zhao = 10¹⁶

jing = 10³²

gai = 10⁶⁴

zi = 10¹²⁸

rang = 10²⁵⁶

gou = 10⁵¹²

jian = 10¹⁰²⁴

zheng = 10²⁰⁴⁸

zai = 10⁴⁰⁹⁶

That's MUCH more efficient than the previous three. With only ten names after "wan" we can be taken all the way up to the incomprehensible 10⁴⁰⁹⁶! Now 10⁴⁰⁹⁶ is a number that's VERY rare to actually hear of. While number enthusiasts will often still refer to, say, 10⁴⁴ as one hundred tredecillion, you just won't hear 10⁴⁰⁹⁶ referred to as anything other than "ten to the power of four thousand ninety six". That system is amazingly efficient, and fell out of use only because the numbers that this system can name are not necessary in ordinary scenarios.

The fourth system serves as the inspiration for Knuth's -yllion system, which we are now ready to explain.

This above is a decyllion written out, which was called "zai" in the fourth Chinese naming system.

Now a decyllion is a pretty awesome number. It's so big that all its digits might not fit on your computer screen. It's only the tenth -yllion, and yet it easily transcends the whopping millillion, the largest -illion with a commonly agreed-upon name among number enthusiasts! See how much more efficient Knuth's system is than all the other systems?

One thing you may be wondering is: Why don't we use a system like Knuth's? That's actually a good question, which is exactly what we'll get to next.

As I said, it's a good question why we don't have a system like Knuth's. With Knuth's system we need far fewer names to get all the way past astronomical (I usually consider that range to be upper-bounded by 10¹⁰⁰⁰) numbers than we need in a traditional English system. You can get all the way past 10¹⁰⁰⁰ with only seven -yllion names, in contrast to 332 -illion names needed to get there with English! So why do we use the -illion system when it's easy to make a far more efficient yllion-like system? Here I will give my speculation as to why that is.

For one thing, when people design number systems they are not geared to take such powerful far-reaching approaches as Knuth did. It seems to be much more common to create systems like the -illions or the modern Chinese system (where zai is 10⁴⁴). I believe this is partly because it's not immediately obvious that a system where each named power of 10 is the square of the previous actually works, and because in systems where each name is 1000, 10,000, or whatever times the previous it's easier to name a number than with systems like Knuth's. For example, take the number 10¹⁴. Naming this number in English is easy: you find the smallest power of 1000 smaller than it (in this case a trillion, 10¹²), and multiply it by 100 to get one hundred trillion. In Chinese it's also easy: you can follow just about the same process with powers of 10,000 to get "bai zhao" for 10¹⁴. But in Knuth's system you need to first figure out the smallest number that can be expressed as 10²ⁿ less than 10¹⁴ (in this case 10⁸, a myllion), divide 10¹⁴ by 10⁸ to get 10⁶, then repeat the process to get a hundred myriad for 10⁶, and combine the name with "myllion" to get one hundred myriad myllion. That is quite a lot harder than naming numbers in regular English.

And it only gets worse with larger numbers. For example, let's say you want to name 10⁵⁵ in terms of -illions. With normal English you can look up a table of -illions, find 10⁵⁴ is called a septendecillion, and bam, you have ten septendecillion. But with Knuth's -yllions you need to repeat the smallest-number-of-the-form-10²ⁿ process several times to get the more confusing name ten hundred myriad byllion tryllion.

Even though Knuth didn't explicitly name -yllions past a vigintyllion, it is possible to continue without much difficulty. We can have names like:

unvigintyllion = 10^8,388,608 = 10²²³

duovigintyllion = 10^16,777,216 = 10²²⁴

trevigintyllion = 10^33,554,432 = 10²²⁵

...

trigintyllion = 10^{4,294,967,296} = 10²³²

...

quadragintyllion = 10^{4,398,046,511,104} = 10²⁴²

...

all the way up to centyllion = 10^{5,070,602,400,912,917,605,986,812,821,504} = 10²¹⁰²

Now what should we call the myllionth -yllion? Knuth himself had a brilliant idea on how it should be named, which brings us to the next level of Knuth's yllions.

