2.4.4
Filling in the Great -illion Gap
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INTRODUCTION
When I was a kid I became fascinated by mathematics mainly through the notion of the infinite. It seemed strange to me how an infinite number of numbers could exist. One could also construct an infinite number of addition problems. Each one of these would have a solution, in principle, yet since there was an infinite number of them, no one would ever be able to answer them all ! How could this be unless the solutions existed on their own? Once I was aware of the extendability of mathematics and the infinite, it began to take on a life of its own, almost like it was a completely dependent and abstract reality.
This is how I first became interested in large numbers. If there are an infinite number of numbers, and they existed independently of thought, then there had to be really large numbers that I had never thought of. What would these numbers be like?
The first thing I learned about big numbers was the extended series of canonical -illions introduced in the first article of this chapter. I had found them in a websters dictionary I had at home under the entry for number. In my class the furthest they ever got was 999,999. A "million" was not mentioned, yet I had learned about it somewhere. I was enthralled to discover that there were more number names after this. I discovered that after a million came a billion, then a trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, and a personal favorite, the decillion. The names continued sequentially until vigintillion; then there was a break, and the next named number was the mind boggling "Centillion", which was 1 followed by an amazing 303 zeroes ! For me this was like learning about the "googol". Ironically I never learned about the "googol" as a kid. To this day I have this list of -illion numbers memorized, because I read and wrote them over and over again obssessively as a kid. I also never had problems with the 3 times table because of these numbers. I also think I lost some respect for my mathematical instruction because the way they talked about arithematic it felt like they were denying these kinds of numbers even existed ! I don't think I'm the only kid who found that mathematics was actually a lot more fascinating and mysterious than school was letting on.
Many things about mathematics puzzled me. For me, I could not reconcile the discrepancy between the infinite and the finite. I became determined to find out what exactly lay between the finite and the infinite, and this was what first lead me to large numbers. Having become fed up with the infathomableness of the infinite I decided that "I've had it with this infinity nonsense. I'm just going to reach it and settle this once and for all, even though everyone says it's impossible". I failed to reach it through the finite of coarse, but the result was the discovery of amazing large numbers. I'll discuss more about my discoveries on this first large numbers project later in Section III. For now I'd like to discuss one of the first steps in this project.
FILLING IN THE GREAT -ILLION GAP
The huge gap between vigintillion (10^63) and a centillion (10^303) could not have escaped my notice; it was too glaring. It bothered me. So the very first thing I did on my first large numbers project was to fill in this gap.
What was the point of so many missing names? I understood that the -illion names were essentially an extension of counting. In order for it to provide names for the intermediate numbers from vigintillion and centillion, there would have to be "in-between" -illions. This was before I had access to any kind of internet ( prior to 1993 ), and I also didn't have access to much other literature. It seemed that what the websters dictionary was telling me was that people had named -illions up to vigintillion, and had named centillion, but no one had bothered to name any others. Well the solution seemed clear to me: I'd just have to name them myself.
I provided a name for every power of 1000 up to a centillion. Of coarse being a kid my system was far from very logical, yet it was not merely random either.
I accepted all the -illions up to vigintillion as "real" -illions. To continue I observed this pattern. Every name was formed by appending a word to -illion, thus providing a sequence of names, million, billion, trillion, quadrillion, etc. Now vigintillion was the largest -illion. So I reasoned I could continue the system by appending words to -igintillion. After vigintillion would be "migintillion". Since vigintillion was 10^63, a migintillion would be 10^66. I noticed that a migintillion, was simply a million times 10^60, thus I could exploit this new pattern to extend the system much further. Just like million was followed by billion, migintillion would be followed by bigintillion (10^69). Logically this would continue with trigintillion (10^72), quadrigintillion (10^75), quintigintillion (10^78), sexigintillion (10^81), septigintillion (10^84), octigintillion (10^87), nontigintillion (10^90), decigintillion (10^93), and so on. There are a few spelling mishaps in my earliest version, but the basic idea is there. Of coarse this system would exhaust itself when I ran out of standard extensions. I went as far as Novemdecigintillion which was 10^120, but I was barely even half way to a centillion!
Rather than continue with "vigintigintillion", I changed tactics to something even more extendable. Next I began with "Chexillion" for 10^123. Yes this name was based on Chex cereal, which I thought was a great cereal as a kid. Now my idea was to recycle the names by appending "Chex" to them.
