View source for -yllion

{{short description|Mathematical notation}}
{{Numeral systems}}
'''-yllion''' (pronounced {{IPAc-en|aI|lj|@n}})<ref>{{Cite web|url=http://mrob.com/pub/math/largenum-2.html#yllion|title = Large Numbers (Page 2) at MROB}}</ref> is a proposal from [[Biography:Donald Knuth|Donald Knuth]] for the terminology and symbols of an alternate [[Decimal|decimal]] superbase{{cfn|date=October 2022}} system. In it, he adapts the familiar English terms for [[Large numbers|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.

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#Large numbers|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<sup>''n''</sup></sup>, 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 10<sup>4</sup>, 10<sup>8</sup>, 10<sup>12</sup>, 10<sup>16</sup>, and so on.

==Details and examples==
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")
*10<sup>4</sup> to 10<sup>8</sup>&nbsp;− 1 are divided before the 4th-last digit and named "''foo'' [[Myriad|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."
*10<sup>8</sup> to 10<sup>16</sup>&nbsp;− 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."
*10<sup>16</sup> to 10<sup>32</sup>&nbsp;− 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 <var>n</var>-yllion" is <math>10^{2^{n+2}}</math>. "One trigintyllion" (<math>10^{2^{32}}</math>) would have 2<sup>32</sup> + 1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional "trigintillion" has merely 94 digits &mdash; not even a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, "one centyllion" (<math>10^{2^{102}}</math>) would have 2<sup>102</sup> + 1, or 507,0602;4009,1291:7605,9868;1282,1505, or about 1/20 of a tryllion digits, whereas a conventional "[[Centillion|centillion]]" has only 304 digits.

