The -yllion System and its Extensions

Last updated 28/12/23

(Note: The "long scale" doesn't exist. It's a lie by the government because they're evil and they do that sort of thing.)

The -illions (million, billion, trillion, etc.) have been around for a long time. Like, since the middle ages. "Million" dates back to the 13th century, with the others coming over the course of a few hundred years, and they form the large number system most people are familiar with today. Let's assess it:

A few things to note here: First, it's relatively easy to determine at a glance what "-illion number" a name is since the prefixes stem from latin (Quintillion must be the 5th since it stems from the Latin "Quinque", Decillion must be the 10th since it stems from "Decem", and so on). Second, each -illion number n is exactly equal to 10³ⁿ⁺³, or 10³⁽ⁿ⁺¹⁾. The "+1" there is what leads me to the third remark; "Thousand" ruins everything. It means that despite a million being the first -illion number, it's the second power of 1,000. It also means that the 100th -illion (centillion, stemming from the latin "Centum") is equal to 10³⁰³ instead of 10³⁰⁰. This is a result of the fact that both idea of a thousand and the word "thousand" well predate the idea of an extendable power system like the -illions, which makes sense considering most people rarely if ever need to use numbers beyond the millions. The -illion system has been tacked on to the end more recently, creating a weird frankensystem. Just like most of English, really.

The easily extendible nature of the -illions has led to an explosion of homemade extensions, many continuing well past the 1,000th and 1,000,000th -illion numbers. But what if we could come at the idea in a new way? In 1981 Donald Knuth published an essay titled Supernatural Numbers (Get it? "Super-natural" numbers, as in large natural numbers), in which he lays out a new extendible system: The -yllions. Under the -yllion system numbers are the same up to 10,000 (10⁴), which is known as a myriad. This term has actually existed since the time of the ancient Greeks, who had symbols for numbers up to 10,000 and thus considered it their "million" (they even used the symbol M for it). Numbers beyond 10,000 are named using multiples of a myriad instead of multiples of a thousand or million. For example, 999,999 is no longer "Nine hundred ninety-nine thousand nine hundred ninety-nine" but instead "Ninety-nine myriad ninety-nine hundred ninety-nine". Note that "thousand" is no longer present in the name, the thousands are now named as tens of hundreds. Thus, the largest number below a myriad has the name "Ninety-nine hundred ninety-nine" meaning that the largest number we can currently name below a myriad myriads (99,999,999) is "Ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine". This has a nice symmetry to it.

And Knuth points out that we already often refer to thousands as tens of hundreds, like how we call 1981 "nineteen hundred eighty-one" (or just "nineteen eighty-one") because let's be honest, calling it "One thousand nine hundred eighty-one" would suck. And indeed we can shorten this new naming scheme similarly to how we name years by dropping the word "hundred", giving the number before a myriad myriads the name "Ninety-nine ninety-nine myriad ninety-nine ninety-nine". It takes a bit to wrap your head around it, but once you're used to it it can be quite intuitive. And since we're now working with powers of the myriad, it makes sense to move the commas in our notation to every 4 digits instead of every three. This makes the number before a myriad myriads 9999,9999 instead of 99,999,999.

(Side note, Knuth describes the current thousand-million system as "Menschenwerk", which translates literally as "work of man". In German this is used to imply the fallacy or imperfection of something that has been developed by people. I'm inclined to agree. Systems built on older systems typically don't fit together quite right, we just get used to them. Take QWERTY keyboards for example, their main goal was to separate frequent key combinations to prevent typewriter jams, but modern keyboards don't jam. There are many more ergonomic keyboard layouts, but QWERTY is and will remain the standard.)

