Appendix:Knuth numeric system

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The suffix -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, and so on.

Details and examples[edit]

For a more extensive table, see Myriad system.

Value Name Notation
100 One 1
101 Ten 10
102 Hundred 100
103 Ten hundred 1000
104 Myriad 1,0000
105 Ten myriad 10,0000
106 Hundred myriad 100,0000
107 Ten hundred myriad 1000,0000
108 Myllion 1;0000,0000
1012 Myriad myllion 1,0000;0000,0000
1016 Byllion 1:0000,0000;0000,0000
1024 Myllion byllion 1;0000,0000:0000,0000;0000,0000
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


In Knuth's -yllion proposal:

  • 1 to 99 have their usual names.
  • 100 to 9999 are divided before the 2nd-last digit and named "blah hundred blah". (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 "blah myriad blah". Knuth also introduces at this level a grouping symbol (comma) for the numeral. So, 382,1902 is "3 hundred 82 myriad 19 hundred 2".
  • 108 to 1016-1 are divided before the 8th-last digit and named "blah myllion blah", and a semicolon separates the digits. So 1,0002;0003,0004 is "1 myriad 2 myllion, 3 myriad 4".
  • 1016 to 1032-1 are divided before the 16th-last digit and named "blah byllion blah", and a colon separates the digits. So 12:0003,0004;0506,7089 is "12 byllion, 3 myriad 4 myllion, 5 hundred 6 myriad 70 hundred 89".
  • 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!)

See also[edit]

  • Alternatives to Knuth's proposal that date back to the French Renaissance came from Nicolas Chuquet and Jacques Peletier du Mans.
  • A related proposal by Knuth is his up-arrow notation.

References[edit]

  • Donald E. Knuth. Supernatural Numbers in The Mathematical Gardener (edited by D. A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325.
  • Robert P. Munafo. Large Numbers, 1996-2008.