1.09. Extensions to the -illions III: Knuth's Yllions and More Extensible Systems

Introduction

We are now familiar with a variety of illion schemes: Henkle's, Conway's, Rowlett's, and Bowers' systems. All of those pose advantages and disadvantages, and they vary greatly in extensibility. However, there's a clever trick that fixes many of those problems, which was devised by Donald Knuth when he presented his -yllion scheme. Although that trick can be applied to the usual -illions, since it was devised together with the -yllions I will discuss the two in tandem.

Donald Knuth (born 1938) is a famed mathematician, computer scientist, and former professor at Stanford University. He is one of the most well-known computer scientists, considered the "father of the analysis of algorithms". He has written many books and papers, including a series "The Art of Computer Programming" and many others.^[1] Among other things, he is also the inventor of the famous up-arrow notation, one of the most well-known large number notations. Not only do up-arrows provide one of the most natural and intuitive ways to denote large numbers, they also have an additional claim to fame as they're used in the definition of the popular large number Graham's number, and the two are often discussed together in popular large number discussions; we'll learn about both of them in section 2 of this site.

In his essay Supernatural Numbers, part of the recreational mathematics book The Mathematical Gardner which is made of several passages written by various people, Knuth explains that when we examine the English numbers as we know them today, they clearly could have been designed much better.^[2] To show how the English naming system can be changed to be much more efficient with names, he invented an -yllion scheme that names numbers much more efficiently than the normal -illions do. But to find out how he does that, let's first look at the history behind schemes like Knuth's, particularly the Chinese numerals.

Chinese Numerals

Most English speakers think of Chinese as a strange and exotic language, one with no aspects that one could hope to relate to the familiar English. While it is true that Chinese is radically different from English with its entirely different system for writing and its unusual pronunciation systems like tones which can distinguish words (a well-known example is that tones are the only difference between the words for "four" and "death", leading to 4's unlucky connotation in Chinese culture), the numbers in Chinese are actually a lot simpler than English numbers!

Chinese, like English, gives unique names to the numbers one through ten (yi, er, san, si, wu, liu, qi, ba, jiu, shi respectively). But after this point it starts showing its difference between the English numbers, combining names more so than English does. Compare the names for numbers in English and Chinese:

In Chinese, the very next unique name for numbers after "shi" for ten is "bai" for 100. Then you can form names such as "bai yi" = 101, "er bai" = 200, or "liu bai liu shi liu" = 666. Then you can do just about the same thing with "qian" for 1000. For example, in Chinese 1729 is "qian qi bai er shi jiu". Now, unlike in English, Chinese has its own word for 10,000—it is "wan". But after "wan", there was at one point variance in the names, and this is where we get to four different systems.

After "wan", in all four Chinese systems, comes the names "yi" (differentiated in tone from the word for one), "zhao", "jing", "gai", "zi", "rang", "gou", "jian", "zheng", and "zai". What varies between the systems is what numbers the names refer to.

In the first system, each name is ten times the previous, leading to the numbers:

yi = 100,000

zhao = 1,000,000

jing = 10,000,000

gai = 100,000,000

zi = 10⁹

rang = 10¹⁰

gou = 10¹¹

jian = 10¹²

zheng = 10¹³

zai = 10¹⁴

This system was proposed in the 6th century book "Wujing suanshu" (Arithmetic in Five Classics). Clearly, it's not very efficient, and it has today fallen out of use.

In the second system, each name is 10,000 times the previous, leading to the numbers:

yi = 10⁸

zhao = 10¹²

jing = 10¹⁶

gai = 10²⁰

zi = 10²⁴

rang = 10²⁸

gou = 10³²

jian = 10³⁶

zheng = 10⁴⁰

zai = 10⁴⁴

Now this is a lot more efficient than the previous system. The largest name, zai, refers to 10⁴⁴ (known as 100 tredecillion in English), which is already so large that it's just not a number we hear of much in life at all. Note that this system mirrors the short scale of -illions. The second system won out the other systems, and it's the system that is used in modern Chinese.

In the third system, each name is 100,000,000 times the previous number, leading to the names:

yi = 10⁸

zhao = 10¹⁶

jing = 10²⁴

gai = 10³²

zi = 10⁴⁰

rang = 10⁴⁸

gou = 10⁵⁶

jian = 10⁶⁴

zheng = 10⁷²

zai = 10⁸⁰

This system is to the second system as the long scale is to the short scale. Although it's still more efficient with names than the second, like the first it eventually fell out of use.

