1.7. Extensions to the -illions I: Henkle, Conway, and Rowlett




Introduction

We've looked at the -illions in earlier in this site (AB), the -illions that are actually part of the English language. They're more than enough to get by in our world, but some people had the question: how far can you logically extend the -illion idea? This article serves to look at how people attempted to answer that, and examine the various systems people have used, ranging from simple extensions like Conway and Guy's to complete revamps like Russ Rowlett's.

Actually, I've already discussed some of the non-canonical -illions in the article on the usage of the -illions, the -illions used to fill the gap between vigintillion and centillion. But there are actually a whole world of different -illion systems out there, some simple and traditional but some exceptionally unique.

Here I'm sorting the -illions by a mix of chronological order and extensibility, so let's get to it.

1.7A. Professor Henkle's one million -illions

First I'll discuss the extension to the -illions invented by someone who is called Professor Henkle. According to Robert Munafo, Henkle is most likely William Downs Henkle, from Ohio and born 1828. Henkle's -illions first appeared in an 1860 edition of the Ohio Educational Monthly, and they were later published by someone else in a 1903 article.

Henkle's Names

Henkle's -illions were likely the first attempt to extend upon the traditional -illion naming system. Up to duodecillion he used the same names as in the official -illions, but after that he took a diferent approach. Here are the new number names (table taken from Sbiis Saibian's page on Henkle's -illions):

 n 10^(3n+3) 
Henkle 
 13 10^42 
tertio-decillion 
 14 10^45 
quarto-decillion 
 15 10^48  quinto-decillion
 16 10^51 
 sexto-decillion
 17 10^54 
 septimo-decillion
 18  10^57  octo-decillion
 19  10^60  nono-decillion
 20  10^63  vigillion
 21 10^66 
primo-vigillion 
 22  10^69  secundo-vigillion
 23  10^72  tertio-vigillion
 24  10^75  quarto-vigillion
 25  10^78  quinto-vigillion
 26  10^81  sexto-vigillion
 27  10^84  septimo-vigillion
 28  10^87  octo-vigillion
 29  10^90  nono-vigillion
 30  10^93  trigillion
 40 10^123 
quadragillion 
 50  10^153  quinquagillion
 60  10^183  sexagillion
 70  10^213  septuagillion
 80  10^243  octogillion
 90  10^273  nonagillion
 100  10^303  centillion
 101 10^306 
 primo-centillion
 110 10^333 
decimo-centillion 
 111 10^336 
undecimo-centillion 
 112  10^339  duodecimo-centillion
 113  10^342  tertio-decimo-centillion
 114  10^345  quarto-decimo-centillion
 120 10^363 
vigesimo-centillion 
 121  10^366 primo-vigesimo-centillion
 130  10^393  trigesimo-centillion
 140  10^423  quadragesimo-centillion
 150 10^453 
quinquagesimo-centillion 
 160  10^483  sexagesimo-centillion
 170  10^513  septuagesimo-centillion
 180  10^543  octogesimo-centillion
 190  10^573  nonagesimo-centillion
 200  10^603  ducentillion
 300  10^903  trecentillion
 400  10^1203  quadringentillion
 500  10^1503  quingentillion
 600  10^1803  sexcentillion
 700  10^2103  septingentillion
 800  10^2403  octingentillion
 900  10^2703  nongentillion
 1000  10^3003  millillion
 1100 10^3303 
 centesimo-millillion
 1200  10^3603  ducentesimo-millillion
 1300  10^3903  trecentesimo-millillion
 1400  10^4203  quadringentesimo-millillion
 1500  10^4503  quingentesimo-millillion
 1600  10^4803  sexcentesimo-millillion
 1700  10^5103 septingentesimo-millillion 
 1800  10^5403  octingentesimo-millillion
 1900  10^5703  nongentesimo-millillion
 2000  10^6003  bi-millillion
 3000  10^9003  tri-millillion
 4000  10^12,003  quadri-millillion
 5000  10^15,003  quinqui-millillion
 6000  10^18,003  sexi-millillion
 7000  10^21,003  septi-millillion
 8000  10^24,003  octi-millillion
 9000  10^27,003  novi-millillion
 10,000  10^30,003  deci-millillion
 11,000  10^33,003  undeci-millillion
 12,000  10^36,003  duodeci-millillion
 13,000  10^39,003  tredeci-millillion
 14,000  10^42,003  quatourdeci-millillion
 15,000  10^45,003  quindeci-millillion
 16,000  10^48,003  sexdeci-millillion
 17,000  10^51,003  septi-deci-millillion
 18,000  10^54,003  octi-deci-millillion
 19,000  10^57,003  novi-deci-millillion
 20,000  10^60,003  vici-millillion
 21,000  10^63,003  semeli-vici-millillion
 22,000  10^66,003  bi-vici-millillion
 23,000  10^69,003  tri-vici-millillion
 24,000  10^72,003  quadri-vici-millillion
 25,000  10^75,003  quinqui-vici-millillion
 26,000  10^78,003  sexi-vici-millillion
 27,000  10^81,003  septi-vici-millillion
 28,000 
 10^84,003  octi-vici-millillion
 29,000  10^87,003  novi-vici-millillion
 30,000  10^90,003 trici-millillion 
40,000 
 10^120,003  quadragi-millillion
50,000 
 10^150,003  quinquagi-millillion
 60,000  10^180,003  sexagi-millillion
 70,000  10^210,003  septuagi-millillion
 80,000  10^240,003  octogi-millillion
 90,000  10^270,003  nonagi-millillion
 100,000  10^300,003 centi-millillion 
 101,000  10^303,003  semeli-centi-millillion
102,000 
 10^306,003  bi-centi-millillion
 200,000  10^600,003 ducenti-millillion 
 300,000  10^900,003  trecenti-millillion
 400,000  10^1,200,003  quadringenti-millillion
 500,000  10^1,500,003 quingenti-millillion 
 600,000  10^1,800,003  sexcenti-millillion
 700,000  10^2,100,003  septingenti-millillion
 800,000  10^2,400,003  octingenti-millillion
 900,000  10^2,700,003  nongenti-millillion
 1,000,000 10^3,000,003 
 milli-millillion

