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{{short description|Mathematical notation}}

[[Donald Knuth]]

{{More footnotes|date=August 2019}}

adapted the familiar naming scheme to handle much larger numbers, dodging ambiguity by changing the -illion to -yllion.

{{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 [[Donald Knuth]] for the terminology and symbols of an alternate [[decimal]] superbase{{cfn|date=October 2022}} 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.

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 3three or 6six 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 10⁴, 10⁸, 10¹⁶, 10³², ..., 10^{2^''n''}, 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 10⁴, 10⁸, 10¹², 10¹⁶, and so on.

==Details and examples==

*1 to 99 have their usual names. (In fact 1&ndash;999 have their usual names, and will be used below to save space; but to emphasize the pattern, this group is separate.)

{{cleanup lang|date=September 2021}}

*100 to 9999 are divided before the 2nd-last digit and named "''blah'' hundred ''blah''". (eg 1234 is "twelve hundred thirty-four"; 7623 is "seventy-six hundred twenty-three")

{{Wiktionary|-yllion}}

*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 "382 myriad 19 hundred 2".

In Knuth's ''-yllion'' proposal:

*10⁸ to 10¹⁶-1 are divided before the 8th-last digit and and named "''blah'' myllion ''blah''", and a semicolon separates the digits. So 1,0002;0003,0004 is "1 myriad 2 myllion 3 myriad 4"

*1 to 999 still have their usual names.

*10¹⁶ to 10³²-1 are divided before the 16th-last digit and and named "''blah'' byllion ''blah''", and a colon separates the digits. So 12:0003,0004;0506,7089 is "12 byllion 3 myriad 4 myllion 506 myriad 70 hundred 89"

*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")

*etc

*10⁴ to 10⁸&nbsp;− 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&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."

*10¹⁶ to 10³²&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 2³² + 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 2¹⁰² + 1, or 507,0602;4009,1291:7605,9868;1282,1505, or about 1/20 of a tryllion digits, whereas a conventional "[[centillion]]" has only 304 digits.

The corresponding [[Chinese numerals#Large numbers|Chinese "long scale" numerals]] are given, with the [[traditional Chinese|traditional]] form listed before the [[simplified Chinese|simplified form]]. Same numerals are used in the Ancient Greek numeral system, and also the Chinese "short scale" (new number name every power of 10 after 1000 (or 10^3+''n'')), "myriad scale" (new number name every 10^4''n''), and "mid scale" (new number name every 10^8''n''). Today these Chinese 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 '''[[Numeral_(linguistics)#Myriad,_Octad,_and_-yllion_systems|Myriad system]]'''.

See also [[Knuths up-arrow notation]].

{| class="wikitable"

|-

! Value !! Name !! Notation

!Standard English name (short scale)!! Ancient Greek !! Chinese ("long scale") || Pīnyīn ([[Standard Chinese|Mandarin]]) !! Jyutping ([[Standard Cantonese|Cantonese]]) !! Pe̍h-ōe-jī ([[Hokkien]])

|-

| 10⁰

| align="center" | One

| 1

|One

| εἷς (heîs)

| 一

| yī

| jat¹

| it/chit

|-

| 10¹

| align="center" | Ten

| 10

|Ten

| δέκα (déka)

| 十

| shí

| sap⁶

| si̍p/cha̍p

|-

| 10²

| align="center" | One hundred

| 100

|One hundred

| ἑκατόν (hekatón)

| 百

| bǎi

| baak³

| pah

|-

| 10³

| align="center" | Ten hundred

| 1000

|One thousand

| χίλιοι (khī́lioi)

| 千

| qiān

| cin¹

| chhian

|-

| 10⁴

| align="center" | One myriad

| 1,0000

|Ten thousand

| μύριοι (mýrioi)

| 萬, 万

| wàn

| maan⁶

| bān

|-

| 10⁵

| align="center" | Ten myriad

| 10,0000

|One hundred thousand

| δεκάκις μύριοι (dekákis mýrioi)

| 十萬, 十万

| shíwàn

| sap⁶ maan⁶

| si̍p/cha̍p bān

|-

| 10⁶

| align="center" | One hundred myriad

| 100,0000

|One million

| ἑκατοντάκις μύριοι (hekatontákis mýrioi)

| 百萬, 百万

| bǎiwàn

| baak³ maan⁶

| pah bān

|-

| 10⁷

| align="center" | Ten hundred myriad

| 1000,0000

|Ten million

| χιλιάκις μύριοι (khiliákis mýrioi)

| 千萬, 千万

| qiānwàn

| cin¹ maan⁶

| chhian bān

|-

| 10⁸

| align="center" | One myllion

| 1;0000,0000

|One hundred million

| μυριάκις μύριοι (muriákis mýrioi)