Now the fourth system is particularly interesting: each name is the square of the previous number. This leads to the names:

yi = 10⁸

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.

Knuth's Naming Systen

Knuth's -yllion naming system begins with the usual English number names up to 999 (nine hundred ninety nine). However, after this point things start to change up. In the -yllion system, 1000 is not called "one thousand", but rather "ten hundred". All numbers from 100 to 9999 are named "x hundred y". For example:

1001 = ten hundred one

2048 = twenty hundred forty eight

7776 = seventy-seven hundred seventy-six

Then, "ten thousand" is instead referred to as a "myriad". It is denoted 1,0000 in this system; note that the commas group digits by fours instead of by threes. Then, from 1,0000 to 9999,9999 we name numbers "x myriad y". For example:

1,9683 = one myriad ninety-six hundred eighty-three

100,0000 = one hundred myriad (sounds a lot like the Greek word for "million", doesn't it?)

1677,7216 = sixteen hundred seventy-seven myriad seventy-two hundred sixteen

Now after this point Knuth's system starts to get interesting. Knuth proposes the name "myllion" (pronounced /mɑɪljən/ (mile-yun) to distinguish from "million") for what we'd call 100 million. This is the same as "yi" in the fourth Chinese naming system. In the -yllion system a myllion is denoted 1;0000,0000. Note the usage of a semicolon here. This is so that you can easily read a semicolon as "myllion" and a comma as "myriad". For example, 3;0021,0000 is three myllion twenty-one myriad. All numbers from a myllion to 9.999...*10¹⁵ (called 9.999... quadrillion using normal English numbers) are then named "x myllion y", where x and y can be any number less than a myllion. Here are some examples:

11;0011,0011 = eleven myllion eleven myriad eleven

1,0000;0000,0000 = one myriad myllion

1,3000;0023,0000 = one myriad thirty hundred myllion twenty-three myriad

1350,5245;7019,3626 = thirteen hundred fifty myriad fifty hundred forty-five myllion seventy hundred nineteen myriad thirty-six hundred twenty-six

Analogous to a myllion, a byllion, pronounced /bɑɪljən/ (bile-yun), is 10¹⁶, which is known in English as 10 quadrillion and in the fourth Chinese system as "zhao". In the -yllions, a byllion is denoted 1:0000,0000;0000,0000. Just like we use a semicolon to group myllions, we use a colon to group byllions. Here are some examples of numbers we can name with Knuth's system:

1:0001,0001;0001,0001 = one byllion one myriad one myllion one myriad one — think of "one myriad one myllion" as multiplying "one myriad one" by "myllion".

1;0000,0000:0003,0000;0000,0202 = one myllion byllion three myriad myllion two hundred two

2000,0000;0000,0000:0000,0000;0000,0000 = twenty hundred myriad myllion byllion

9999,9999;9999,9999:9999,9999;9999,9999 = ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine myllion ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine byllion ninety-nine hundred ninety-nine ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine myllion ninety-nine hundred ninety-nine myriad ninety-nine hundred ninety-nine

Also, note that this system also takes away the ambiguity between the short and long scales!

Continuing the pattern, a tryllion is 10³² (also known as "100 nonillion" in English or "jing" in the fourth Chinese system). You can tell that these names are very efficient, and they still work. There are several punctuation marks that have been used to separate tryllions by different people. Knuth himself has used a double semicolon (;;),^[4] so that a tryllion would be denoted 1;;0000,0000;0000,0000:0000,0000;0000,0000. However, it is also popular to use apostrophes to separate tryllions, so that a tryllion would be denoted 1'0000,0000;0000,0000:0000,0000;0000,0000. Some examples of numbers we can name are:

1;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000 — one myllion tryllion

1000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000 — ten hundred myriad myllion byllion tryllion. In English this is known as a vigintillion.

Now think about this for a second. With Knuth's system, we need only four names for powers of 10 past 100 to get all the way to 64-digit numbers, in contrast with 21 such names in English. And 64-digit numbers are something that we just don't hear of much in life at all! But this is still the beginning of what numbers Knuth's system can easily take us to...

A quadryllion is 10⁶⁴, denoted

1::0000,0000;0000,0000:0000,0000;0000,0000;;0000,0000;0000,0000:0000,0000;0000,0000 using Knuth's system, using double colons to separate quadryllions.^[4] However I suggest using '⁴ to separate quadryllions, since the name "quadryllion" comes from "quadri-", the Latin root for 4. With that idea we can write a quadryllion as:

1'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

Using -yllions up to quadryllions we can already give a name to a googol. In Knuth's yllions a googol is called one myriad tryllion quadryllion. Written out using this system a googol is:

1,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000

:0000,0000;0000,0000

A quintyllion is of course 10¹²⁸, which we can denote:

1'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000

:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

Although Knuth's essay implies that quintyllions would be separated with ;;; following the pattern of ; : ;; ::, for the sake of this article we'll continue the pattern of apostrophes followed by a superscript number.