There are a few interesting things to point out. First off, he skips names once the pattern becomes apparent. This is common practice among googologists who make naming systems, since once we see the pattern we don't need to list out every single name.

Next, note that his system is very systematic if a little arbitrary: for example, 13, 14, 15, and 16 get unique names among the roots that multiply n, but not among the roots that add n. But an important systematic concept is that roots that add to n end in -o while roots that multiply n end in -i. Here's a quick table listing all the roots:

n Ones addition root Ones multiplier root Tens addition root Tens multiplier root Hundreds addition root Hundreds multiplier root
1 primo- semeli- decimo- deci- centesimo- centi-
2 secundo- bi- vigesimo- vici- ducentesimo- ducenti-
3 tetrio- tri- trigesimo- trici- trecentesimo- trecenti-
4 quarto- quadri- quadragesimo- quadragi- quadringentesimo- quadringenti-
5 quinto- quinqui- quinquagesimo- quinquagi- quinquagentesimo- quinquagenti-
6 sexto- sexi- sexagesimo- sexagi- sexcentesimo- sexcenti-
7 septimo- septi- septuagesimo- septuagi- septingentesimo- septingenti-
8 octavo- octi- octogesimo- octogi- octingentesimo- octingenti-
9 nono- novi- nonagesimo- nonagi- nongentesimo- nongenti-
10 decimo- deci- centesimo- centi-
milli-
11 undecimo- undeci-



12 duodecimo- duodeci-



13
tredeci-



14
quatuordeci-



15
quindeci-
 

16
sexdeci-




(if there is nothing listed in a spot on the table, that means there is no root for that, and one should combine roots)

Hopefully the mechanics of Henkle's -illions are pretty clear now. I will now go on to make some remarks on his system.

