2.4.7 - Russ Rowlett's Greek Based -illions

I should note that the names in the short scale after a vigintillion are non-canonical. Rowlett decides to use the standard latin extensions used by most large number enthusiasts and supported by Conway & Guys system in "The book of Numbers".

Rowlett also says that the system can be extended indefinitely but that one must stop somewhere. If your familiar with my site you know I take the word indefinitely with a large grain of salt every time its used. None the less the greek based system can certainly be extended at least as far as Conway and Guys system while remaining strictly within greek prefixes.

It should be obvious from the above pattern established that a triacontatrillion could be continued with triacontatetrillion, triacontapentillion, triacontahexillion, triacontaheptillion, triacontaoktillion, and triacontaennillion. After this we must look into the greek numbers a little closer to see how best to proceed. Here is a table of greek number elements that we can use to create our prefixes up to 999:

In addition 1,000 in greek is "chillia". I should also note that greek doesn't use the english alphabet, but uses greek letters. The word elements above therefore represent english approximations based on pronunciation. This practice is known as "transliteration". Anybody who has watched subbed animes should be familiar with it. Transliterated japanese text is often used in the opening credits so that it is easier for english speakers to hear the words begin spoken in japanese. Transliteration is an imperfect art, in part because certain sounds in other languages don't exist in english. I have also taken some liberty in my transliteration here to smooth over finer details. Linguists will probably groan, but I'm not a linguist and I'm more concerned with the mathematics and concepts involved.

In any case these will not directly be used as prefixes. Note that they differ considerably from Rowletts usage. For example, he uses triaconti, instead of trianta. The prefixes Rowlett uses more closely match those used for polygons, and the more general polytopes. We therefore will construct number elements to better match this:

For 1000 we can use the prefix "kilo". Elements are listed in the normal order with hundreds first, tens next, and ones last. Note that this order is the reverse of Conway and Guys system. In any case we can now name numbers in Rowletts system up to the 1,000th member. Here is a table with some examples...

(I should note that a lot of these names have already been used by Jonathan Bowers for much larger powers of a thousand. We will be discussing his illion system a little later in this chapter.)

Can we extend it even further? It turns out that it is easier to extend Rowletts system past the 1000th member than it is for Conway & Guys system. This is because, unlike latin, which is a dead language, greek is still spoken today. Because of this greek is both an ancient and modern language, and therefore has unique names for numbers past a million. Latin on the other hand has no real word for a "million" or any higher power of a thousand, and this is one of the major road blocks to progress with Conway & Guys system. The use of -illi- as a block seperator is not really a viable solution and does not allow for any truly sweeping extensions. Let's see what we can do with Rowlett's system instead.

According to google translate, modern greek at least has names for powers of a thousand up to a septillion. The following table shows the value, english name, and transliterated greek name obtained from google translate:

After a septillion, entering any higher power of a thousand into google translate simply returns the english name. This suggests there is no higher names for numbers in greek. Actually that's quite untrue, but just like dictionaries there has to be some point at which the writers don't accept certain number names as either canonical, or of any practical use. As we will see in a later article, greek extends much further, and Harry Foundalis has even extended it much further. In any case we can see the pattern and use this to extend at least up to the canonical vigintillion...

So how does this help us extend Rowlett's system to dizzying heights? Well we can use these greek numbers as seperators between blocks of digital triplets in the same way that we do in english. To extend beyond a kilillion we can use...

kilohenillion, kilodillion, kilotrillion, kilotetrillion, kilopentillion, kilohexillion, kiloheptillion, kilooktillion, kiloennillion, kilodekillion, kilohendekillion, kilododekillion, kilotrisdekillion, kilotetradekillion, kilopentadekillion, kilohexadekillion, kiloheptadekillion, kilooktadekillion, kiloenneadekillion, kiloicosillion, kiloicosihenillion, kiloicosidillion, kiloicositrillion, kiloicositetrillion, ...