Note that we have now already gone way beyond numbers that the English short scale allows us to name, and we've even gone beyond what we can name using the canonical long scale (limited at 9.999...*10¹²⁵ = 999 vigintilliard 999 vigintillion ... ... 999 thousand 999).

A sextyllion is 10²⁵⁶ or:

1'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

Sextyllions are enough to give a name to the tremendous centillion (10³⁰³). A centillion is ten hundred myriad myllion tryllion sextyllion.

Septyllion = 10⁵¹² =

1'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

Octyllion = 10¹⁰²⁴ =

1'⁸0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

Nonyllion = 10²⁰⁴⁸ =

1'⁹0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁸0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

And decyllion = 10⁴⁰⁹⁶ =

1'¹⁰0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁸0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁹0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁸0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁷0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁶0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁵0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000'⁴0000,0000;0000,0000:0000,0000;0000,0000'0000,0000;0000,0000:0000,0000;0000,0000

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?

In his essay, Knuth continues his system with the -yllion names:

undecyllion = 10⁸¹⁹²
duodecyllion = 10^16,384
tredecyllion = 10^32,768
quattuordecyllion = 10^65,536
quindecyllion = 10^131,072
sexdecyllion = 10^262,144
septendecyllion = 10^524,288
octodecyllion = 10^1,048,576
novemdecyllion = 10^2,097,152
vigintyllion = 10^4,194,304 (the largest -yllion name he mentions in his list of names)

These are some pretty nice numbers for several reasons. First off these numbers, are really HUGE. They serve as some nice benchmarks for numbers we've examined earlier. For example:

• my upper-bound of all possible chess games is comparable to a tredecyllion

• my version of the Hamlet monkey number (with a 47-key typewriter) is between a sexdecyllion and a septendecyllion

• Borges' number (the number of books in the Library of Babel) is between an octodecyllion and a novemdecyllion

• Henkle's milli-millillion is between a novemdecyllion and a vigintyllion

Now think about that last one for a moment. When we started examining extensions to the -illions, the milli-millillion seemed like quite a mighty number, didn't it? Well, turns out that using Knuth's system we only need twenty -yllions to transcend the milli-millillion! The vigintyllion is such a different kind of number from its measly -illion counterpart which has merely 63 zeros.

There's one last point I'd like to make before we move on. Since powers of two are a popular thing to memorize among number enthusiasts, the -yllions are easy to remember if you know your powers of two. That forms a notable advantage to the many of us who have memorized powers of two. I myself have them memorized up to the first twenty and a few higher ones. If you know what the xth power of 2 is, then that's the number of zeros in the (x-2)th -yllion, (from the sequence myllion, byllion, tryllion ... ).

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.

Intermission: Why don't we use a system like Knuth's?

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 10^{2^n} 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.

As you can see, a system like Knuth's has both advantages and disadvantages. Although the advantages allow for quite an elegant system, the disadvantages are big enough that such a system has not really become mainstream anywhere.

Let's continue with how we can logically extend Knuth's -yllions.

Extending Knuth's -yllions

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 = 10^{2^23}
duovigintyllion = 10^16,777,216 = 10^{2^24}
trevigintyllion = 10^33,554,432 = 10^{2^25}
...
trigintyllion = 10^{4,294,967,296} = 10^{2^32}
...
quadragintyllion = 10^{4,398,046,511,104} = 10^{2^42}
...
all the way up to centyllion = 10^{5,070,602,400,912,917,605,986,812,821,504} = 10^{2^102}

Now a centyllion is another pretty cool number. It has about 5.07 nonillion digits. This means that if you want to write it out in full, you'd need something like 500 sextillion dictionaries. To put that in perspective, if you were to stack up all these dictionaries (assuming each is about 3 in / 7.5 cm thick) they'd reach 1.58 million light-years high, vastly exceeding puny things like the orbit of Pluto or the diameter of the Milky Way. Such a stack of dictionaries would reach about halfway to the Andromeda galaxy!

What should we do after a centyllion? Many have extended Knuth's -yllions analogous to systems like Conway and Guy's, Bowers', etc (example), making names such as millyllion or micryllion. However, I find that usage doesn't really match with the idea of -yllions so much as a system where you give unique names to the myriadth, myllionth, byllionth, etc, yllions. How will we do that? Here is my proposal.

