P12_2-4-10 : Sbiis Saibian's -illions

Sbiis Saibian's

Multi-Tier -illion Series

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NON_PUBLIC

UNDER CONSTRUCTION 2012

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Introduction

We have considered the extensions of the canonical illions by Prof. Henkle, Mr. Ondrejka, Conway&Guy, and Jonathan Bowers. Now I'd like to present a system that I devised back in 2008 to continue the illion series higher into the "tetrational" range.

Here are some of my guiding principles for construction...

1. There are no "gaps" allowed, no matter how far the system extends.

2. Every name must end in -illion and represent a power of 1000.

3. Every name must name a unique power of 1000.

Note: I am an amateur mathematician, not a linguist. Although I will likely incur criticism on this point, I am not particularly interested in keeping number names strictly "latin", nor am I interested in absolute linguistic fidelity. The use of foreign languages is merely used to provide word elements for our system, it is not meant to bog down that system with particulars of language. For our purposes "spelling" will be far more important than "pronunciation". I will however, try and make an effort to have names which are sufficiently distinguishable in pronunciation ( a problem with other systems such as Conway&Guys use of trescentillion and trecentillion).

This system is primarily constructed for the short scale. However it can be adapted for the long scale with the addition of a few more word elements. For our purposes a "million" will be defined as the 1st -illion, "billion" as the 2nd, and so on. It should be understood that the letters prior to -illion represent a number, n, and the value of n-illion is 10^(3n+3). I will be making extensive use of the function H(n) = 10^(3n+3).

To even further simplify this we will use (n) = H(n), and (n,k) = H(H( ... H(H(n)) ... )) with "k" "H"s.

"Reordering Principle"

I will begin my system by assuming the 21 official recognized -illions as canonical. I am not interested in correcting the latin here; these -illions as they stand, are well established enough for our purposes.

Unlike most extensions however, I find the backwards use of word elements unappealing. Louis Epstein has said this is not a concern of his, but I find it an important issue for consistency.

Most extensions, including that of Conway & Guy, continue after vigintillion with unvigintillion, continuing the pattern established by decillion, undecillion, duodecillion, etc. Essentially Conway and Guys system places the ones first, the tens next, and the hundreds last. This order is the reverse of what we are usually used to in counting. This leads to some problems. For example, is trecentillion the 300th or 103rd illion? In Conway&Guys system a "trecentillion" is the 300th illion, and "trescentillion" is the 103rd illion. While is slight distinction solves the spelling problem, it doesn't solve the pronunciation problem. How are we to distinguish these names when spoken?

The issue of reverse order becomes even more problematic once we move beyond the 1000th illion. Are we to continue the reverse order? Or do we follow Conway&Guys lead, and only reverse triplets? Why do we have to list them in this order in the first place?

It is true when we count from 13 to 19, we are actually placing the ones unit ahead of the tens. For example thirteen is essentially "three-ten", fourteen is "four-ten", etc. This standard is quite consistent across languages. I assume this has something to do with the use of base 20 by very early cultures. This pattern is also observed in the latin, which is why our -illion numbers follow this pattern as well.

However, the important thing to note is that after 20, the order flips. We don't say one-twenty, we say twenty-one for 21. Why then should we continue with unvigintillion? Furthermore, if we look at latin, do the numbers continue this way? If they don't what justification do we have for continuing this way? Yes it is convenient, if only because it keeps the order consistent through the whole system, and it works well for Conway and Guy, as far as they extend it. It will become apparent later on however that this approach will force us to choose between unappealing alternatives. For now let's us just explore the ramifications of my suggestion so I can discuss how my system begins.

So how do I decide to continue after Vigintillion? Well first we will use "viginti-" as our prefix for 20. To continue we place viginti after a million to obtain "vigintimillion". This will be followed by vigintibillion, vigintitrillion, vigintiquadrillion, etc. up to vigintinonillion. When viginti- is followed by a vowel, we can place a dash between viginti and the rest of the name as in "viginti-octillion". Essentially we can think of this as two connected words. This is essentially how it would be pronounced if spoken.

We continue with trigintillion for the 30th -illion (10^93), just like in Conway and Guys system. Obviously next comes trigintimillion, trigintibillion, etc. Using the appropriate latin units of ten, followed by the canonical first 9 -illions, we thus establish a system which is very much like counting. Our units of ten will be viginti-, triginti-, quadraginti-, quinquaginti-, sexaginti-, septaginti-, octoginti-, and nonaginti-. Note that I will not be ending the groups of ten after viginti- with -a- like in Conway and Guys system. This may be twisting latin, but I prefer the look and sound of ending in -i- instead of -a-, and it also seems more consistent with the use of -illion.

I find this means of extension more satisfying. In some ways vigintimillion is the logical completion of what I had attempted as a kid with migintillion. It is far more logical of coarse. It also doesn't sound too bad, and could sound canonical if you roll it around your brain for awhile. I realize it fly's in the face of the established nomenclature, but bare with me. I will show why it comes in handy later down the very long -illion road.

Thus with the system as I've just established it, we can reach 10^300, or the 99th -illion with the name "nonagintinonillion". As stated before, I am keeping with the established -illions, so the next term will of coarse be a "centillion". Where would we go from here? It should be obvious ... centimillion, centibillion, centitrillion, centiquadrillion, etc. Again, this is very reminescent of my old system, but with much stronger justification.

To continue we merely establish the hundreds come first, then tens , and then ones, just like english numbers. The only exception to this comes when we go through the 10s. When we try to append centi- to a vowel, can again use a dash. Thus after centidecillion, would come centi-undecillion. It follows with centiduodecillion, centitredecillion, centiquattuordecillion,etc.

We can thus continue until we reach centinonagintinonillion, the 199th -illion. What next? This is where the reordering starts to come in handy. In Conway&Guys system, they create unit of a hundred terms, but they can get kind of confused with appending a unit of one term, because of the poor ordering. For example, what is the difference between ducentillion, and duocentillion? Ducentillion is the 200th illion, where duocentillion is the 102nd. Only 1 letter distinguishes these from each other. How about trescentillion and trecentillion? Again trescentillion would be the 103rd, while trecentillion would be the 300th. Beginning to see a problem? The problem arises because the ones term, and the multiplier in front of the the hundred are occupying the same location. This necessitates the use of spelling subtleties to distinguish the cases. At least in the Conway&Guy system these subtleties are addressed by a careful application of their rules. The irony is that most people overlook them! It is not uncommon on a typical list of the first thousand -illions (created by your typical large number enthusiast), to include the same name for two different numbers! The most common mistake is the use of trecentillion for both the 103rd and 300th illion.

In my system, we don't need a special set of terms of the hundreds, because these spelling ambiguities never arise. Instead we can recycle the prefixes, duo, tre, quattuor, quin, sex, septen,octo, and novem. For example, the 200th illion will simply be duocentillion. This can NOT be confused with the 102nd illion, which is actually centibillion in my system. In Conway&Guys system however, this would not read normally as "two - hundred" but as "two and a hundred", which is unusual in our counting system. Again my logic follows closer to how we actually count in english. We do not continue after a hundred with "one and a hundred", "two and a hundred" etc. Yet this is exactly how the Conway and Guy system works.

After duocentillion would be duocentimillion, duocentibillion, etc. We would eventually reach, trecentillion, quattuorcentillion, quincentillion, sexcentillion, septencentillion, octocentillion, and novemcentillion. This would get us all the way to novemcentinonagintinonillion, the 999th -illion or 1E3000. We achieved this with considerably little effort. We also did it with very few word components. Here is a table containing all the word components (excluding -illion) used to build these names. Only a few rules are needed for their construction.

