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### 2.4.6 - Conway & Guy's Latin based -illions

John H. Conway 1987

2.4.6
Conway & Guys Latin Based -illion Series

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Previous article 2-4-4 : Prof. Henkles One million illions and Beyond

The Quasi-official illions of Conway and Guy

Perhaps motivated by the gap between vigintillion and centillion, two mathematicians , John Horton Conway & Richard K. Guy , sought to bring the illion sequence to it's natural conclusion. Their system, which is now widely accepted and used by large number enthusiasts was first published in "The book of Numbers", a book the two mathematicians wrote collaboratively and released back in 1996 ( John Horton Conway is also the inventor of the well known "Chain arrow notation" published in the same book beginning on page 61. We will be talking more about him later in Section III). Their system provides a logical resolution to the problem posed in the last article.

Their idea was simple and sensible. Why not extend the latin prefixes up to the limit of the latin language. Romans made common use of all counting numbers up to a thousand, and had a regular system of names. One could then theoretically create a scheme for naming the first 1000 illions without much academic ambiguity.

On page 15 of "The book of numbers", the authors provide this table along with this footnote ...

(note: the text has been altered slightly for greater clarity )

The first nine -illions are million , billion , trillion , quadrillion , quintillion , sextillion , septillion , octillion , and nonillion. To form names for the 10th to 999th -illion  do so by listing the unit component first, then the tens, and finally hundreds component. Then replace the last vowel with illion to form the denomination.

Units                                          Tens                                           Hundreds

1                    un                                                 (n)deci                                          (nx)centi

2                   duo                                              (ms)viginti                                   (n)ducenti

3                   tre(*)                                           (ns)triginta                                 (ns)trecenti

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

* note : when tre occurs immediately before a component which includes s or x within parathesis, use tres inplace of tre . when se occurs immediately before a component which includes s within parathesis, replace with ses , and when x is within the parathesis, replace with sex . when septe occurs immediately before a component which includes m within parathesis , replace with septem , and when n is within the parathesis, replace with septen . when nove occurs immediately before a component which includes m within parathesis , replace with novem , and when n is within the parathesis , replace with noven.

In addition it is widely accepted that the 1000th illion should be "millillion", on the grounds that "mille" is latin for thousand. This would mean a millillion should stand for 10^3003.

Conway and Guy have taken some degree of liberty here. Perhaps in the interest of keeping the system more strictly latin they had made small tweaks to the system. For example, by a careful application of the rules above, one would conclude that according to the Conway nomenclature, the 19th illion should be "novendecillion", not "novemdecillion" which is more common. Also the 16th illion would be "sedecillion" instead of the usual "sexdecillion".

The system is none the less invaluable to the number hobbyist who can now give name for any illion from the 1st to 1000th. Also one can be comfortable that the notation is relatively logical being firmly built on top of latin numbers.

Here are some example constructions to help clarify the above rules. The key to remember is that the s and x apply to how to modify tre and se , and the m and n apply to how to modify septe and nove. Everything else is simply combining the components in order of units, tens, and hundreds along with the illion suffix.

Order                                             Name                                                                          Value

876th                                           seseptuagintaoctingentillion                                              10^2631

120th                                           viginticentillion                                                               10^363

59th                                              novenquinquagintillion                                                  10^180

587th                                          septemoctogintaquingentillion                                         10^1764

The value can be obtained by this nifty function. Let H(n) = 10^3(n+1). n is the order of the -illion. So in the first example, to find the value of the 876th -illion you have to find H(876). this will be 10^3(876+1) = 10^2631. To see whether this makes sense consider a "million". A million is the "first -illion". So a million is the same as H(1). Now observe that H(1) = 10^3(1+1) = 10^6 = 1,000,000. Now consider a billion, the second -illion. It would be H(2) which is 10^3(2+1) = 10^9 = 1,000,000,000. Each time we go to the next order, the value gets multiplied by 1000.

Now note that conway said nothing about what to do when the tens component is followed by the hundreds component octingenti , the only hundreds component beginning with a vowel. This leads to a double vowel as in seseptuagintaoctingentillion. Thankfully this doesn't seem to lead to a pronouncation problem, as one simply reads out the components almost as if they were separate words. Attempting to run the vowels into one word however leads to difficulty. prehaps even worse are cases where the is no tens component, such as the 802nd illion "duooctingentillion" and the 808th illion "octooctingentillion", and my favorite the 88th illion "octooctogintillion". This double oh seems out of place. Again we naturally pronounce it as a single oh, but if we attempt to pronounce it as we do in "took" most peoples tongues end up tripping on themselves.

Perhaps Conway and Guy were aware of these relatively trivial ambiguities and decided to gloss over them because such languistic nuances might bore their audience. Also the problem could easily be solved with the addition of a few extra rules. For example, when a "oo", "ao" , or "io" occurs as a result of combining components replace them with "o" . So for example the 820th illion would be "vigintioctingentillion"  but we can correct it as "vigintoctingentillion"

A little bit of study of the table will reveal the only vowel combos that can result are "oo" , "ao" , and "io" . "eo" cannot occur because the four cases of components ending in e ( tre , se , septe , and nove ) all have special rules to cover them when connecting to either octoginta and octingenti , namely they become tres , sex ,septem, and novem respectively.

Despite these minute criticisms, the Conway nomenclature is quite an impressive tool , and it has brought a much larger class of illions into the mainstream. It is possible that someday this nomenclature in it's entirety may become mainstream. But it is also likely that the 21 canonical illions will remain alone as officially recognized. The reason is that a 1000th illions is a luxury of whim. There is no use for such an extended system. Listing each one individually would be a waste of space in any dictionary, and a full explanation of the rules would seem unneccesarily technical. So perhaps all of the "great illions" are doomed to obscurity.

What is important here however, is not whether webster or oxford chooses to recognize these numbers. What's important is that the system exists for anyone to use, it works and refers to actual numbers, and is almost universally used (with some variation) by number hobbyists (the people who actually find these issues interesting). It is "official enough".

Conway and Guy paved the way for the general public to use such numbers. However their system has not been adopted down to the rules. Instead people have recognized the logicalness of using unit tens and hundreds components in tandum. What has been generally lost is the specifics. On the internet you can find several places where the illions are listed up to millillion ( this is the popular list these days, in fact it occupies a large portion of the discussion of large numbers ). As is to be expected of such a large and obscure list, their are a number of variations among the names. In general though, the usual lists represent a simplification of Conway&Guys nomenclature.

For example no one seems to use "sedecillion", and people almost always use "novem" instead of "noven" regardless of what component follows. One might complain that this is "twisting latin", but there are other considerations besides a literal languistic translation of latin into prefixes for illions.

First off, a system with too many rules, is liable to be simplified. People will generally know what number your referring to when you type "novemtrigintillion" even though it should be "noventrigintillion", and it's easier to remember. Also it must be admitted that language itself is an imperfect art. Furthermore in extending the system our primary interest should be the theory behind it. It is more important that we have a name at all, than what specifically that name is. Basically we simply want a way to turn the order of the -illion into latin so that we can name them sequentially.
So is that the end of the -illions? Conway and Guy offer one further suggestion for how to continue.
A millillion, could easily be followed by a "milli-million", then a "milli-billion", "milli-trillion", and so on. By appending milli- to all the -illions so far we can eventually reach milli-novemnonagintanongentillion (the 1999th -illion or 10^6000). Next would be billillion. Then we continue with billi-million, billi-billion, billi-trillion, etc.
You may now see the pattern. We can simply use -illi- to seperate the thousands from the hundreds, tens and ones. The -illi- then operates as a comma. We can also use -nilli- when we want to specify an empty "block".
Thus the one millionth -illion in Conway and Guy's system would be milli-illi-illion, the one billionth -illion would be milli-illi-illi-illion, and so on. In this way every power of a thousand could be named. Of coarse, given a large enough power the notation becomes quite cumbersome. For example the decillionth -illion would be a milli-illi-illi-illi-illi-illi-illi-illi-illi-illi-illi-illion.
Others such as Landon Curt Noll offer similiar solutions for continuing indefinitely. We can use millia- to continue in this way. First would be a millillion (10^3,003), next a millia-millillion (10^3,000,003), then a millia-millia-millillion (10^3,000,000,003) and so on...
As you can see we are more or less hitting a brick wall here. We have essentially exhausted the latin based numbers. This kind of brings us full circle in a way, because that is exactly where our conversation began. The whole reason there aren't many latin numbers above a thousand was because people didn't have much use for them.
So is that it? Well ... after this it becomes much more difficult. The reason is that it starts to get difficult to organize the various components and the complexity starts to really mount beyond this point. We will consider the issue of extension in great detail alittle later.
First however I'd like to return to the issue of the short verses long scale. In the next article we will consider the system proposed Russ Rowlett to resolve this dilemma. Later I will offer my own solution this problem, as well as a solution to extending beyond the limits of latin.