First, I suggest that the 101st -yllion (equal to 10²¹⁰³, the square of a centyllion) be called a centimyllion, followed by centibyllion, centitryllion, and so on, and in general, centi-x-yllion is the 100+xth -yllion. We can do the same to the 200th -yllion, which I'll call "duocentyllion". Then we can continue with the 300th, 400th, etc, -yllions being named tricentyllion, quadricentyllion, quinticentyllion, sexticentyllion, and so on, continuing with the same roots we used to name the -yllions. For example the 4096th -yllion would be a quadraginticentisexnonagintyllion.

Then for the myriadth -yllion I propose the name myryllion. That number is equal to 10^{2^10,002}, or about 10^{8*10^3010}. It has roughly a millillion digits, or about an octyllion nonyllion digits in terms of Knuth's -yllions. Yes, even the number of digits is now an unfathomable number, greater than the number of Planck volumes in the observable universe, dwarfed by a factor of itself fifteen times! But this is still just the beginning ...

After a myryllion we have names like myrimyllion, myribyllion ... myridecyllion ... myricentyllion ... myridecicentyllion ... you get the idea. In general, x-myri-y-yllion is the x*10,000+yth -yllion, so we can have, say, viginticentivigintimyrisextyllion for the 20,200,006th (2020,0006th using Knuth's notation) -yllion.

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.

Higher-tiered -yllions: The latin-x-yllion system

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^{2^1;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".

It's strange how obscure this "latin-x-yllion" system really is. Although Knuth's -yllions are themselves a popular topic, before this article was written "latinmyllionyllion" had no results on Google! I can't really find an explanation for why the latin-x-yllion system is so much less known than the -yllion system itself, but nonetheless it provides a useful way to continue to more insane heights.

But before we do so, let's take a moment to think about how clever this system is. It totally dodges the problems of systems with a similar idea. For example, think back to the extension of Rowlett's system, and the problems with a name like "tetradekakisekatommyrillion"—does that refer to the 4*10³³th -illion or the 10⁴⁵th -illion in that system? The only way we could fix that is by using hyphens, which creates notable pronunciation issues.

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".

And that system can take us to insane heights alright. For example, a "latinquadryllionyllion" is 10^{2^(10^64+2)}, which is already larger than Skewes' number! And that's only the beginning ... just imagine names like "latincentyllionyllion" = 10^{2^(10^2^102+2)}, which itself transcends even the largest -illion nameable in Bowers' system! Everything that seemed mighty just a little while ago is now totally left in the dust!!! Just imagine crazy names like:

latinduocentyllionyllion

latinlatintwomillionthreemyriadyllionyllion

latinlatinmyryllionyllionyllion

and a whole bunch of complex names, for example:

latintwolatinfivemyllionyllionthreelatintwomyllionthreehundredmyriadtwohundredtwentytwoyllionfiveduocentitryllionlatinmyllionyllionthreemyryllionseventeensexdecylliontwohundredtwentytwomyllionsextyllionseventymylliontwomyriadeightyninehundredfortyoneyllion

That is exactly equal to:

10^{2^(2*10^2^5;0000,0002 + 3*10^2^2;0300,0024 + 5*10^(2^1;0000,0002+2^205) + 3*10^2^1,0002 + 17*10^26,2144 + 222*10^264 + 70;0002,8943)}

or approximately:

10^{10^10^10^150,000,000}

Wow ... that's just a number that utterly transcends anything we've previously encountered! There are 150 million digits in the number of digits in the number of digits in the number of digits in this number. If that's head-spinning, then that's good! Almost anyone would consider this a strange monstrous number, as it's bigger than a googolplex, or a googolduplex, or even the mind-boggling googoltriplex! And that itself pales in comparison to something like:

latinlatinlatin ... (100 latins total) ... latinlatinlatinmyllionyllionyllion ... (101 yllions total) ... yllion

which is equal to about:

10^{10^10^... ...^10^301,030,000}

where there are 200 10's.

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.

Applying the "latin" idea to the -illions

The latin-x-yllion is all in all a cleverly designed system, but its one problem to those who want to extend the -illions as far as they can is that it's applied to the -yllions. So can we apply that idea to the -illions? Of course we can.

