1,001,000

micro-millillion

megakillillion

2,000,000

duomicrillion

micremegillion

10,000,000

decimicrillion

vecemegillion

100,000,000

centimicrillion

hectemegillion

1,000,000,000

nanillion

gigillion


This is quite good now! Up next we have the 4th tier, the final tier of Bowers' system...

The Fourth Tier

The final tier of Bowers' -illion scheme is the fourth tier. The fourth tier, unlike the previous ones, is not a full complete tier, but nonetheless it's still the final tier of the system. Let's jump right into the naming system:

The first few names after the 1000th tier 3 root, kala-, are just "kal" appended with ena, oda, tra, tera, peta, ecta, zeta, yota, xena, and daka. These are almost the same as the option 3 tier 3 ones place roots, but dropping the starting "t" from ena, oda, and ecta. After this point you can name the 1011th through 1999th tier-3 roots by appending "kal" with the 11th through 999th tier-3 roots.

Then, we can do the same thing to "dala-" (the 2000th tier 3 root), "trala-" (3000th), tala- (4000th), pala- (500th), exala- (6000th), zala- (7000th), yala- (8000th), and nala- (9000th). After nala-, to get the x*1000th tier 3 root (x≤999) you just use the xth tier 3 root (dropping the a) followed by "ala". For example, the 56,000th tier 3 root is pekectala-.

The next milestone is the millionth tier 3 root, "meja-". You can append "mej" with any of the first 999,999 tier 3 roots to get the 1,000,001th to 1,999,999th tier 3 roots. Then you can do to "mej" what you can do with "kal" to get (for example) deja-, treja-, teja-, etc., as the 2,000,000th, 3,000,000th, etc. tier 3 roots. For example, yoonecpetejtractralatera- is the 895,033,004th tier 3 root. See if you can figure out which root "botexekenejzotikodalahonectera-" refers to.

Then we can do the same to gija- (billionth to 999 billionth roots), dropping the "g" as needed, asta-, luna- (dropping l as needed), ferma- (dropping f as needed), jova- (drop j as needed), sola- (drop s as needed), beta- (drop b as needed), gloca- (drop gl as needed), gaxa- (drop g as needed), supa- (drop s as needed), versa- (drop v as needed), and finally multa- (dropping the "m" as needed). So we're almsot to the end of the journey through Bowers' -illions! To review all this here's a table of multiplier tier 3 roots (e.g. turning "kala" into "dala") and the tier 4 roots:

n Multiplier ones (option 1) Ones (option 4) Tier 4 root
1 letter in ( ) in tier 4 root ena (k)al[a]
2 d oda (m)ej
3 tr tra (g)ij
4 t tera ast
5 p peta (l)un
6 ex exa (f)erm
7 z zeta (j)ov
8 y yota (s)ol
9 n xena (b)et
10
10 to 19 are same as ones option 3 (gl)oc
11

(g)ax
12

(s)up
13

(v)ers
14

(m)ult

Note: The multiplier roots other than the topion 1 ones are the same roots as the tier 3 roots. Also the ones (option 4) are used when the tens and hundreds are both 0 but you're naming numbers larger than a kalillion.

So now, we're quite close to the end of our journey! With that said, let's find out the largest numbers the system lets us name, and then discuss how to go further.

Limit of Bowers' -illions

Now we'll find out the limit of the system. First off, what is the largest tier 3 separator we can name using roots up to multa-? It is:

nonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxen
unnonecxenastnonecxenijnonecxenejnonecxenalnonecxena-

That's the 999,999,999,999,999,999,999,999,999,999,999,999,999,999th (999 tredecillion 999 duodecillion ... 999 thousand 999th) tier 3 root.

But this isn't entirely the largest separator we can name. Think back to the tier 2 separators, where (for example) giga- is the 3rd tier 3 separator but gigo- is the billionth tier 2 separator, and the two are nearly equivalent except that one is tier 3 and the other is tier 2. Then, we can name the largest tier 2 separator as well. To do this, we list all possible tier 3 separators in decending order (999,999,999,999,999,999,999,999,999,999,999,999,999,999th,
999,999,999,999,999,999,999,999,999,999,999,999,999,998th,
999,999,999,999,999,999,999,999,999,999,999,999,999,997th, etc., all the way down to the first) and insert enneenneaconteennahecte before each of them. This gives us the largest tier 2 root:

enneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecx

enovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenneennaconteennahectenon

ecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunn

onecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxenultnonecxenersnonecxenu

pnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxene

jnonecxenalnoneczetaenneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxe

netnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekexaenneennac

onteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecx

enirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecpetaenneennaconteennahectenonecxenultnon

ecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastn

onecxenijnonecxenejnonecxenalnonecteraenneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenax

nonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenaln

onectraenneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenol

nonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecodaenneennaconteennahec

tenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxe

nunnonecxenastnonecxenijnonecxenejnonecxenalnonekenaenneennaconteennahectenonecxenultnonecxenersnonec

xenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonec

xenejnonecxenalnonecaenneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonec

xenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnoyocxena


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahecteicaenneennaconteennahectenedakaenneennaconteennahecteyodakaenneennacontee

nnahectezedakaenneennaconteennahecteexdakaenneennaconteennahectepedakaenneennaconteennahecteted

akaenneennaconteennahectetradakaenneennaconteennahectedokaenneennaconteennahectehendaenneennaco

nteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconteennahectezett

aenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraenneennaconteenn

ahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto-





What about the largest tier 1 separator, which would itself form the largest -illion (e.g. milli, the 1000th tier 1 separator, forms millillion, the 1000th -illion)? To form such a separator we take all possible tier 2 separators from the largest (shown above) down to the smallest, preceding each with novemnonagintinongenti. Thus we obtain:




novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaocteenneaconteennahecto


novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilahepteenneaconteennahecto


novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilahexeenneaconteennahecto


novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilapenteenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimillnovemnonagintinongentillion




But that isn't the largest number we can name. To find the largest number Bowers' system lets us name, we list all possible -illions we name in decreasing order, and precede each with "nine hundred ninety nine". Therefore the largest number we can name is:





nine hundred ninety nine novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimillnovemnonagintinongentillion


nine hundred ninety nine novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennaconte

ennahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraennee

nnaconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimilloctononagintinongentillion


nine hundred ninety nine novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennacontee

nnahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraenneen

naconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimillseptennonagintinongentillion


nine hundred ninety nine novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennacontee

nnahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraenneen

naconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimillsexnonagintinongentillion


nine hundred ninety nine
novemnonagintinongentienneennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocno

necxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonecxenaenn

eennaconteennahectenonecxenultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxeno

vnonecxenirmnonecxenunnonecxenastnonecxenijnonecxenejnonecxenalnonekyotaenneennaconteennahectenonecxe

nultnonecxenersnonecxenupnonecxenaxnonecxenocnonecxenetnonecxenolnonecxenovnonecxenirmnonecxenunnonec

xenastnonecxenijnonecxenejnonecxenalnoneczeta


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


enneennaconteennahectedakaenneennaconteennahectexennaenneennaconteennahecteyottaenneennacontee

nnahectezettaenneennaconteennahecteexaenneennaconteennahectepetaenneennaconteennahecteteraenneen

naconteennahectegigaenneennaconteennahectemegaenneennaconteennahectekilaenneenneaconteennahecto


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


novemnonagintinongentiicosonovemnonagintinongentienneconovemnonagintinongentiocteconovemnonagin

tinongentihepteconovemnonagintinongentihexeconovemnonagintinongentipenteconovemnonagintinongenti

tetreconovemnonagintinongentitreconovemnonagintinongentidueconovemnonagintinongentimeconovemno

nagintinongentiveconovemnonagintinongentixononovemnonagintinongentiyoctonovemnonagintinongentize

ptonovemnonagintinongentifemtonovemnonagintinongentipiconovemnonagintinongentinanonovemnonagint

inongentimicronovemnonagintinongentimillquinnonagintinongentillion


... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...








... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...


nine hundred ninety nine vigintillion nine hundred ninety nine novemdecillion nine hundred ninety nine octodecillion nine hundred ninety nine septendecillion nine hundred ninety nine sexdecillion nine hundred ninety nine quindecillion nine hundred ninety nine quattuordecillion nine hundred ninety nine tredecillion nine hundred ninety nine duodecillion nine hundred ninety nine undecillion nine hundred ninety nine decillion nine hundred ninety nine nonillion nine hundred ninety nine octillion nine hundred ninety nine septillion nine hundred ninety nine sextillion nine hundred ninety nine quintillion nine hundred ninety nine quadrillion nine hundred ninety nine trillion nine hundred ninety nine billion nine hundred ninety nine million nine hundred ninety nine thousand nine hundred ninety nine


That above, represents the limit of Bowers' system. Now that we know exactly what the limit of this intricate system is, we are now ready to analyze the advantages and drawbacks of this system.


Evaluating Bowers' -illions


Bowers' -illion scheme, unlike schemes such as Conway and Guy's and the extended version of Rowlett's, is technically a closed-ended scheme, a scheme with an actual limit to what -illions you can name. Conway and Guy's, by contrast, is a much simpler scheme that can technically name any number. So what advantage does Bowers' system pose?


First off, imagine a language where the only way to name numbers is by reading all the digits. Since every integer is made of a finite set of digits, it would indeed be able to name all numbers, right? Yes, it technically could name any numbers, but it QUICKLY becomes very unwieldy. For example, take the number:


1,000,000,000,000


That's a number small enough that it has everyday use in our modern world. In our read-all-the-digits language, however, it would read as:


one zero zero zero zero zero zero zero zero zero zero zero zero


You'd need to listen carefully to know how many zeros were said, and even then the name would be much more cumbersome than simply doing what the English language does by calling it


one trillion


which is obviously a much shorter name.