Instead of a million, under this system a myriad myriads is called a myllion (10⁸). The similarity between the words "myriad" and "myllion" is a lot nicer than the jump from "thousand" to "million". Instead of just using another comma, Knuth implements the semicolon (;) to separate numbers before and after a myllion. For example, 10 myllion (10⁹) would be denoted as 10;0000,0000 and a myriad myllion (10¹²) is 1,0000;0000,0000. Well a myriad myllion must be a byllion, right? Just like how a thousand million is a billion? Aha! And that's where the -yllion system begins to differ from the -illions. We already know that 10¹² can be called a myriad million, so why not make a byllion a myllion myllion (10¹⁶) instead? We can still denote any number below it, the number before a byllion would thus be "Ninety-nine ninety-nine myriad ninety-nine ninety-nine myllion ninety-nine ninety-nine myriad ninety-nine ninety-nine", or 9999,9999;9999,9999. Under the -illion system this would be "Nine quadrillion nine hundred ninety-nine trillion nine hundred ninety-nine billion nine hundred ninety-nine million nine hundred ninety-nine thousand nine hundred ninety-nine", which is 58 characters longer.

We need a new separating symbol for a byllion: Knuth uses a colon (:). Of course, the astute among you might realise that we can currently represent any number up to a byllion byllion (10³²), which we can then call a tryllion. Let's assess the -yllion system so far:

As you can tell, each -yllion is equal to the last one squared, instead of the -illion system where each is equal to the last multiplied by a thousand. This means it grows much much quicker:

To be exact, each -yllion number n is equal to 10²ⁿ⁺². Knuth gives an example to demonstrate larger numbers: The total number of ways to shuffle a pack of cards (52!, or 52×51×50×49...) is 8065::8175,1709;4387,8571:6606,3685;6403,7669;;7528,9505;4408,8327:7824,0000;0000,0000. This is equal to 8.066×10⁶⁷, or around eighty hundred sixty-five quadryllion. Under the -illion system it would be around 80 unvigintillion, which is the 21st -illion.

I know you're keen to see how far we can take this. The 20th -yllion, known as a vigintyllion, is equal to 10²²², or 10^419,4304. The 30th -yllion, known as a trigintyllion, is equal to 10²³², or 10^42;9496,7296.

We can determine from this that every 10th -yllion is roughly equal to 10¹⁰³ⁿ, and likewise every 100th -yllion is roughly equal to 10¹⁰³⁰ⁿ, although it's worth noting that Knuth only went as far as the 20th in his original source. The 1000th -yllion, known as a millyllion, would be approximately equal to 10¹⁰³⁰¹, or a 1 followed by over a sextyllion digits.

At this point the commonly accepted extention of the -illions is from Jonathan Bowers, who uses the SI prefixes milli-, micro-, nano- and so on to represent powers of 1000 for the -illion number. Thus the millionth -illion is commonly accepted to be a micrillion (10^3,000,003), the billionth -illion is commonly accepted to be a nanillion (10^{3,000,000,003}), and so on. Since the -yllion system is based on powers of the myriad, it would be wise to instead devise our own system. The 10,000th -illion is already known as a myrillion, and so it makes sense for the myriadth -yllion to be a myryllion, approximately equal to 10¹⁰³⁰¹⁰. Since a myriad myriads is a myllion, the myllionth -yllion could be named a myllyllion, approximately equal to 10^103010,3000.

You could of course extend it again with a myryllyllion (the myriadth -yllion number) and a myllyllyllion (the myllionth -yllion number), and then again with a myllyllyllyllion, and so on. This sequence grows tetrationally; Each member of the sequence is roughly equal to 10 to the power of 10 to the power of the last. For example, a myllyllyllyllyllion (what a name) would be approximately equal to 10↑↑10, or 10 tetrated to the 10th.

Update 28/12/23: After going back through Knuth's original essay on the topic I realised I had missed his own solution to tetrational growth, although it is a bit more unwieldy. He names numbers of the form 10²ⁿ⁺² as "latin{n}yllion", where {n} refers to the name for n with spaces removed. This means that "Latinoneyllion" would be 10²¹⁺² = 10²³, which is a myllion. Likewise, "Latintwoyllion" would be equal to a byllion, and so on. Knuth gives the example of "Latinbyllionyllion", or 10^{210,000,000,000,000,002}, which is identical to the Byllyllion named earlier. The final example he gives is a "latinlatinlatinbyllionyllionyllionyllion", approximately equal to 10↑↑8 with "byllyllyllyllion" as its equivalent name in the above system.

And I think that's a good place to stop. Feel free to devise your own extension to this system if you'd like!