Commentary on Henkle's -illions

Note that first of all, Henkle's system is more complicated than the English number name system, because we have no different multiplier names for numbers.

To understand what that means, think of this. To form the name of 8000 you just need combine eight, the word for 8, and thousand, the word for 1000, to get eight thousand for 8000. But in Henkle's -illions, to form the name for the 8000th -illion you need to combine the multiplier root for 8 with the name for the 1000th -illion to get octi-million.

Why does Henkle need to do that? Because of his reversed order in naming. To get what that means look at the 4096th -illion in Professor Henkle's system vs the name for 4096 in English:

sexto-nonagesimo-quadri-millillion vs. four thousand ninety-six

This shows that in Henkle's -illion system has the numbers practically reversed: sexto-nonagesimo-quadri-milli could be imagined as six and ninety and four-thousand. The name "six and ninety and four-thousand" is quite different from "four thousand ninety-six". But why does that matter?

In the name "six and ninety and four-thousand", you need to listen to the end to know that the number is approximately 4000. But in "four thousand ninety-six", right when you hear the word "thousand" you know right away that the number is approximately 4000.

Henkle's system, all in all, is a simple convenient system, but not quite as modern as many of the extended -illions we see today which follow better with the order used in English numerals. However, the system still is important because it serves as a pioneering for more modern and extensible -illion systems, as discussed in the next section.

1.7B. Conway and Guy's -illions

The next -illion system to discuss is a system both more well-known and more extensible than Henkle's. It is a system created by famed mathematicians John Horton Conway and Richard K. Guy, in their Book of Numbers, a book about recreational mathematics. The book is the same book that introduced the world to Conway's chained arrows as well as a mysterious alternate version of Graham's number, both of which I'll discuss in section 2.

Conway and Guy's system is very popular, and there are many pages on the Internet (here, here, and here for example) that simply list names of -illions fom the system (though the first of those pages also has miscellaneous numbers, some of them erroneous). Conway and Guy's -illions have been described as given a sort of "cult significance".

 Conway and Guy's -illion names

Here is how Conway and Guy propose to name numbers from a decillion to 10^3003 (taken from Wikipedia's article on names of large numbers):

Units Tens Hundreds
1 Un N Deci NX Centi
2 Duo MS Viginti N Ducenti
3 Tre (*) NS Triginta NS Trecenti
4 Quattuor NS Quadraginta NS Quadringenti
5 Quinqua NS Quinquaginta NS Quingenti
6 Se (*) N Sexaginta N Sescenti
7 Septe (*) N Septuaginta N Septingenti
8 Octo MX Octoginta MX Octingenti
9 Nove (*) Nonaginta Nongenti
(*) ^ When preceding a component marked S or X, “tre” increases to “tres” and “se” to “ses” or “sex”; similarly, when preceding a component marked M or N, “septe” and “nove” increase to “septem” and “novem” or “septen” and “noven”.

This leads to the following names below a centillion:


(click picture above to zoom in)

The table has some coloring to make some points:

Red names are -illions exactly as they are called in the English language.

Blue names are -illions that have names in the English language, but are modified in this system: quindecillion is renamed to quinquadecillion, sexdecillion to sedecillion, and novemdecillion to novendecillion.

Black names are -illions that don't have names in the English language and are named in this system.

The s's, n's, m's, and x's in the system are colored and bolded to indicate where they appear: green s's, orange n's, pink m's, teal x's.

About those s's, n's, m's, and x's, I find it a little weird that they are used. It seems to be so much unnecessary work to know where to put a s, n, m, or x. If you say, for example, sexvigintillion instead of sesvigintillion, people acquainted with the -illions will still know what you're talking about. Also, most people seem to prefer the dictionary names for 10^48, 10^51, and 10^60 to Conway and Guy's names. Nonetheless, they provide a useful and pretty simple system.