Next would come "Chexmillion", for 10^126. Again this would be a million times 10^120. Next would come "Chexbillion", "Chextrillion", "Chexquadrillion", ... , "Chexdecillion", and so on as you'd expect. Eventually I would reach "Chexnovemdecillion". Instead of continuing with "Chexvigintillion", I began a new pattern. Now I would use an altered form of chex-, and use "chexin-" instead.
Next would be "Chexintillion" for 10^183. After this would be "Chexinmillion" for 10^186, thus establishing the beginning of a new pattern, "Chexinbillion", "Chexintrillion", "Chexinquadrillion", etc.
So what happens after I inevitably reach Chexinnovemdecillion? In fact, I originally don't even have this number name! Instead it ends at "Chexinoctodecillion" standing for 10^237. It's here that I break slightly with the established logic as I now prepare to reach the goal of Centillion.
I try to establish a more logical extension beginning at 10^240. I call this power of a thousand, a "Gkbrillion". Seemingly impossible to pronounce I probably would have said it as "Gibrillion". After this comes a "Thoubrillion". Next comes "millbrillion", thus bringing us back to the usual type of extension. This provides names all the way up to what should be "Novemdecibrillion", but which I actually called "Novabrillion", which was 10^300. Finally after this would come a "Centillion", and thus the gap was finally filled, even if somewhat haphazardly.
It should be noted that I didn't know that the -illion names were based on latin numbers at the time, so I couldn't have extended them with latin. Oddly enough, even though I bothered to provide names for all of these -illions, I also kind of understood it wasn't that important what they were called. It was more important to me that they were at least called something. I more or less came up with these names as I went along, with no preconceived plan of how to go about it.
Amazingly I still have the original paper I wrote these names on, and when I decided to create a large numbers site I decided to preserve these names and also resolve a few of their pecularities. I have since made alterations to the names so that they follow a much more logical pattern.
For completeness I will now present a table showing my original nomenclature, and also a "corrected" version.
As a kid I was happy merely filling in the gap. I didn't have much interest in continuing the series after that, and instead switched to considering large numbers in the abstract as decimals and scientific notation rather than by "names". None the less I can easily imagine how I would have continued...
After a centillion, can come "centimillion", "centibillion", "centitrillion", etc. following the same logic. We can use all of the -illion numbers, but this time they would begin with the prefix centi-, until we eventually reached centinovemdecibrillion. After this would logically have to come centicentillion. Conveniently enough this would be 10^603, or roughly a centillion centillions...
but we could continue then with centicentimillion, centicentibillion, centicentitrillion, etc. Eventually we would reach names like centicenticenticenti ... ... ... centicentilllion of arbitrary length. In a sense this provides a name for EVERY power of 1000. However it is not terribly practical for naming very large -illions. For example, the name for 10^3003 would be centicenticenticenticenticenticenticenticenticentillion !!!
Apparently I wasn't the only kid who saw a need to fill in the great -illion gap. Robert Munafo (author of another excellent site on large numbers) mentions a system he used to extend beyond vigintillion [1] . Logically he continued with unvigintillion, duovigintillion, trevigintillion, ... following the trend established by decillion, undecillion, duodecillion, etc. After novemvigintillion, he introduces "bigintillion" (10^93). The prefixes un-, duo-, tre-, etc. can then be appended to this new base. To create new terms he appends the letters of the alphabet to -igintillion. Thus we would have "cigintillion" (10^123), "digintillion" (10^153), "eigintillion" (10^183), etc. Munafo acknowledges himself that there are problems with this. For one, how do you distinguish the pronoucation of cigintillion with sigintillion? I also wonder what happens when we go past "uigintillion" and find ourselves with "vigintillion" again. This system would provide names up to "novemzigintillion" which would be 10^840. That would be the 279th -illion, and the 280th power of a thousand.
So how should we provide names for the -illions between a vigintillion and a centillion and beyond ? It turns out there is a perfectly logical way to extend it. The system proposed by Conway and Guy is essentially a refined version of a basic way to extend the pattern. We will learn about this system in the next article. It will allow us to easily provide logical names for the first 1000th -illion numbers !
Next : Prof. Henkles One Million illions and Beyond
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SOURCES:
[1] http://mrob.com/pub/math/ln-notes1-2.html#adhoc_chuquet : This page on Robert Munafo's Large Number site discusses various extensions of the Chuquet -illion series. It also contains a mention of a system he used to extend past a vigintillion when he was a kid.