Basically we can append every name we've already generated with kilo. Once this is exhausted an we inevitably reach kiloenneahectaenneacontaennillion, we continue with duokilillion, then duokilohenillion, duokilodillion, duokilotrillion, duokilotetrillion, ... etc.

As you can see, the "kilo" is working the same way a thousand within english number names, only this time within the name of the illion! This allows us to name some very large powers of a thousand and enter into the hyper-exponential range of numbers. Here are some more examples reaching up to the 999,999,999th illion in Rowletts system:

Amazing, and we can go further still with Rowletts system. We now want to name the billionth member of Rowletts illion series. That would be "disekatommyrillion". We can of coarse continue with disekatommyriahenillion, disekatommyriadillion, etc. The elements "disekatommyria" , "ekatommyria", and "kilo" act in the same way that the words, billion, million, and thousand act when naming numbers. They act as a way to seperate blocks of digital triplets and also to keep track of the magnitude. Here are some examples:

disekatommyriaekatommyriakilohenillion = th(1,001,001,001) = 10^3,003,003,003

duodisekatommyrillion = th(2,000,000,000) = 10^6,000,000,000

trisdekadisekatommyriapentahectatriacontahexaekatommyriaheptacontatetrakiloenneahectadillion =

th(13,536,074,902) = 10^ 40,608,224,706

enneahectaenneacontaenneadisekatommyriaenneahectaenneacontaenneaekatommyria-enneahectaenneacontaenneakiloenneahectaenneacontaennillion = th(999,999,999,999) =

10^2,999,999,999,997

One can of coarse continue with the higher powers of a thousand in greek to extend Rowletts system even further. Here is how we would construct some of them...

So now we can extend Rowletts system up to the limit of canonical modern greek. Can we go further... yes, but the system is actually already has serious problems. First and foremost, the modern greek numbers follow the convention established by the short scale. This is the very thing we were trying to avoid and yet it has creeped back into the system. Another more serious problem is that because greek uses greek prefixes to name its powers of a thousand, there is now some ambiguity in the terms we could generate. For example:

Is a tetradekakisekatommyrillion the quattuordecillionth power of a thousand, or ... is it simply the 4 decillionth power of a thousand.

Note that the decillionth power of a thousand would be dekakisekatommyrillion. To say the 4 decillionth member we would append the prefix tetra to it to obtain tetradekakisekatommyrillion. This is the same as the quattuordecillionth power of a thousand. Also note that these are very different numbers:

th(4*10^33) = 10^(1.2*10^34)

th(10^45) = 10^(3*10^45)

What can we do to get around this problem? Well remember that these numbers are based on the modern greek numbers. This means that the greeks probably have a way to distinguish it themselves, most likely by using either an accent or an audible pause that indicates whether the prefix is applied to the block separator as a multiplier or as a modifier. We can use a dash to represent a pause in speech and distinguish cases.

The dash can be thought of as a multiplier. So for example we will let tetra-dekakisekatommyrillion mean the 4 decillionth power of a thousand. If the dash is not present it means the prefix is part of the block seperator and is modifying it. So tetradekakisekatommyrillion would mean the quattuordecillionth power of a thousand. We can also include a dash at the end of a block seperator to help it stand out from the rest of the name. Here are some examples...

pentacontahexa-dekakisekatommyria-oktahectatria-hexakisekatommyria-

hepta-tetrakisekatommyria-henillion

= th(56,000,000,000,803,000,007,000,000,000,000,001) = 10^168,000,000,002,409,000,021,000,000,000,000,003

This works pretty well. It also allows us to continue with the greek past the canonical numbers into an extended system of greek numbers. Obviously we can simply increase the modifier to the seperator block by using higher and higher greek prefixes. Where will this lead? Let's find out:

We can extend further than this. It should be obvious now that we can take the same greek prefixes we had originally used to get this far as a modifier to the block seperator. This means we can go beyond the kilokisekatommyrillion and eventually reach the ekatommyriakisekatommyrillion which would be 10^(3*10^3,000,003). We could go further still with disekatommyriakisekatommyrillion which would be 10^(3*10^3,000,000,003). Next would be trisekatommyriakisekatommyrillion which would be 10^(3*10^3,000,000,000,003), and so on.