Modifications of Conway and Guy's -illions

Many people simplified the roots to make them more like the standard dictionary -illions:

Units Tens Hundreds
1 Un Deci Centi
2 Duo Viginti Ducenti
3 Tre Triginta Trecenti
4 Quattuor Quadraginta Quadringenti
5 Quin Quinquaginta Quingenti
6 Sex Sexaginta Sescenti
7 Septen Septuaginta Septingenti
8 Octo Octoginta Octingenti
9 Novem Nonaginta Nongenti

Now we have an even simpler system! We now don't need to memorize when to put a s, x, m, or n where, AND it's faithful to the dictionary -illions!

Of course, usage of s, x, m, and n was put for a reason: to make the -illions easier to pronounce. For example, in the simplified system 10^60 is called novemdecillion, but in Conway and Guy's it's called novendecillion. Novendecillion is easier to pronounce, because a "nd" sound cluster (e.g. bond) is a bit easier to pronounce in a word than a "md" cluster (e.g. bombed). However this isn't a very big deal, which is why the simplified version is accepted so much.

Continuing the System

In any case, Conway and Guy give a suggestion to continue the system to far greater heights:

The thousandth -illion (10^3003) can be called millinillion. Then you can continue with millimillion for the 1001st -illion, and in general x-illi-y-illion is equal to the 1000x+y-th -illion. For example we can have`

10^3009 = millibillion
10^3012 = millitrillion
10^3033 = millidecillion
10^3063 = millivigintillion
10^3303 = millicentillion
10^6003 = billinillion
10^6006 = billimillion
10^6009 = billibillion
10^6030 = billidecillion
10^9003 = trillinillion
10^12,003 = quadrillinillion
10^30,003 = decillinillion
10^300,003 = centillinillion
10^2,999,997 = novenonagintanongentillioctononagintanongentillion
10^3,000,000 = novenonagintanongentillinovenonagintanongentillion

Then what? We can easily continue with: x-illi-y-illi-zillion = the 1,000,000x+1,000y+z-th -illion. So we can have:

10^3,000,003 = millinillinillion — that's equal to Henkle's "milli-millillion"
10^3,000,006 = millinillimillion
10^3,000,303 = millinillicentillion
10^3,003,003 = millimillinillion
10^3,003,006 = millimillimillion
10^3,006,003 = millibillinillion
10^3,333,333 = milliundecicentillidecicentillion
10^6,000,003 = billinillinillion
10^30,000,003 = decillinillinillion
10^300,000,003 = centillinillinillion

and we can continue in a similar fashion:

10^3,000,000,003 = millinillinillinillion
10^6,000,000,003 = billinillinillinillion
10^3,000,000,000,003 = millinillinillinillinillion
10^3,000,000,000,000,003 = millinillinillinillinillinillion

and so on ...

Limit of the System?

So we can go on forever, and ever, just making longer and longer names, right? Well, only in a theoretical sense. For example, let's get back to how a googolplex is named in that system:

googolplex = 1010,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000 =

ten trilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­trestriginta­trecentilli­duotriginta­trecentillion

That's a rather long name, with 769 letters. It's a name long enough that it takes more than a moment to say (it took 42.56 seconds for me to say the name), and a bit of effort to write.

What is significant about the name of a googolplex in that system? It shows that eventually, the names will necessarily get so long that they can no longer be used at all. For example, a googolduplex will have a name like googolplex's name, but with trestrigintatrecentilli repeated not 32 times, but 3,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,
333,333,333,333,333,332 times. That number is just under a third of a googol.

The name for a googolduplex in this system would have 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,001 letters, not counting the space. That number is itself equal to eighty duotrigintilllion and one in Conway and Guy's -illions, or just eight times a googol, plus one. Since the number of letters is about 8*1020 (800 quintillion) times the number of atoms in the observable universe, we can safely say that writing or saying the name of that number is just not going to happen.
So you could go further and further, but the system does have a limit on what numbers can be actually named, and we need more versatile -illion systems.

1.7C. Rowlett's Greek -illions and Sbiis Saibian's extension

Up next in line to discuss is an -illion proposal by Russ Rowlett. Russ Rowlett is a retired mathematics educator at the University of North Carolina at Chapel Hill, and he has a website that you can find here. One thing he has on his site is a units of measurement page. That page has many subpages, including an alphabetized dictionary of measurement terms and many longer pages relating to measurement.

