2.4.5 - Prof. Henkles million illions

Filling in the Blanks

Borgmanns goes on to give a critique of Henkles system. He makes one important note about the system. When an element ends in an "o" the element is additive, and when it ends in an "i" its a multiplier to the following elements. Borgmann goes on to say that while it would be tempting to accept this list as authoritative it is far too riddled with peculiarities. Henkle uses no less than three different latin systems in his naming scheme. He uses the cardinal, ordinal and distributive latin numbers. These each vary slightly.

None the less, it may be that Henkles pioneering work provided both the impetus and blue print for the more modern extensions of the illion series. One important question is whether the in between terms can be constructed, and if so, whether they are distinguishable from each other, and easy enough to identify for someone familiar with the system.

The first test is to see how we can, in theory, construct a full set of a million Henkle illions. Technically he provides a direct name for the first 30, but then he begins to skip around once each level of the pattern becomes readily apparent.

To a trigillion, the 30th illion, he appends the latin ordinals, primo-, secundo-, tertio-, quarto-, quinto-, sexto-, septimo-, octo-, and nono-. These act as elements (or roots) which can be appended as needed. Each represents a specific value which is added to the rest of the roots. Henkle then provides us with the tens units elements. We have quadragillion, quinquagillion, sexagillion, septuagillion, octogillion, and nonagillion. Removing the -illion suffix, we can obtain these roots: deci-, vigi-, trigi-, quadragi-, quinquagi-, sexagi-, septuagi-, octogi-, and nonagi-. These are all the tens units. Next comes centillion, which comes from the root centi-. So far so good? Obviously any illion up to the hundredth can be created by appending a ones root to a tens root and adding illion. For example the 53rd illion would be tertio-quinquagillion.

After a centillion comes primo-centillion. This suggests that we can just continue to the 199th illion by appending roots to centillion. Unfortunately there is a small technicality. When the tens root are not appended to illion, they have an extended version. The 110th illion in Henkles system is decimo-centillion. This means we need to learn the full versions of the tens roots. They are decimo-, vigesimo-, trigesimo-, quadragesimo-, quinquagesimo-, sexagesimo-, septuagesimo-, octogesimo-, and nonagesimo-. Henkle then provides us with the hundreds roots. Namely, ducenti-, trecenti-, quadringenti-, quinquagenti-, sexcenti-, septuagenti-, octingenti-, nongenti-. It turns out however that these too are shorten versions of the full roots. Once we get past a millillion, we find that the 1100th illion is centesimo-millillion, not centi-millillion as might be expected. The full hundreds roots are centesimo-, ducentesimo-, trecentesimo-, quadringentesimo-. quinquagentesimo-, sexcentesimo-, septuagentesimo-, octingentesimo-, and nongentesimo-. To simplify matters we can create a table with all of these roots:

Note that all of these roots end in "o". There values are therefore added. These roots are always listed from ones to tens to hundreds to give a name to some illion. For an example the 432nd illion would be secundo-trigesimo-quadringentillion.

This allows us to name the first 999 illion without any real ambiguity. The reason for this is that each of the 27 roots is unique and can be told apart from any other.

To continue we have millillion, where the root would be millio. We can append all of the pre-existing roots to millillion to obtain names up to the 1999th illion. The 2000th illion is listed as bi-millillion. In this case bi- is a multiplier of the millio root. What follows after this is a series of multipliers. We have tri-millillion, quadri-millillion, quinqui-millillion, sexi-millillion, septi-millillion, octi-millillion, and novi-millillion. This provides us with our "ones multiplier roots". Namely, bi-, tri-, quadri-, quinqui-, sexi-, septi-, octi-, and novi-. Next comes deci-millillion. After this we append a ones root to deci, however these differ considerably from the ones multiplier roots we just reviewed. Henkle uses undeci-, duodeci-, tredeci-, quatuordeci-, quindeci-, sexdeci-, as special cases of his multipliers. After this he returns to something more expected, such as septi-deci-, octi-deci-, and novi-deci-. It is interesting to note that although ending in an "i" should signify a multiplier, septi-deci- does not mean 7 times ten, but rather 7 plus 10 with the total multiplied by the following root.

After this we reach vici-millillion, vici- being the multiplier for 20. Here a new root is introduced: semeli-vici-millillion. At first this seems unfitting, but you might notice that there was no term for one on the first pass. On the second pass through the tens we used un- as the root for one. So semeli- is part of the alternative sequence, bi-, tri-, quadri-, etc. We can see this because after semeli-vici-millillion comes bi-vici-millillion, tri-vici-millillion, quadri-vici-millillion, etc.

What remains now are the tens multiplier roots. These are deci-, vici-, trici-, quadragi-, quinquagi-, sexagi-, septuagi-, octogi-, and nonagi-. Nothing else of interest apparently happens after a centi-millillion, and all we have left to know is the hundreds multiplier roots. These are centi-, ducenti-, trecenti-, quadringenti-, quinquagenti-, sexcenti-, septuagenti-, octingenti-, and nongenti-. Astute readers may notice these are identical to the hundreds roots from the original set. However they only appear in this form when appended to illion. When appended to any other element they take on their fuller forms, centesimo, ducentesimo, etc. Thus there is never a situation where a additive root could be mistaken for a multiplier root. We now construct the following table for the multiplier roots:

By comparing each member of the roots table with its corresponding multiplier equivalent we find that each table has unique terms. This means that none of the terms should become confused with each other.