The corresponding [[Chinese numerals#Large numbers|Chinese "long scale" numerals]] are given, with the traditional form listed before the simplified form. Same numerals are used in the Chinese "short scale" (new number name every power of 10 after 1000 (or 10<sup>3+''n''</sup>)), "myriad scale" (new number name every 10<sup>4''n''</sup>), and "mid scale" (new number name every 10<sup>8''n''</sup>).  Today these numerals are still in use, but are used in their "myriad scale" values, which is also used in [[Japanese numerals#Powers of 10|Japanese]] and in [[Korean numerals#Cardinal numerals|Korean]].  For a more extensive table, see '''[[Philosophy:Numeral (linguistics)#Myriad, Octad, and -yllion systems|Myriad system]]'''. 
{| class="wikitable"
|-
! Value !! Name !! Notation 
!Standard English name (short scale)!! Chinese ("long scale") || Pīnyīn ([[Social:Standard Chinese|Mandarin]]) !! Jyutping (Cantonese) !! Pe̍h-ōe-jī ([[Social:Hokkien|Hokkien]])
|-
| 10<sup>0</sup>
| align="center" | One
| 1
|One
| 一
| yī
| jat<sup>1</sup>
| it/chit
|-
| 10<sup>1</sup>
| align="center" | Ten
| 10
|Ten
| 十
| shí
| sap<sup>6</sup>
| si̍p/cha̍p 
|-
| 10<sup>2</sup>
| align="center" | One hundred
| 100
|One hundred
| 百
| bǎi
| baak<sup>3</sup>
| pah
|-
| 10<sup>3</sup>
| align="center" | Ten hundred
| 1000
|One thousand
| 千
| qiān
| cin<sup>1</sup>
| chhian
|-
| 10<sup>4</sup>
| align="center" | One myriad
| 1,0000
|Ten thousand
| 萬, 万
| wàn
| maan<sup>6</sup>
| bān
|-
| 10<sup>5</sup>
| align="center" | Ten myriad
| 10,0000
|One hundred thousand
| 十萬, 十万
| shíwàn
| sap<sup>6</sup> maan<sup>6</sup>
| si̍p/cha̍p bān
|-
| 10<sup>6</sup>
| align="center" | One hundred myriad
| 100,0000
|One million
| 百萬, 百万
| bǎiwàn
| baak<sup>3</sup> maan<sup>6</sup>
| pah bān
|-
| 10<sup>7</sup>
| align="center" | Ten hundred myriad
| 1000,0000
|Ten million
| 千萬, 千万
| qiānwàn
| cin<sup>1</sup> maan<sup>6</sup>
| chhian bān
|-
| 10<sup>8</sup>
| align="center" | One myllion
| 1;0000,0000
|One hundred million
| 億, 亿
| yì
| jik<sup>1</sup>
| ek
|-
|10<sup>9</sup>
| align="center" | Ten myllion
|10;0000,0000
|One billion
| 十億, 十亿
| shíyì
| sap<sup>6</sup> jik<sup>1</sup>
| si̍p/cha̍p ek
|-
| 10<sup>12</sup>
| align="center" | One myriad myllion
| 1,0000;0000,0000
|One trillion
| 萬億, 万亿
| wànyì
| maan<sup>6</sup> jik<sup>1</sup>
| bān ek
|-
| 10<sup>16</sup>
| align="center" | One byllion
| 1:0000,0000;0000,0000
|Ten quadrillion
| 兆
| zhào
| siu<sup>6</sup>
| tiāu
|-
| 10<sup>24</sup>
| align="center" | One myllion byllion
| 1;0000,0000:0000,0000;0000,0000
|One septillion
| 億兆, 亿兆
| yìzhào
| jik<sup>1</sup> siu<sup>6</sup>
| ek tiāu
|-
| 10<sup>32</sup>
| align="center" | One tryllion
| 1'0000,0000;0000,0000:0000,0000;0000,0000
|One hundred nonillion
| 京
| jīng
| ging<sup>1</sup>
| kiaⁿ
|-
| 10<sup>64</sup>
| align="center" | One quadryllion
|
|Ten vigintillion
| 垓
| gāi
| goi<sup>1</sup>
| kai
|-
| 10<sup>128</sup>
| align="center" | One quintyllion
|
|One hundred unquadragintillion
| 秭
| zǐ
| zi<sup>2</sup>
| chi
|-
| 10<sup>256</sup>
| align="center" | One sextyllion
|
|Ten quattuoroctogintillion
| 穰
| ráng
| joeng<sup>4</sup>
| liōng
|-
| 10<sup>512</sup>
| align="center" | One septyllion
|
|One hundred novensexagintacentillion
| 溝, 沟
| gōu
| kau<sup>1</sup>
| kau
|-
| 10<sup>1024</sup>
| align="center" | One octyllion
|
|Ten quadragintatrecentillion
| 澗, 涧
| jiàn
| gaan<sup>3</sup>
| kán
|-
| 10<sup>2048</sup>
| align="center" | One nonyllion
|
|One hundred unoctogintasescentillion
| 正
| zhēng
| zing<sup>3</sup>
| chiàⁿ
|-
| 10<sup>4096</sup>
| align="center" | One decyllion
|
|Ten milliquattuorsexagintatrecentillion
| 載, 载
| zài
| zoi<sup>3</sup>
| chài
|}

==Latin- prefix==
In order to construct names of the form <var>n</var>-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 <math>10^{2^{202}}</math>.

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

For instance, <math>10^{-4}</math>is a ''myriadth.''
<math>10^{-16777216}</math> is a ''vigintyllionth.''

==See also==
* {{Annotated link|Nicolas Chuquet|space_cat=no}}
* {{Annotated link|Jacques Pelletier du Mans}}
* {{Annotated link|Knuth's up-arrow notation}}
* {{Annotated link|The Sand Reckoner}}

==References==
{{Reflist}}
* Donald E. Knuth. ''Supernatural Numbers'' in The Mathematical Gardener (edited by [[Biography:David A. Klarner|David A. Klarner]]). Wadsworth, Belmont, CA, 1981. 310&mdash;325.
* Robert P. Munafo. ''[http://mrob.com/pub/math/largenum-2.html#yllion The Knuth -yllion Notation]'' ( {{Webarchive|url=https://web.archive.org/web/20120213070143/http://mrob.com/pub/math/largenum-2.html#yllion |date=2012-02-13 }} 2012-02-25), 1996–2012.

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