Of interest here is a page called "Names of Large Numbers", which discusses the names of the -illions, the issue of short vs long scale, and a system to resolve the ambiguity by revamping the -illions in their entirety. He explained it, gave a list of all names below a googol, and told how you can go further. Rowlett's system has gained some recognition, even finding its way to Wikipedia. But just how does it work?

Rowlett's System and Names

Rowlett uses a variant of the Greek number root system. For some background on those roots, here are the names of numbers 1 through 10 in modern Greek (using Google Translate, transliterated from the Greek alphabet to the Latin alphabet):

enasdyotriatesserapenteexieptaoktoenneadeka

From those come the Greek number roots (which vary somewhat in usage):

1 — henmono (mono actually comes from Greek "monos" meaning alone)
2 — didyduo
3 — tri
4 — tetra
5 — penta
6 — hexa
7 — hepta
8 — octaogda
9 — ennea (sometimes replaced with Latin nona-)
10 — dekadeca

The first few names have something a little special: thousand and million are kept since they mean the same number in the short and long scales, and 10^9 is not called trillion with the Greek roots because trillion is the name of an existing number (10^12 in the short scale, 10^18 in the long scale; rather, it's called gillion from the SI prefix "giga" meaning one billion.

But from here on out Rowlett uses Greek prefixes, giving the following names for numbers (table once again taken from Sbiis Saibian's site):

 n  th(n) Short  
Long 
Greek 
 1 10^3 
 thousand thousand
thousand 
 2 10^6  million million 
million 
 3 10^9 
 billion milliard 
 gillion
 4  10^12  trillion  billion  tetrillion
 5  10^15  quadrillion  billiard pentillion 
 6  10^18  quintillion  trillion  hexillion
 7  10^21  sextillion  trilliard  heptillion
 8  10^24  septillion  quadrillion  oktillion
 9  10^27  octillion  quadrilliard  ennillion
 10  10^30  nonillion  quintillion  dekillion
 11  10^33  decillion  quintilliard  hendekillion
 12  10^36  undecillion  sextillion  dodekillion
 13  10^39  duodecillion  sextilliard  trisdekillion
 14  10^42  tredecillion  septillion  tetradekillion
 15  10^45  quattuordecillion  septilliard  pentadekillion
 16  10^48  quindecillion  octillion  hexadekillion
 17  10^51  sexdecillion octilliard 
heptadekillion 
 18  10^54  septendecillion  nonillion  oktadekillion
 19  10^57  octodecillion  nonilliard  enneadekillion
 20  10^60  novemdecillion  decillion  icosillion
 21  10^63  vigintillion  decilliard  icosihenillion
 22  10^66  unvigintillion  undecillion  icosidillion
 23  10^69  duovigintillion  undecilliard  icositrillion
 24  10^72  trevigintillion  duodecillion  icositetrillion
 25  10^75  quattuorvigintillion  duodecilliard  icosipentillion
 26  10^78  quinvigintillion  tredecillion  icosihexillion
 27  10^81  sexvigintillion  tredecilliard  icosiheptillion
 28  10^84  septenvigintillion  quattuordecillion  icosioktillion
 29  10^87  octovigintillion  quattuordecilliard  icosiennillion
 30  10^90  novemvigintillion  quindecillion  triacontillion
 31  10^93  trigintillion  quindecilliard  triacontahenillion
 32  10^96  untrigintillion  sexdecillion  triacontadillion
 33  10^99  duotrigintillion  sexdecilliard  triacontatrillion


Let's take a look at what Rowlett decided to do with the Greek roots. Things go as you would expect up until the name "oktillion" instead of "octillion" for 10^24. It's pretty obvious why he renamed it: so it would not be confused with octillion. But there's another problem: they would both be pronounced /ok-til-yun/, leading to confusion. Luckily that's an easy problem to fix: pronounce octillion /ok-til-yun/ and oktillion /oak-til-yun/.