Borgmanns states that "anyone who attempts to fill in the intermediate names omitted from Henkle's table will soon run into difficulties. One difficultly is that some of the intermediate names are so long as to be unwieldy." Saying that some of the names are "long" is of coarse subjective, unless the name is so long that it can not be written out in full for all practical purposes. There is no such name in Henkles system, at least up to the millionth member. The longest name would be the one that used all of the longest roots possible. The longest possible name should therefore be:

septimo-quinquagesimo-quinquagentesimo-quinqui-quinquagi-quinquagenti-millillion

This is the 555,557th illion. It contains 74 letters. Its sizable, but it is not impossible to write out or say. One objection might be that it seems to long for an illion name. Given that this names the number 10^1,666,674, which is a ridiculously huge number, is it so inappropriate that it should have a long name? More to the point, if we are to provide names for a million illions, it is unavoidable that some of those names are going to be quite long if we are going to create a system which is at all intelligible. After all "five hundred fifty five thousand five hundred fifty seven" is a long name in english. To provide short names for every number from 1 to a million would require us to increase the number of roots to the point where we'd practically need a unique name for each number! As I showed at the end of chapter 1-1 this is not in least bit practical. I therefore find this complaint a rather moot point. In fact when we go to even more advanced systems well find that intermediate names can be long enough to stretch across the observable universe, but lets not get ahead of ourselves. The bottom line is that these names are more exercises of construction, and not everyone of them will be able to be said. Just as there are english numbers with so many non-zero digits that they can't be named, why can't there be powers of ten with so many zeroes that they also can not be named?

As to the difficulties of the system Borgmanns says that a "millillion" is difficult to pronounce. It is a little tricky to read, but I don't find it too difficult to pronounce. I usually pronounce it as a "mil-lil-illion". Said this way it has a nice ring to it.

He also mentions that a related difficultly is avoiding ambiguity. He cites that a sexcentillion might be both 10^321, and 10^1803 in a given system. That is, it might be either the 106th illion, or the 600th. Henkle however avoids such an ambiguity. the 106th illion would be sexto-centillion, where the 600th would simply be sexcentillion. He also finds fault with the hyphens. I don't really see that they are a big problem, and they aid a great deal in readability.

Borgmann ends his article stating: "This has been a sampling of the problems encountered by anyone who attempts to formulate a wholly rational system of number names. so far, no one has succeeded. The challenge remains...". To this I would have to ask what is a "wholly rational system of number names"? Can some definition be provided?

I suspect what Borgmann means, possibly without realizing it, is a system which extends to all numbers, doesn't run into ambiguities and always provides short names for at least some numbers, no matter how large.

The first condition is a simple one to satisfy. For example, any whole number can be named by saying it digits in order. So 214 would just be "two one four". Such a "system of number names" is complete, in that given an unlimited amount of space any number name could be written, and given an unlimited amount of time any number name could be spoken. Is this not wholly rational? Of coarse certain names are going to be a bother, such as 10,000 which would have to be "one zero zero zero zero". One problem with this might be if someone has to say a string of random numbers, say 23, 16, 43, etc. They would have to say "two three one six four three". They would need some way to verbally separate the digits or else it might sound like a single long number. Any word can be inserted between them to clarify. For example "and" could be used. Then they could just say " two three and one six and four three ". Perhaps this system is too simple for your liking. There are other perfectly simple ways to extend number names indefinitely. Another trick would be to use the standard names up to some standard illion. Even a million will work fine. Once we inevitably get up to 999 million 999 thousand 999 we can simply continue with one thousand million. Eventually we can reach a million million, a million million million, and so on for every power of a million. This system is also complete and provides a name for every number. There is not much that is illogical about it and it is not ambiguous.

Perhaps this isn't what Borgmann means. Perhaps hes only referring to extending the illion series. However here we can use the same tricks to extend indefinitely without problem. All we would need to do is used the same tricks within the latin language.

Notice that for every proposed system, although it can theoretically name every number, in practice it can't. Not only will there be intermediate terms too long, but there will also be numbers so large that even they can not be expressed. The idea that there will always be a very big number with a short name somewhere down the road is not realizable. As I've shown, names are inevitably going to become longer no matter what we do. This means that the proposed definition can not be realized. You can't have a logical complete system that also manages to keep pumping out short names. To have short names new rules need to be introduced into the system which in turn increase the complexity and increase the likelihood of ambiguities. I suppose in this sense Borgmann is correct in asserting that no one has yet succeeded. However I would go one step further and say that such a task is literally impossible. The best we can do is create a complete and logical system.

A Further Extension by Mr. Ondrejka

The article on Henkles illions in word ways sparked a few follow up submissions on how to continue the names even further. The first of these improvements was described by Mr. Rudolf Ondrejka. His system is very similar to Henkles, though he makes a few improvements and extends the system to the billionth illion. By 1968 the names up to vigintillion had been so well established that Ondrejka decided to abandon Henkles names below the 21st illion and go with the canonical names. After this however he extends, just as Henkles had, using the latin ordinals. Namely: after vigintillion, would be primo-vigintillion, secundo-vigintillion, tertio-vigintillion, etc. using the same prefixes as Henkles. He then uses trigintillion for the 30th. This is identical to our modern equivalent. One wonders whether our modern names come from the work of these individuals. In any case, Ondrejka's system is also introduced primarily through a long list of names[3]. For convenience, here is a list with all of the Ondrejka illions included in the article: