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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 of large numbers|names for much larger numbers]]. In addition to providing an extended range, ''-yllion'' also dodges the [[long and short scales|long and short scale]] ambiguity of -illion.
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| Knuth's digit grouping is [[Scientific notation|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 numerals|Chinese numeral systems]], in which units stand for 10<sup>4</sup>, 10<sup>8</sup>, 10<sup>16</sup>, 10<sup>32</sup>, ..., 10<sup>2''n''</sup>, and so on.
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| {{Numeral systems}}
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| ==Details and examples==
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| For a more extensive table, see '''[[List_of_numbers#Myriad_system|Myriad system]]'''.
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| {| class="wikitable"
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| |-
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| ! Value !! Name !! Notation
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| |-
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| | 10<sup>0</sup>
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| | align="center" | One
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| | 1
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| |-
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| | 10<sup>1</sup>
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| | align="center" | Ten
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| | 10
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| |-
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| | 10<sup>2</sup>
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| | align="center" | Hundred
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| | 100
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| |-
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| | 10<sup>3</sup>
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| | align="center" | Ten hundred
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| | 1000
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| |-
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| | 10<sup>4</sup>
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| | align="center" | Myriad
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| | 1,0000
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| |-
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| | 10<sup>5</sup>
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| | align="center" | Ten myriad
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| | 10,0000
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| |-
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| | 10<sup>6</sup>
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| | align="center" | Hundred myriad
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| | 100,0000
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| |-
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| | 10<sup>7</sup>
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| | align="center" | Ten hundred myriad
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| | 1000,0000
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| |-
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| | 10<sup>8</sup>
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| | align="center" | Myllion
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| | 1;0000,0000
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| |-
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| | 10<sup>12</sup>
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| | align="center" | Myriad myllion
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| | 1,0000;0000,0000
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| |-
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| | 10<sup>16</sup>
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| | align="center" | Byllion
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| | 1:0000,0000;0000,0000
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| |-
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| | 10<sup>24</sup>
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| | align="center" | Myllion byllion
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| | 1;0000,0000:0000,0000;0000,0000
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| |-
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| | 10<sup>32</sup>
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| | align="center" | Tryllion
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| | 1 0000,0000;0000,0000:0000,0000;0000,0000
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| |-
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| | 10<sup>64</sup>
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| | align="center" | Quadryllion
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| | 1'0000,0000;0000,0000:0000,0000;0000,0000 0000,0000;0000,0000:0000,0000;0000,0000
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| |-
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| | 10<sup>128</sup>
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| | align="center" | Quintyllion
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| |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
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| |-
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| | 10<sup>256</sup>
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| | align="center" | Sextyllion
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| |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
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| |-
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| |}
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| In Knuth's ''-yllion'' proposal:
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| *1 to 999 have their usual names.
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| *1000 to 9999 are divided before the 2nd-last digit and named "''[[Placeholder name|foo]]'' hundred ''bar''." (e.g. 1234 is "twelve hundred thirty-four"; 7623 is "seventy-six hundred twenty-three")
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| *10<sup>4</sup> to 10<sup>8</sup>-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."
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| *10<sup>8</sup> to 10<sup>16</sup>-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."
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| *10<sup>16</sup> to 10<sup>32</sup>-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."
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| *etc.
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| 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.
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| Abstractly, then, "one <var>n</var>-yllion" is <math>10^{2^{n+2}}</math>. "One trigintyllion" (<math>10^{2^{32}}</math>) 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).
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| ==See also==
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| *Alternatives to Knuth's proposal that date back to the [[French Renaissance]] came from [[Nicolas Chuquet]] and [[Jacques Peletier du Mans]].
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| *A related proposal by Knuth is [[Knuth's up-arrow notation|his up-arrow notation]].
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| *[[The Sand Reckoner]]
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| == References ==
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| * Donald E. Knuth. ''Supernatural Numbers'' in [[The Mathematical Gardener]] (edited by [[David A. Klarner]]). Wadsworth, Belmont, CA, 1981. 310—325.
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| * Robert P. Munafo. ''[http://mrob.com/pub/math/largenum-2.html#yllion The Knuth -yllion Notation]'' ([http://www.webcitation.org/65ijZePQD Archived] 2012-02-25), 1996-2012.
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| {{Donald Knuth navbox}}
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| {{DEFAULTSORT:Yllion}}
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| [[Category:Suffixes]]
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| [[Category:Numerals]]
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| [[Category:Mathematical notation]]
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| [[Category:Large integers]]
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| [[Category:Donald Knuth]]
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