A similar thing is done with "dekillion" instead of "decillion". This time pronunciation is an even easier problem to fix: pronounce decillion /des-ill-yun/ and pronounce dekillion /deck-ill-yun/.

After that continuation would go similar to expected, with a few quirks. To summarize them let's compare numbers in Greek vs Rowlett's -illions.

n n in Greek nth -illion in Rowlett's system
11 enteka hendekillion
12 dodeka dodekillion
13 dekatria trisdekillion
14 dekatessera tetradekillion
15 dekapente pentadekillion
16 dekaexi hexadekillion
17 dekaepta heptadekillion
18 dekaokto oktadekillion
19 dekaennea enneadekillion
20 eikosi * icosillion
21 eikosi ena icosihenillion
22 eikosi dyo icosidillion
23 eikosi tria icositrillion
24 eikosi tesseris icositetrillion
25 eikosi pente icosipentillion
26 eikosi exi icosihexillion
27 eikosi epta icosiheptillion
28 eikosi okto icosioktillion 
29 eikosi ennea icosiennillion
30 trianta triacontillion
31 trianta ena triacontahenillion
32 trianta dyo triacontadillion
33 trianta tria triacontatrillion
* pronounced /ee-koh-see/

As you can tell, the names aren't a perfect match, but they work quite well and are in fact easier to memorize than the real Greek numbers. Triacontatrillion is as far as Rowlett goes with the names, but it's easy to go further: Sbiis Saibian does just that on his article on Rowlett's -illions, and from here on out I'll cover how they're extended.

Extending Rowlett's -illions

To extend Rowlett's -illions, you start with the Greek names for 40, 50, 60, 70, 80, 90, which are:

sarantapenintaexintaevdomintaogdontaeneninta

But those don't really match with the pattern Rowlett used. To match with the pattern we use these roots instead:

tetracontapentacontahexacontaheptacontaoktacontaennaconta

This can give us the names:

tetracontillion = 10^120
pentacontillion = 10^150
hexacontillion = 10^180
heptacontillion = 10^210
oktacontillion = 10^240
ennacontillion = 10^270

(a lot of these and later names intersect with Bowers' -illions, but we'll discuss those later)

After that root, we won't use "ekato", the Greek name for 100, but we'll use the name "hectillion" for 10^300. Thus it's easy to continue:

hectahenillion = 10^303 — this is equal to a centillion in the short scale
...
hectadekillion = 10^330
hectaicosillion = 10^360
...
duohectillion = 10^600
duohectahenillion = 10^603
...
triahectillion = 10^900
tetrahectillion = 10^1200
pentahectillion = 10^1500
...
ennahectillion = 10^2700
...
ennahectaennacontaennillion = 10^2997

Good so far. But now we need a name for 10^3000, the thousandth -illion in this system. Even though the Greek name for a thousand is chilias (/keel-ee-as/), Sbiis Saibian instead suggests the name "kilillion", using "kilo" from the SI prefix for 1000. Thus we can continue:

kilillion = 10^3000
kilohenillion = 10^3003
...
kilodekillion = 10^3030
...
kilohectillion = 10^3300
...
duokilillion = 10^6000
triakilillion = 10^9000
tetrakilillion = 10^1200
...
dekakilillion = 10^30,000
icosikilillion = 10^60,000
triacontikillillion = 10^90,000
...
hectakilillion = 10^300,000
...
the largest name we can have:
ennahectaennacontaennakiloennahectaennacontaennillion = 10^2,999,997

A natural question is: how do you go further? Simple: just use the Greek word for a million. According to Google Translate the word is "ekatommyrio"—this is pronounced /eh-kah-toh-MEE-ree-oh/. The name combines "ekato" meaning 100 with "myriades", the ancient Greek word for 10,000; therefore, "ekatommyrio" can be thought of meaning "hundred myriad".