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Sbiis Saibian's Tier 1 Roots

Rules:

1. If hundreds,tens=0 : use first option ones root followed by illion

2. If hundreds = 0, tens = 1 : use second option ones root followed by tens root and then illion. Place a dash between roots if there is a double vowel, and drop i from deci.

3. If hundreds = 0, tens > 1 : tens root followed by first option ones root followed by illion. Place dash between roots if there is a double vowel.

4. If hundreds > 0, tens =! 1 : hundreds root followed by tens followed by first option ones.

5. If hundreds > 0 , tens = 1 : hundreds root followed by 2nd option ones followed by deci with i dropped.

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I hope that the system is simple and intuitive. Just to eliminate any possible ambiguity I will also list a large set of examples in the following table...

Here we have finally run out of latin words. This is also the point of divergence of several systems that extend beyond this range. For the millionth -illion, some suggest milli-millillion, Bowers suggests micrillion, based on the SI prefix micro- for one millionth, and Louis Epstein suggests megillion for the millionth power of a million in the long scale. For my system I propose a new starting point.

Rather than use the SI prefixes micro or mega, we should try to establish a logic. The problem is that ancient latin didn't have denominations above a thousand. Can we simply use the word "million" as our root for the millionth -illion?

If so we can call the millionth illion, a millionillion (10^3,000,003). Since english has it's roots in latin it's conceivable that if latin were a modern language they would adopt our number system. Thus we can say a "million" in latin would simply be a "million". Using this word element we can continue without much confusion...

After a millionillion, would be a millionmillion (10^3,000,006), a millionbillion (10^3,000,009), a milliontrillion (10^3,000,012), etc.

Basically we can treat a "million-" as a new block seperator. Thus, if we want to name the 1 million 234 thousand 567th -illion, it would be millionduocentitrigintiquattuormilliquincentisexagintiseptillion.

We can append prefixes to -million- to go even further. the 2 millionth illion would be duomillionillion,

the 10 millionth illion would be decimillionillion, etc.

We can also create many more block seperators. Next we can use billion- , trillion-, quadrillion-, quintillion-, sextillion-, septillion-, octillion-, and nonillion-.

With this we can get as far as ...

novemcentinonagintinovemnonillionnovemcentinonagintinovemoctillionnovemcentinonagintinovem-septillionnovemcentinonagintinovemsextillionnovemcentinonagintinovemquintillionnovemcentinonaginti-novemquadrillionnovemcentinonagintinovemtrillionnovemcentinonagintinovembillionnovemcentinonaginti-novemmillionnovemcentinonagintinovemmillinovemcentinonagintinonillion (10^(3x10^33) )

This can get alittle confusing, but there are surpisingly no serious problems so far. This is because the roots used for million through nonillion, are different from the roots used in the blocks for the ones unit. There are a few examples that get alittle tangled. For example, how would you interpret sexsextillionillion? This would be the 6 sextillionth -illion. What about octooctillionillion? This would be the 8 octillionth -illion.

Problems start to arrise when we reach past decillion- however. For example how do we interpret "duodecillionillion" ? Is this the 2 decillionth -illion, or is this simply refering to the duodecillionth -illion?

To resolve this issue we will use a seperating term. We will use "de", which is roughly latin for "of".

We will treat de- as a multiplier. For example , if we insert it between duo and decillion we get...

duodedecillionillion. This we will interpret as the 2 decillionth -illion. You can think of duodedecillion as reading "two of decillion", or "two times decillion". If a de- is not present we will interpret duodeci as a whole unit, so that duodecillionillion will be the duodecillionth -illion. We can omit the de- when possible.

In this manner we can continue with decillionillion, undecillionillion, duodecillionillion, tredecillionillion, etc. and know that the "inbetweens" are taken care of. In this manner we can easily name the first centillion -illions. The centillionth -illion would be a "centillionillion" or 10^(3x(10^303)+3). In fact, I actually gave this name to the number 10^(10^303) as a kid when I was exploring very large numbers. I sensed there was something kind of off about this, but couldn't figure out how to correct it. Besides, I wanted a name for 1 followed by a centillion zeroes.

