-yllion

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-yllion is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase 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. Today the corresponding characters are used for 104, 108, 1012, 1016, and so on.

Details and examples[edit]

For a more extensive table, see Myriad system. The corresponding Ancient Chinese numerals are given, with the traditional form listed before the simplified form. Today these numerals are still in use, but are used for different values.

Value Name Notation Ancient Chinese Pīnyīn (Mandarin) Jyutping (Cantonese) Pe̍h-ōe-jī (Hokkien)
100 One 1 jat1 it/chit
101 Ten 10 shí sap6 si̍p/tsa̍p
102 Hundred 100 bǎi baak3 pah
103 Ten hundred 1000 qiān cin1 chhian
104 Myriad 1,0000 萬, 万 wàn maan6 bān
105 Ten myriad 10,0000 十萬, 十万 shíwàn sap6 maan6 si̍p/tsa̍p bān
106 Hundred myriad 100,0000 百萬, 百万 bǎiwàn baak3 maan6 pah bān
107 Ten hundred myriad 1000,0000 千萬, 千万 qiānwàn cin1 maan6 chhian bān
108 Myllion 1;0000,0000 億, 亿 jik1 ik
1012 Myriad myllion 1,0000;0000,0000 萬億, 万亿 wànyì maan6 jik1 bān ik
1016 Byllion 1:0000,0000;0000,0000 zhào siu6 tiāu
1024 Myllion byllion 1;0000,0000:0000,0000;0000,0000 億兆, 亿兆 yìzhào jik1 siu6 ik tiāu
1032 Tryllion 1'0000,0000;0000,0000:0000,0000;0000,0000 jīng ging1 kiann
1064 Quadryllion gāi goi1 gai
10128 Quintyllion zi2 tsi
10256 Sextyllion ráng joeng4 liōng
10512 Septyllion 溝, 沟 gōu kau1 kau
101024 Octyllion 澗, 涧 jiàn gaan3 kán
102048 Nonyllion zhēng zing3 tsiànn
104096 Decyllion 載, 载 zài zoi3 tsài

In Knuth's -yllion proposal:

  • 1 to 999 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 10^{2^{n+2}}. "One trigintyllion" (10^{2^{32}}) would have 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).

See also[edit]

References[edit]