| 億, 亿

| yì

| jik¹

| ek

|-

|10⁹

| align="center" | Ten myllion

|10;0000,0000

|One billion

| δεκάκις μυριάκις μύριοι (dekákis muriákis mýrioi)

| 十億, 十亿

| shíyì

| sap⁶ jik¹

| si̍p/cha̍p ek

|-

|10¹⁰

| align="center" | One hundred myllion

|100;0000,0000

| Ten billion

| ἑκατοντάκις μυριάκις μύριοι (hekatontákis muriákis múrioi)

| 百億, 百亿

| bǎiyì

| baak³ jik¹

| pah ek

|-

|10¹¹

| align="center" | Ten hundred myllion

|1000;0000,0000

| One hundred billion

| χῑλῐάκῐς μυριάκις μύριοι (khīliákis muriákis múrioi)

| 千億, 千亿

| qiānyì

| cin¹ jik¹

| chhian ek

|-

| 10¹²

| align="center" | One myriad myllion

| 1,0000;0000,0000

|One trillion

| μυριάκις μυριάκις μύριοι (muriákis muriákis mýrioi)

| 萬億, 万亿

| wànyì

| maan⁶ jik¹

| bān ek

|-

| 10¹³

| align="center" | Ten myriad myllion

| 10,0000;0000,0000

| Ten trillion

| δεκάκις μυριάκις μυριάκις μύριοι (dekákis muriákis muriákis mýrioi)

| 十萬億, 十万亿

| shíwànyì

| sap⁶ maan⁶ jik¹

| si̍p/cha̍p bān ek

|-

| 10¹⁴

| align="center" | One hundred myriad myllion

| 100,0000;0000,0000

| One hundred trillion

| ἑκατοντάκις μυριάκις μυριάκις μύριοι (hekatontákis muriákis muriákis mýrioi)

| 百萬億, 百万亿

| bǎiwànyì

| baak³ maan⁶ jik¹

| pah bān ek

|-

| 10¹⁵

| align="center" | Ten hundred myriad myllion

| 1000,0000;0000,0000

| One quadrillion

| χιλιάκις μυριάκις μυριάκις μύριοι (khiliákis muriákis muriákis mýrioi)

| 千萬億, 千万亿

| qiānwànyì

| cin¹ maan⁶ jik¹

| chhian bān ek

|-

| 10¹⁶

| align="center" | One byllion

| 1:0000,0000;0000,0000

|Ten quadrillion

| μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis mýrioi)

| 兆

| zhào

| siu⁶

| tiāu

|-

| 10²⁴

| align="center" | One myllion byllion

| 1;0000,0000:0000,0000;0000,0000

|One septillion

| μυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis muriákis mýrioi)

| 億兆, 亿兆

| yìzhào

| jik¹ siu⁶

| ek tiāu

|-

| 10³²

| align="center" | One tryllion

| 1'0000,0000;0000,0000:0000,0000;0000,0000

|One hundred nonillion

| μυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis muriákis muriákis muriákis muriákis mýrioi)

| 京

| jīng

| ging¹

| kiaⁿ

|-

| 10⁶⁴

| align="center" | One quadryllion

|

|Ten vigintillion

|

| 垓

| gāi

| goi¹

| kai

|-

| 10¹²⁸

| align="center" | One quintyllion

|

|One hundred unquadragintillion

|

| 秭

| zǐ

| zi²

| chi

|-

| 10²⁵⁶

| align="center" | One sextyllion

|

|Ten quattuoroctogintillion

|

| 穰

| ráng

| joeng⁴

| liōng

|-

| 10⁵¹²

| align="center" | One septyllion

|

|One hundred novensexagintacentillion

|

| 溝, 沟

| gōu

| kau¹

| kau

|-

| 10¹⁰²⁴

| align="center" | One octyllion

|

|Ten quadragintatrecentillion

|

| 澗, 涧

| jiàn

| gaan³

| kán

|-

| 10²⁰⁴⁸

| align="center" | One nonyllion

|

|One hundred unoctogintasescentillion

|

| 正

| zhēng

| zing³

| chiàⁿ

|-

| 10⁴⁰⁹⁶

| align="center" | One decyllion

|

|Ten milliquattuorsexagintatrecentillion

|

| 載, 载

| zài

| zoi³

| chài

|}

==ExternalLatin- 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>.

[http://home.earthlink.net/~mrob/pub/math/largenum.htm Large Numbersl]

== Negative powers ==

''I really wish I had enough money to write a cheque to put one of these names on, and watch the bank's spinny&hellip;''

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 [[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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