In any case we can extend quite far with this system. it is more convenient than the Conway&Guy extension which uses -illi- repeatedly to sepeate blcks. For example, to name the decillionth -illion, Conway and Guy would use milli-nilli-nilli-nilli-nilli-nilli-nilli-nilli-nilli-nilli-nilli-nillion. In my system this would simply be a decillionillion. Consider the advantage of a centillionillion over the equivalent term in Conway and Guys system.

Another advantage is that we don't run out of terms so quickly like the SI prefixes. After Jonathan Bowers inevitably reaches yoctillion, he is forced to invent names almost on the fly (to be fair he starts using greek number prefixes). Louis Epstein simply stops at Yottillion, and uses other means to continue.

So how far can this system extend? ...

Well we can certainly go as far as millillionillion without to much trouble. What happens next? Next we would start with millionillionillion the millionillionth -illion or 10^(3x(10^3,000,003)+3). Things seem alright until we reach terms like duodedecillionillionillion. In this case does this simply mean the 2 decillionillionth -illion, or the duodedecillionillionth -illion? One way to solve this problem would be to use multiple of's to distinguish between various levels of meaning. So we will define duodedecillionillionillion as the duodedecillionillionth -illion. We define duodededecillionillionillion as the 2 decillionillionth -illion. To avoid using verbal repetition of this sort, we will use greek prefixes with de- to form multiple levels of de-. This second level we will use dode- instead of dede-. You might guess what happens next. Eventually we can wrap around one more time to reach a millionillionillionillion.

Again we don't have problems until we reach around decillionillionillionillion. We now use triade- to represent the third level of seperation.

So far so good. We have a logical alternative to the system of Conway&Guy. We want to do more than suggest an alternative however. We want to extend way way beyond this system in the same spirit as Jonathan Bowers, Harry Foundalis, and Louis Epstein.

After novemcentinonagintinonillion will come a millillion. I'll consider a "millillion" canonical even though it's not listed by any formal dictionary simply because many number enthusiasts have adopted it and it has more or less become a non-ambiguous standard. Jonathan Bowers even came up with it independently. That should be proof enough that it is a logical and obvious choice.

How do we continue after a millillion? We establish the new root milli-, and the rule of dropping the last i when followed by a vowel. Thus we continue with millimillion, millibillion, millitrillion, etc.

When we reach millidecillion, we continue with millundecillion, dropping the i. Next would come milliduodecillion, millitredecillion, etc. Obviously we can simply add milli- as a prefix to every -illion name less than order 1000 to form names up to the 1999th -illion, or millinovemcentinonagintinonillion (10^6000).

How do we continue? With duomillillion of coarse. We can simply recycle our ones units to form ...

tremillillion (10^9003), quattuormillillion (10^12,003), quinmillillion (10^15,003), sexmillillion (10^18,003), septenmillillion (10^21,003), octomillillion (10^24,003), and novemmillillion (10^27,003).

We needn't stop at novemmillinovemcentinonagintinonillion (10^30,000) however. We can continue with decimillillion, then undecimillillion, duodecimillillion, ... , vigintimillillion, vigintunmillillion, vigintiduomillillion, ... trigintimillillion, trigintunmillillion, trigintiduomillillion, ... centimillillion, etc.

Basically we can use all the existing word elements to reach the 999,999th -illion. This -illion would be called the novemcentinonagintinovemmillinovemcentinonagintinonillion (10^3,000,000).

The term -milli- is operating here very much like a comma; the same way the word "thousand" works when we count to seperate the "first block" from the second.(Recall our discussion of counting from Chapter 1-1).

Here is a table of examples to help you get the idea...