Now let's take a different number:


2,636,137,034,582


In the read-all-the-digits language, this number would be called:


two six three six one three seven zero three four five eight two


Needless to say this is a very unwieldy name. Strangely English has a longer name for this number:


two trillion six hundred thirty six billion one hundred thirty seven million thirty four thousand five hundred eighty two


Although it's longer, the English name for this number is far more convenient, since right away you hear "two trillion" which already lets you know approximately how big it really is.


Conway and Guy's system has the same problem. For example, take Conway and Guy's system's name for a googolplex:


ten trilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­duotriginta­trecentillion


and compare it with the name in Bowers' system:


ten tretriotriaconto-tretrigintitrecentiduetriaconto-tretrigintitrecentimetriaconto-tretrigintitrecentitriaconto-tretrigintitrecentienneicoso-tretrigintitrecentiocteicoso-tretrigintitrecentihepteicoso-tretrigintitrecentihexeicoso-tretrigintitrecentipenteicoso-tretrigintitrecentitetreicoso-tretrigintitrecentitrioicoso-tretrigintitrecentidueicoso-tretrigintitrecentiicoso-tretrigintitrecentienneco-tretrigintitrecentiocteco-tretrigintitrecentihepteco-tretrigintitrecentihexeco-tretrigintitrecentipenteco-tretrigintitrecentitetreco-tretrigintitrecentitreco-tretrigintitrecentidueco-tretrigintitrecentimeco-tretrigintitrecentiveco-tretrigintitrecentixono-tretrigintitrecentiyocto-tretrigintitrecentizepto-tretrigintitrecentiatto-tretrigintitrecentifemto-tretrigintitrecentipico-tretrigintitrecentinano-tretrigintitrecentimicro-tretrigintitrecentimilli-doetrigintitrecentillion.


Although the first name is a little shorter than the second, with the second it's much easier to tell what kind of large number we're talking about (assuming you're familiar with Bowers' system of course). Now that's where we start to hit an advantage of Bowers' system.


English itself allows you to give some pretty big numbers (say 1063) short names such as "vigintillion". Such short names are known as milestone names, in contrast to the long names for most numbers, such as ten vigintillion two hundred ninety-three novemdecillion five hundred seventy-three octodecillion two hundred forty-eight septendecillion seven hundred sixty sexdecillion one hundred twenty-eight quindecillion seven hundred thirty-nine quattuordecillion five hundred three tredecillion nine hundred forty-eight duodecillion seven hundred fifty-one undecillion eight hundred ninety decillion two hundred thirty-five nonillion seven hundred nine octillion one hundred forty-three septillion two hundred eighty-five sextillion six hundred twenty quintillion five hundred eighty-two quadrillion one hundred three trillion two hundred fifty-eight billion nine hundred seventy-two million three hundred fifty-four thousand two hundred thirty-five. And Bowers' system, like the English language itself, is largely all about such milestone names. By contrast, Conway and Guy's system, when you think about it, is more reminiscent of the "read-all-the-digits" language.


So in this sense, Bowers' system is more extensible than Conway and Guy's! But now think about systems like Bowers' a little more. Bowers' system is built upon milestone names, so can we extend the idea of milestone names further? Well, it depends what exactly you mean. It's easy to extend upon the list of milestone names, as some people have done (example), often creating rather absurd names along the way (the person in the example named some -illions after months and after cartoons). However, it's quite the challenge to get the name system to really work to create large numbers. This is what Sbiis Saibian successfully did on his article on Bowers' -illions (though with a bit of variance from how I do it in this article).


With the root tables and rules clearly listed, doing this might not seem like such a challenge. But think back to the various vowels we needed to use so that we wouldn't confuse names like killo-millillion vs. killamillillion. Is it really possible to do all this complicated stuff and milestone names as far as you want? Tediousness from the complications aside, there's one important thing to realize: there are only a finite number of combinations of letters that can create short milestone names, and of course you'd need to explicitly list the system for each tier of milestone names. Imagine how complicated a list of even the milestone names for a 100-tier system would be, let alone the system itself.


So if we really want to strive for a system as extensible as we can really get, we'd need to take quite a different approach from Bowers. Instead of making a whole set of milestone names, we could do something easier! What would that be?! The next article is where we will find that out.


Conclusion


Bowers' -illions have a spirit that matches with the entire idea of Bowers' work: using creative, varied, whimsical names for the polytopes or large numbers he studies. His -illions, much like Conway and Guy's, have therefore gotten a sort of cult significance, where people would list the names for numbers on webpages, or extend upon them—maybe even make their own Bowers-esque "system" of just listing up milestone names. However, a crucial snag of such systems, if we really want to follow the idea of "how high can we extend the idea of -illions?", is that such systems tend to be closed-ended and/or insanely complicated if we extend them high enough. Even this 4-tiered system takes up a pretty long article to fully go over, and Sbiis Saibian's article on them is even longer!


There is a better approach proposed by famed mathematician and computer scientist Donald Knuth. He first invents an -yllion system entirely different from the -illions we use, and then gives a clever and surprisingly elegant extension to that idea which we can apply to our regular -illions. That's exactly what we'll look at next.


1.9. Knuth's -yllions and Further Extensible Systems

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