This works pretty well until we reach terms in which there is once again ambiguity as to which level the greek prefix is applied. For example:

We know that a tetradekakisekatommyriakisekatommyrillion would be 10^(3*10^(3*(10^45)+3)). That is the 10^(3*(10^45+3)) power of a thousand. We also know that we can use a dash to multiply the block seperator within the block seperator as in tetra-dekakisekatommyriakisekatommyrillion which would be 10^(3*10^(12*(10^33)+3)). However, how do we apply an ordinary multiplier to the entire block seperator of dekakisekatommyriakisekatommyria? To resolve this we need more than the single dash. We can use a double dash to specify yet another level of separation. By this logic we can say that:

tetra--dekakisekatommyriakisekatommyrillion = 10^(12*10^(3*(10^33)+3)

It is also possible to apply prefixes at several levels (tiers). For example we can say:

tetra--tetra-tetradekakisekatommyriakisekatommyrillion = 10^(12*10^(12*(10^45)+3))

In this way we can name not thousands or millions of illions but literally hyper exponential numbers of terms. Some of these "names" however will be increadibly massive.

For example we could have ...

enneahectaenneacontaennea--hectakisekatommyriakisekatommyria--enneahectacontaennea--enneahectaenneacontaennea-enneacontaenneakisekatommyria-enneahectaenneacontaennea-enneacontaoktakisekatommyria-enneahectaenneacontaennea-enneacontaheptakisekatommyria- ... ... ... ... ... ... ... ... -enneahectaenneacontaennea-disekatommyria-enneahectaenneacontaennea-ekatommyria-enneahectaenneacontaennea-kilo-enneahectaenneacontaennea-kisekatommyria--enneahectaenneacontaennea--enneahectaenneacontaennea-enneacontaenneakisekatommyria- ... ... ... ... ... ... ... ... -enneahectaenneacontaokta-kisekatommyria--enneahectaenneacontaennea--enneahectaenneacontaennea-enneacontaenneakiskeatommyria- ... ... ... ... ... ... ... ... -enneahectaenneacontahepta-kisekatommyria-- ... ... ... ... ... ... ... ... -- ... ... ... ... ... ... ... ... -- ... ... ... ... ... ... ... ... -- ... ... ... ... ... ... ... ...

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

... ... ... ... ... ... ... ... ... --enneahectaenneacontaennillion

This would be the term just before hectahenakisekatommyriakisekatommyrillion which is equal to 10^(3*10^(3*(10^306)+3)) Therefore the above name would apply to the name for the number equal to 10^(3*10^(3*(10^306))). Already this is an example of a name that couldn't be written within the observable universe even if every particle was a letter! Confused yet? This is always the cost of extended a system to its limits. The complexity increases. This means that continuing is not just an issue of endurance, but the ability to deal with complexity ( a point often missed by the average person when considering naming larger and larger numbers).

Now we can continue in this manner and as you might imagine we can keep taking any greek number we name and plugging it back as a modifier to the original block. In order to distinguish the further tiers produced in this manner we can use triple dashes, quadruple dashes, or as many as we need.

This leads us to the following sequence of terms:

Eventually we would reach terms like...

ekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakis-ekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakis-

ekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakis-

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

ekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakis-ekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyriakisekatommyrillion

with any number of chains of "ekatommyriakis". Eventually even these names would become so long that even they would not fit in the universe! Then what? Well we have finally arrived at the limit of Rowletts system. In order to continue I would need to invent something new. However Rowlett only said we could extend using the greek numbers. Anything new would be outside the scope of his proposal.