Although it's not well-known, there is an idea Knuth came up with to extend upon his -yllions, described later in "Supernatural Numbers". Robert Munafo describes this idea on his number list^[5], introducing people to that obscure but clever extension. I can't gather very much information on it, other than that he extends his -yllions with "latin[name-of-n-with-spaces-deleted]yllion" to denote the nth -yllion. From that, I can guess that this "latin-x-yllion" system starts somewhere around the strange number known as the "latinmyllionyllion", equal to 10^21;0000,0002, a number with a 30.1-million-digit number of digits. For our purposes that's exactly what I'll do, call the myllionth -yllion "latinmyllionyllion".

Knuth's system, on the other hand, totally dodges this problem. If it were not for inserting the word "latin" at the beginning of such names, would "twomyllionyllion" when spoken refer to the two-myllionth -yllion, or two times a "myllionyllion"? However, with using "latin" we can easily distinguish names like "two latinmyllionyllion" vs "latintwomyllionyllion".

latinduocentyllionyllion

latinlatintwomillionthreemyriadyllionyllion

latinlatinmyryllionyllionyllion

and a whole bunch of complex names, for example:

latintwolatinfivemyllionyllionthreelatintwomyllionthreehundredmyriadtwohundredtwentytwoyllionfiveduocentitryllionlatin

myllionyllionthreemyryllionseventeensexdecylliontwohundredtwentytwomyllionsextyllionseventymylliontwomyriadeighty

ninehundredfortyoneyllion

That is exactly equal to:

10^{2(2*1025;0000,0002 + 3*1022;0300,0024 + 5*1021;0000,0002+2205 + 3*1021,0002 + 17*1026,2144 + 222*10264 + 70;0002,8943)}

which is equal to about:

Now that above is a truly insane number!!! It's a number that just mops the floor with the weeny little multillions and Poincaré recurrence times. Those can be expressed sort of like above with only four tens, but here we'll need 200! If each 10 was a centimeter in height, then that whole expression when written out would be two meters tall, a little taller than the average person! Now think about that. Even writing out this number using exponents is quite a sizable expression. Even the number of digits in the number of digits in the number of the digits ... ... ... in the number of digits (say that 170 times) in the number is just such a new kind of number from all those other tiny little numbers, even Skewes' number which seemed so out-of-this-world a little while ago. Now THAT is insanity. Now we can safely say it's an UTTERLY ABSURD understatement to say that all its digits would not fit in the observable universe, and yet that is again and again said even for MUCH LARGER numbers like Graham's number, whose number of digits is really indistinguishable from the number itself!!

Now that we have that, you might notice that it isn't hard to apply this idea to the normal -illions. This is where we'll get to next.

With that idea, I would suggest that up to 10^3,000,003 we use the similified version of Conway and Guy's naming scheme (e.g. 10^3,000,000 = novemnonagintinongentillinovemnonagintinongentillion) and after that point we use the latin-x-illion system. For example, a googolplex in that system would be named:

Techincally speaking, Knuth's system (even if you apply the "latin-x-yllion" idea to the -illions) has no limit, much like Conway and Guy's; they are both open-ended systems. However, they both have limits to numbers we can practically name—after the limit is passed, we would find infinite hordes of incomprehensibly long names far beyond any numbers whose names could fit in the observable universe! Such hordes would begin somewhere around the number:

Of course we can go further with the system! Imagine, instead of long chains of "latin" or "yllion"s, shortening "latinlatin" to "dualatin", "latinlatinlatin" to "trialatin", "latinlatinlatin" to "quadralatin", etc, continuing with the exact same roots as the -yllions. For example "latinlatin ... (10⁸ latins) ... latin" would become "latinmyllionalatin". Similarly we can turn "yllionyllion" to "duayllion", "yllionyllionyllion" to "triayllion". With that we can extend that further to go change the number of latins or yllions to any number, for example "latinmyllionalatinmyllionlatinmyllionayllion", which would be the same as:

But that's probably rather confusing, right? And plus, that name probably does seem pretty unwieldy! Try to imagine a shortener system for THAT and how confusing that would be!! Maybe after that, we could have a third shortener system, then a fourth shortener system, a Nth shortener system where N is the number named above ... I'll leave you to imagine where that could take you.

But wait—first off you need to define how such a system really works, which has GOT to be at least a little confusing. And besides, does a name like

So is extending the -illions really a futile pursuit?! Well, remember that we've already established that the goal for extending -illions is to see how far you can logically take the idea of -illions. In any case, I think we've reviewed enough -illion systems now. So up next we'll recap what we've learned in section 1 with a review page.