-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 Equivalent Ancient Chinese numeral
100 One 1
101 Ten 10
102 Hundred 100
103 Ten hundred 1000
104 Myriad 1,0000 萬, 万
105 Ten myriad 10,0000 十萬, 十万
106 Hundred myriad 100,0000 百萬, 百万
107 Ten hundred myriad 1000,0000 千萬, 千万
108 Myllion 1;0000,0000 億, 亿
1012 Myriad myllion 1,0000;0000,0000 萬億, 万亿
1016 Byllion 1:0000,0000;0000,0000
1024 Myllion byllion 1;0000,0000:0000,0000;0000,0000 億兆, 亿兆
1032 Tryllion 1 0000,0000;0000,0000:0000,0000;0000,0000
1064 Quadryllion 1'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000
10128 Quintyllion 1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
10256 Sextyllion 1 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
10512 Septyllion 溝, 沟
101024 Octyllion 澗, 涧
102048 Nonyllion
104096 Decyllion 載, 载

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 hundred million, and still 7 digits short of a googol).

See also[edit]

References[edit]