# -yllion

-yllion (pronounced /ljən/)[1] is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase[clarification needed] system. In it, he adapts the familiar English terms for large numbers to provide a systematic set of names for much larger numbers. In addition to providing an extended range, -yllion also dodges the long and short scale ambiguity of -illion.

Knuth's digit grouping is exponential instead of linear; each division doubles the number of digits handled, whereas the familiar system only adds three or six more. His system is basically the same as one of the ancient and now-unused Chinese numeral systems, in which units stand for 104, 108, 1016, 1032, ..., 102n, and so on (with an exception that the -yllion proposal does not use a word for thousand which the original Chinese numeral system has). Today the corresponding Chinese characters are used for 104, 108, 1012, 1016, and so on.

## Details and examples

In Knuth's -yllion proposal:

• 1 to 999 still have their usual names.
• 1000 to 9999 are divided before the 2nd-last digit and named "foo hundred bar." (e.g. 1234 is "twelve hundred thirty-four"; 7623 is "seventy-six hundred twenty-three")
• 104 to 108 − 1 are divided before the 4th-last digit and named "foo myriad bar". Knuth also introduces at this level a grouping symbol (comma) for the numeral. So 382,1902 is "three hundred eighty-two myriad nineteen hundred two."
• 108 to 1016 − 1 are divided before the 8th-last digit and named "foo myllion bar", and a semicolon separates the digits. So 1,0002;0003,0004 is "one myriad two myllion, three myriad four."
• 1016 to 1032 − 1 are divided before the 16th-last digit and named "foo byllion bar", and a colon separates the digits. So 12:0003,0004;0506,7089 is "twelve byllion, three myriad four myllion, five hundred six myriad seventy hundred eighty-nine."
• etc.

Each new number name is the square of the previous one — therefore, each new name covers twice as many digits. Knuth continues borrowing the traditional names changing "illion" to "yllion" on each one. Abstractly, then, "one n-yllion" is ${\displaystyle 10^{2^{n+2}}}$. "One trigintyllion" (${\displaystyle 10^{2^{32}}}$) would have 232 + 1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional "trigintillion" has merely 94 digits — not even a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, "one centyllion" (${\displaystyle 10^{2^{102}}}$) would have 2102 + 1, or 507,0602;4009,1291:7605,9868;1282,1505, or about 1/20 of a tryllion digits, whereas a conventional "centillion" has only 304 digits.

The corresponding Chinese "long scale" numerals are given, with the traditional form listed before the simplified form. Same numerals are used in the Ancient Greek numeral system, and also the Chinese "short scale" (new number name every power of 10 after 1000 (or 103+n)), "myriad scale" (new number name every 104n), and "mid scale" (new number name every 108n). Today these Chinese numerals are still in use, but are used in their "myriad scale" values, which is also used in Japanese and in Korean. For a more extensive table, see Myriad system.

Value Name Notation Standard English name (short scale) Ancient Greek Chinese ("long scale") Pīnyīn (Mandarin) Jyutping (Cantonese) Pe̍h-ōe-jī (Hokkien)
100 One 1 One εἷς (heîs) jat1 it/chit
101 Ten 10 Ten δέκα (déka) shí sap6 si̍p/cha̍p
102 One hundred 100 One hundred ἑκατόν (hekatón) bǎi baak3 pah
103 Ten hundred 1000 One thousand χίλιοι (khī́lioi) qiān cin1 chhian
104 One myriad 1,0000 Ten thousand μύριοι (mýrioi) 萬, 万 wàn maan6 bān
105 Ten myriad 10,0000 One hundred thousand δεκάκις μύριοι (dekákis mýrioi) 十萬, 十万 shíwàn sap6 maan6 si̍p/cha̍p bān
106 One hundred myriad 100,0000 One million ἑκατοντάκις μύριοι (hekatontákis mýrioi) 百萬, 百万 bǎiwàn baak3 maan6 pah bān
107 Ten hundred myriad 1000,0000 Ten million χιλιάκις μύριοι (khiliákis mýrioi) 千萬, 千万 qiānwàn cin1 maan6 chhian bān
108 One myllion 1;0000,0000 One hundred million μυριάκις μύριοι (muriákis mýrioi) 億, 亿 jik1 ek
109 Ten myllion 10;0000,0000 One billion δεκάκις μυριάκις μύριοι (dekákis muriákis mýrioi) 十億, 十亿 shíyì sap6 jik1 si̍p/cha̍p ek
1010 One hundred myllion 100;0000,0000 Ten billion ἑκατοντάκις μυριάκις μύριοι (hekatontákis muriákis múrioi) 百億, 百亿 bǎiyì baak3 jik1 pah ek
1011 Ten hundred myllion 1000;0000,0000 One hundred billion χῑλῐάκῐς μυριάκις μύριοι (khīliákis muriákis múrioi) 千億, 千亿 qiānyì cin1 jik1 chhian ek
1012 One myriad myllion 1,0000;0000,0000 One trillion μυριάκις μυριάκις μύριοι (muriákis muriákis mýrioi) 萬億, 万亿 wànyì maan6 jik1 bān ek
1013 Ten myriad myllion 10,0000;0000,0000 Ten trillion δεκάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis mýrioi) 十萬億, 十万亿 shíwànyì sap6 maan6 jik1 si̍p/cha̍p bān ek
1014 One hundred myriad myllion 100,0000;0000,0000 One hundred trillion ἑκατοντάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis mýrioi) 百萬億, 百万亿 bǎiwànyì baak3 maan6 jik1 pah bān ek
1015 Ten hundred myriad myllion 1000,0000;0000,0000 One quadrillion χιλιάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis mýrioi) 千萬億, 千万亿 qiānwànyì cin1 maan6 jik1 chhian bān ek
1016 One byllion 1:0000,0000;0000,0000 Ten quadrillion μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis mýrioi) zhào siu6 tiāu
1024 One myllion byllion 1;0000,0000:0000,0000;0000,0000 One septillion 億兆, 亿兆 yìzhào jik1 siu6 ek tiāu
1032 One tryllion 1'0000,0000;0000,0000:0000,0000;0000,0000 One hundred nonillion jīng ging1 kiaⁿ
1064 One quadryllion Ten vigintillion gāi goi1 kai
10128 One quintyllion One hundred unquadragintillion zi2 chi
10256 One sextyllion Ten quattuoroctogintillion ráng joeng4 liōng
10512 One septyllion One hundred novensexagintacentillion 溝, 沟 gōu kau1 kau
101024 One octyllion Ten quadragintatrecentillion 澗, 涧 jiàn gaan3 kán
102048 One nonyllion One hundred unoctogintasescentillion zhēng zing3 chiàⁿ
104096 One decyllion Ten milliquattuorsexagintatrecentillion 載, 载 zài zoi3 chài

## Latin- prefix

In order to construct names of the form n-yllion for large values of n, Knuth appends the prefix "latin-" to the name of n without spaces and uses that as the prefix for n. For example, the number "latintwohundredyllion" corresponds to n = 200, and hence to the number ${\displaystyle 10^{2^{202}}}$.

## Negative powers

To refer to small quantities with this system, the suffix -th is used.

For instance, ${\displaystyle 10^{-4}}$is a myriadth. ${\displaystyle 10^{-16777216}}$ is a vigintyllionth.