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

(back to 1.06)

Introduction

We've looked at the -illions in earlier in this site (A, B), specifically 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.

I've already discussed some of the non-canonical -illions in the article on the usage of the -illions, namely the -illions used to fill the gap between vigintillion and centillion. But there is 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):

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:

(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 between 4000 and 4999.

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):

(*) When preceding a component marked ^S or ^X, “tre” changes to “tres” and “se” to “ses” or “sex”; similarly, when preceding a component marked ^M or ^N, “septe” and “nove” change to “septem” and “novem” or “septen” and “noven”.

This leads to the following names below a centillion:

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:

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 = 10^{10,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000,­000} =

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 had a website that has been archived here. One thing he had 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 quite a lot of recognition, 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):

enas, dyo, tria, tessera, pente, exi, epta, okto, ennea, deka

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

1 — hen, mono (mono actually comes from Greek "monos" meaning alone)

2 — di, dy, duo

3 — tri

4 — tetra

5 — penta

6 — hexa

7 — hepta

8 — octa, ogda

9 — ennea (sometimes replaced with Latin nona-)

10 — deka, deca

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 after the name "gillion", Rowlett uses the Greek prefixes listed above, leading to the following names (table once again taken from Sbiis Saibian's site):

As you can tell, the names aren't a perfect match, but they work quite well and are probably easier to memorize than the real Greek numbers. if "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 he extended them.

Extending Rowlett's -illions

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

saranta, peninta, exinta, evdominta, ogdonta, eneninta

But those don't really match with the pattern Rowlett used. To match with the pattern better, Saibian used these roots instead:

tetraconta, pentaconta, hexaconta, heptaconta, oktaconta, ennaconta

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
hexahectillion = 10^1800
heptahectillion = 10^2100
oktahectillion = 10^2400
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", Sbiis Saibian instead suggests the name "kilillion", using "kilo" from the SI prefix for 1000. The name "chilillion" would likely be confusing to pronounce; while the letters "ch" are pronounced as "k" in words from Greek, that might not be obvious to everyone with the name "chilillion", and I'm sure some would pronounce it as /tʃɪlɪljən/ (chill-ill-yun) instead of /kɪlɪljən/ (kill-ill-yun). And might make people think the number is named after Mexican cuisine, not the Greek word for a thousand. That wouldn't be any good, would it?

But let's not get distracted thinking about how delicious a good, warm bowl of chili is. Let's continue with some examples of names.

kilillion = 10^3000
kilohenillion = 10^3003
...
kilodekillion = 10^3030
...
kilohectillion = 10^3300
...
duokilillion = 10^6000
triakilillion = 10^9000
tetrakilillion = 10^12,000
...
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. We already learned the word from the first article of this section, but in case you haven't read that one, it's "ekatommyrio". The name combines "ekato" meaning 100 with "myriades", the ancient Greek word for 10,000; therefore, "ekatommyrio" can be thought of meaning "hundred myriad".

As such, 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")

octillion = oktakisekatommyrio

nonillion = enneakisekatommyrio

decillion = dekakisekatommyrio

undecillion = hendekakisekatommyrio

duodecillion = dodekakisekatommyrio

tredecillion = trisdekakisekatommyrio

quattuordecillion = tetradekakisekatommyrio

Here we encounter a major 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 10^{4*3*10^33+3}th -illion or the 4*10^{3*10^33+3}th -illion? We can again devise a fix for this. Just name the 10^{4*3*10^33+3}th -illion "tetra-dekakisekatommyriakisekatommyrillion" and the 4*10^{3*10^33+3}th -illion "tetra--dekakisekatommyriakisekatommyrillion". In other words, you use multiple hyphens to separate tiers, and we'll revisit this topic shortly.

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! That system sounds pretty inescapable, doesn't it?

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...

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