1.08. Extensions to the -illions II: Jonathan Bowers' -illions
Perhaps the best known -illion system amongst the googology community is a system proposed by Jonathan Bowers, an important figure in googology. He's a name we'll run into a lot throughout this site, and our first encounter with his work is his take on extending the -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.
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.
Orange 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" doesn't create conflicts. 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
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 idea using SI prefixes:
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, giving us names like "megillion" and going up to "yottillion". 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 extrapolate from all the names Bowers himself gives, even oddities "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:
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:
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:
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:
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" ... and so on.
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 also 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:
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.
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:
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, and so on. With that the 999,999th tier 2 root is simply enneennaconteennahectakillaenneennaconteennahecto-. Hopefully, this seems straightforward so far.
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:
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 may seem arbitrary until you notice that "c" is changed to "k" before the letters "e" and "y"—presumably, this is to prevent the letter "c" from being pronounced like an "s". I suppose then that in this system "c" would also be changed to "k" before the letter "i".
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.
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:
This is quite an extensive system now, but we aren't done yet! 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, and then move on to the names drawn from astronomy. We have 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 almost 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:
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:
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:
What about the largest tier 1 separator, which would itself form the largest -illion (likw how 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:
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