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:

ten latinthreeduotrigintillionthreehundredthirtythreeuntrigintillionthreehundredthirtythreetrigintillionthreehundredthirtythreenovemvigintillionthreehundredthirtythreeoctovigintillionthreehundredthirtythreeseptenvigintillionthreehundredthirtythreesexvigintillionthreehundredthirtythreequinvigintillionthreehundredthirtythreequattuorvigintillionthreehundredthirtythreetrevigintillionthreehundredthirtythreeduovigintillionthreehundredthirtythreeunvigintillionthreehundredthirtythreevigintillionthreehundredthirtythreenovemdecillionthreehundredthirtythreeoctodecillionthreehundredthirtythreeseptendecillionthreehundredthirtythreesexdecillionthreehundredthirtythreequindecillionthreehundredthirtythreequattuordecillionthreehundredthirtythreetredecillionthreehundredthirtythreeduodecillionthreehundredthirtythreeundecillionthreehundredthirtythreedecillionthreehundredthirtythreenonillionthreehundredthirtythreeoctillionthreehundredthirtythreeseptillionthreehundredthirtythreesextillionthreehundredthirtythreequintillionthreehundredthirtythreequadrillionthreehundredthirtythreetrillionthreehundredthirtythreebillionthreehundredthirtythreemillionthreehundredthirtythreethousandthreehundredthirtytwoillion

This is nothing strange or unnatural at all. And plus, we can take our number names all the way up to things like:

latinlatinlatin ... (100 latins total) ... latinlatinlatinmillionillionillion ... (101 illions total) ... illion

which would create a number equal to about:

10^{10^10^... ...^10^3,000,003}

where there are 100 10's.

Now that diagonal tower of tens is half as tall as the tower for its -yllion counterpart, but of course this isn't any less unfathomable of a number.

Limit of the System?

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:

latinlatinlatin ... (x latins total) ... latinlatinlatinmyllionyllionyllion ... (x+1 yllions total) ... yllion

where x is approximately the maximum number of letters that could fit in the observable universe. Obviously it's really hard to calculate such a maximum, but we can upper-bound it if we let x be the number of Planck volumes in the observable universe. In that case we would get the number:

latinlatinlatin ... (3*10^185 latins total) ... latinlatinlatinmyllionyllionyllion ... (3*10^185+1 yllions total) ... yllion

How big would that number be?! It would be approximately:

10^{10^10^... ...^10^301,030,000}

where there are 6*10^185 10's. That number is simply INSANE, and may be beginning to hint at numbers we'll examine in section 2.

Numbers very much like the number above may be considered the practical limit of the system. You may be wondering: is it possible to go STILL FURTHER?!?! This is exactly what we'll get to next.

Going even further?!

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", 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:

latinlatinlatin ... (myllion latins total) ... latinlatinlatinmyllionyllionyllion ... (myllion yllions total*) ... yllion

and then you can call that number X, and have a number equal to:

latinlatinlatin ... (X latins total) ... latinlatinlatinmyllionyllionyllion ... (X yllions total*) ... yllion

* Once again, not counting the "yllion" that's part of the name "myllion".

That is an utterly insane number alright, probably tricky for you to even get straight! It could be more compactly called:

"latinlatinmyllionalatinmyllionlatinmyllionayllionalatinmyllionlatinmyllionalatinmyllionlatinmyllionayllionayllion"

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

"latinlatinmyllionalatinmyllionlatinmyllionyllionalatinmyllionlatinmyllionalatinmyllionlatinmyllionyllionyllion"

really seem like that short of a name, even if it's one of the shortest names we can make by feeding numbers nameable with the compression system onto itself?! And that brings us to my most important point of all, a review of the general idea of extending -illions.

Conclusion

Before we review what we've learned, you may be wondering: why extend the -illions in the first place?! Extending -illions is obviously by no means a practical endeavor, but we've already established that the main goal of systems like this is as a fun little exercise to see how far you can logically extend the -illions. With that said we can now review the whole idea of -extending -illions.

There are two ways extensions to the -illions must end. The first is with systems like Bowers', where there is a specific limit to numbers the system can name. The second is with systems like Conway and Guy's or Knuth's, where there is no limit to numbers the system can name. However, systems of the second type, as we saw, have a limit to numbers with reasonable-length names. You can extend systems of either type to however big heights you want, but in systems of the second type there must always be a point where the numbers are beyond any names that could actually ever be written out! Besides, there is only a finite number of total possible short names for numbers (let's say "short" refers to a name made of less than 100 characters) anyway, so even if we made a system that continuously makes as-short-as-possible milestone names, we would inevitably at some point have exhausted all possible short names. That isn't even accounting for the fact that most of the names would be unpronounceable or hard to distinguish from one another!

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.

Sources

1.10. Review I