In any case, the Greek word for a million gives us the name "ekatommyrillion" for 10^3,000,000. It's easy to continue with many possible names, like ekatommyriakilillion for 10^3,003,000, or more complex names like duodekaekatommyriatriakilotriacontillion for 10^36,009,090. The largest name we can form now is ennahectaennacontaennaekatommyriaennahectaennacontaennakiloennahectaennacontaennillion for 10^2,999,999,997.

You can guess how we can go further: use the Greek word for a billion. The word is "disekatommyrio", which can be thought of as meaning "second million". This can give us even longer names like:

ennahectaennacontaennadisekatommyriaennahectaennacontaennaekatommyriaennahectaennacontaennakiloennahecta
ennacontaennillion for 10^2,999,999,999,997

After that we can continue with the Greek names for the next few -illions:

trillion = trisekatommyrio (can be thought of as meaning "third million")
quadrillion = tetrakisekatommyrio ("fourth million")
quintillion = pentakisekatommyrio ("fifth million")
sextillion = exakisekatommyrio ("sixth million")
septillion = eptakisekatommyrio ("seventh million")

This is as far as Google Translate goes with Greek number names. However, we have enough number names now that we can continue the pattern:

octillion = oktakisekatommyrio
nonillion = enneakisekatommyrio
decillion = dekakisekatommyrio
undecillion = hendekakisekatommyrio
duodecillion = dodekakisekatommyrio
tredecillion = trisdekakisekatommyrio
quattuordecillion = tetradekakisekatommyrio

Here we encounter the first problem: is "tetradekakisekatommyrillion" the 4 decillionth -illion or the quattuordecillionth -illion? That's not hard to fix. Let tetradekakisekatommyrillion be the quattuordecillionth -illion, and name the 4 decillionth -illion tetra-dekakisekatommyrillion.

Can we continue forever? Sure we can have names like "icosakisekatommyrillion" or "kilakisekatommyrillion" or "disekatommyriakisekatommyrillion", but we encounter yet another problem: the name "tetra-dekakisekatommyriakisekatommyrillion"—does that refer to the 104*3*10^33+3th -illion or the 4*103*10^33+3th -illion? That isn't too hard to fix. Just name the 104*3*10^33+3th -illion "tetra-dekakisekatommyriakisekatommyrillion" and the 4*103*10^33+3th -illion "tetra--dekakisekatommyriakisekatommyrillion". In other words you use multiple hyphens to separate tiers - in a later section the concept of tiers will be examined more fully.

Drawbacks of Rowlett's System

You can imagine the progression:

ekatommyrillion = 10^3,000,000
ekatommyriakisekatommyrillion = 10^(3*10^3,000,003)
ekatommyriakisekatommyriakisekatommyrillion = 10^(3*10^(3*10^3,000,003+3))
ekatommyriakisekatommyriakisekatommyriakisekatommyrillion = 10^(3*10^(3*10^(3*10^3,000,003+3)+3))

... and so on ...

and apply "ekatommyriakis" as many times as you please. Now that easily takes you to dizzying heights! It's ok if you're confused by all this; in a little bit, we'll examine these kinds of -illions in a bit more depth.

But there are several drawbacks of this system:

1. Even though the system is meant to solve the issue of short and long scales, because of the Modern Greek which uses the short scale used in the extension, the short scale has found its way back into the system!

2. Pronunciation: how do you distinguish "tetra-dekakisekatommyriakisekatommyrillion" and "tetra--dekakisekatommyriakisekatommyrillion" pronunciation-wise?

Why do those matter? The -illions Rowlett gave are already more than enough for real-world use; extensions to the -illions are generally designed to see how far we can logically take the idea of -illions. After some point the names will necessarily get ridiculously long, but that's just a quirk of extending the -illions to extreme heights. However, there are some unwritten guidelines to extensions to the -illions: optimally, each number should have a name distinguishable by pronunciation AND spelling, and the system should still have some logical advantages to usage. As we will see each -illion system has its own perks and drawbacks. The next -illion system I will discuss, an unusual system proposed by Jonathan Bowers, solves the pronunciation issue but is even more dizzying as we'll discuss in the next article...

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