Note that, as he said, we can extend this system indefinitely ... but only in theory because we only have a finite amount of space to contain the names. While it is true that Rowletts system can theoretically be used to name any power of a thousand, only a finite number of these can actually be named within our universe, while an infinite number can not. For every verbal or mathematical system for generating large numbers we can extend it indefinitely, but the system will still have this built in limitation because humans are bound to finitude. How then do we transcend such a limitation? We must create a new system which operates outside, above and beyond the old.

It might seem then that we can continue indefinitely simply be creating newer and newer systems. However with the addition of each new system the complexity increases, so even then "indefinitely" is rather unobtainable. When mathematicians say something can be extended "indefinitely" it must be understand that this means only in the abstract, provided we have an infinite amount of time, storage and computational power. In practical terms however we not only "have to stop somewhere" but are more or less forced to by natural limitations.

These are very far reaching ideas which go beyond just a discussion of Rowletts system and apply to all systems. We will find this to be the case over and over again as we progress. We'll learn much more about this phenomena in Section III when we start dealing with recursive functions. This principle will become more clear when we deal with more complex numbering schemes.

Conclusion

There is definitely a lot to like about Rowletts system. It is easy to extend quite far. I also really like the sound of terms of tetrillion, pentillion, hexillion, heptillion, etc. The definitions are also very intuitive. It seems very natural to me that a tetrillion is 1,000,000,000,000 where there are 4 groups of triplets because tetra means 4. A Hexillion would be 1,000,000,000,000,000,000. In a lot of ways this system is actually simpler than both the long and short scales. On the other hand the term gillion bothers me and doesn't fit in with the rest of the system. Furthermore the jump from a million ("m" standing for one in the short scale) to a gillion ("g" standing for three in Rowletts system) is counter-intuitive. It feels like jumping from 1 to 3 instead of the expected 2. One other problem with extension is that modern greek itself has succumbed to the large and short scale debate. The greek itself therefore would need to be corrected and adapted to create a truly intuitive system. This is one of the reasons the terms towards the end start to get kind of confusing.

To resolve these issues I have a few suggestions. Firstly I'd say it's probably best to just abandon a thousand and million to have a more complete logic. We can use "millia", for a thousand. "millia" is simply latin for a thousand. A million was probably named improperly to begin with. Instead of "grand thousand" its name should have implied "thousand thousand", or "2nd level of thousand". If this had been done in the first place we probably wouldn't have the long vs. short scale debate in the first place. Now in Rowletts system a million can simply be duomillia. A "gillion" could be a trimillia, without this getting confused with a trillion, and then we could have tetramillia, pentamillia, hexamillia, etc. These sound very nice and make a lot of sense. The greek prefix is the power a millia is being raised. The plural of millia can be millies. So for example we can talk about Americas debt being in the "tetramillies". I find these terms work pretty well. They also work pretty well with the english number naming system. For example:

384,301,827,918,216 would be read as...

384 tetramillia 301 trimillia 827 duomillia 918 millia 216

The greek numbering system should also be altered to fit the same logic as Rowletts system. We can begin with kilo for a thousand, then use dokillia for a million, triskillia for a billion, tetrakillia for a trillion, and so on. Again dashes can be used to distinguish between levels of application.

I have my own idea for how to resolve the short vs. long scale dilemma. I'll be discussing it towards the end of this chapter.

Although we we're able to create an "open ended" system using Russ Rowlett's proposal as a base, it can not really serve as a viable solution to the "counting problem". Why? Because there is not enough pronouncable difference between the names. Furthermore, how do we pronounce a double dash "--" as opposed to a single dash "-". In the next article we consider the system created by Jonathan Bowers. While it is not an "open" system, it does have the advantage of giving every member of the series a uniquely pronouncable name...

Next Jonathan Bowers' 4 Tiered -illion Series

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*Michael Thomas Greer detected a small typo for the 112,000th member due to the fact that a program he had wrote to produce the names for the first million cases produced "hectadodekakilillion" for the 112,000th member. I originally had it listed as "hectaduodekakilillion". After reviewing the system I realized this was a mistake. Props go to Michael for bringing this to my attention.