-yllion

-yllion (pronounced /aɪljən/) 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 (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

In Knuth's -yllion proposal:

• 1 to 999 still 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 2³² + 1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional "trigintillion" has merely 94 digits — not even a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, "one centyllion" ($10^{2^{102}}$) 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 "long scale" numerals are given, with the traditional form listed before the 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 10³⁺ⁿ)), "myriad scale" (new number name every 10⁴ⁿ), and "mid scale" (new number name every 10⁸ⁿ). Today these Chinese numerals are still in use, but are used in their "myriad scale" values, which is also used in Japanese and in Korean. For a more extensive table, see Myriad system.

Value	Name	Notation	Standard English name (short scale)	Ancient Greek	Chinese ("long scale")	Pīnyīn (Mandarin)	Jyutping (Cantonese)	Pe̍h-ōe-jī (Hokkien)
100	One	1	One	εἷς (heîs)	一	yī	jat1	it/chit
101	Ten	10	Ten	δέκα (déka)	十	shí	sap6	si̍p/cha̍p
102	One hundred	100	One hundred	ἑκατόν (hekatón)	百	bǎi	baak3	pah
103	Ten hundred	1000	One thousand	χίλιοι (khī́lioi)	千	qiān	cin1	chhian
104	One myriad	1,0000	Ten thousand	μύριοι (mýrioi)	萬, 万	wàn	maan6	bān
105	Ten myriad	10,0000	One hundred thousand	δεκάκις μύριοι (dekákis mýrioi)	十萬, 十万	shíwàn	sap6 maan6	si̍p/cha̍p bān
106	One hundred myriad	100,0000	One million	ἑκατοντάκις μύριοι (hekatontákis mýrioi)	百萬, 百万	bǎiwàn	baak3 maan6	pah bān
107	Ten hundred myriad	1000,0000	Ten million	χιλιάκις μύριοι (khiliákis mýrioi)	千萬, 千万	qiānwàn	cin1 maan6	chhian bān
108	One myllion	1;0000,0000	One hundred million	μυριάκις μύριοι (muriákis mýrioi)	億, 亿	yì	jik1	ek
109	Ten myllion	10;0000,0000	One billion	δεκάκις μυριάκις μύριοι (dekákis muriákis mýrioi)	十億, 十亿	shíyì	sap6 jik1	si̍p/cha̍p ek
1010	One hundred myllion	100;0000,0000	Ten billion	ἑκατοντάκις μυριάκις μύριοι (hekatontákis muriákis múrioi)	百億, 百亿	bǎiyì	baak3 jik1	pah ek
1011	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	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	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	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	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	One byllion	1:0000,0000;0000,0000	Ten quadrillion	μυριάκις μυριάκις μυριάκις μύριοι (muriákis muriákis muriákis mýrioi)	兆	zhào	siu6	tiāu
1024	One myllion byllion	1;0000,0000:0000,0000;0000,0000	One septillion		億兆, 亿兆	yìzhào	jik1 siu6	ek tiāu
1032	One tryllion	1'0000,0000;0000,0000:0000,0000;0000,0000	One hundred nonillion		京	jīng	ging1	kiaⁿ
1064	One quadryllion		Ten vigintillion		垓	gāi	goi1	kai
10128	One quintyllion		One hundred unquadragintillion		秭	zǐ	zi2	chi
10256	One sextyllion		Ten quattuoroctogintillion		穰	ráng	joeng4	liōng
10512	One septyllion		One hundred novensexagintacentillion		溝, 沟	gōu	kau1	kau
101024	One octyllion		Ten quadragintatrecentillion		澗, 涧	jiàn	gaan3	kán
102048	One nonyllion		One hundred unoctogintasescentillion		正	zhēng	zing3	chiàⁿ
104096	One decyllion		Ten milliquattuorsexagintatrecentillion		載, 载	zài	zoi3	chài

Latin- prefix

In order to construct names of the form n-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 $10^{2^{202}}$.

Negative powers

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

For instance, $10^{-4}$ is a myriadth. $10^{-16777216}$ is a vigintyllionth.

Disadvantages

Knuth's system wouldn't be implemented well in Polish due to some numerals having -ylion suffix in basic forms due to rule of Polish language, which changes syllables -ti-, -ri-, -ci- into -ty-, -ry-, -cy- in adapted loanwoards, present in all thousands powers from trillion upwards, e.g. trylion as trillion, kwadrylion as quadrillion, kwintylion as quintillion etc. (nonilion as nonnillion is only exception, but also not always), which creates system from 10³² upwards invalid.

See also

• Nicolas Chuquet – Mathematician
• Jacques Pelletier du Mans – Humanist, Poet, Mathematician
• Knuth's up-arrow notation – Method of notation of very large integers
• The Sand Reckoner – Work by Archimedes

References

• Donald E. Knuth. Supernatural Numbers in The Mathematical Gardener (edited by David A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325.
• Robert P. Munafo. The Knuth -yllion Notation ( Archived 2012-02-13 at the Wayback Machine 2012-02-25), 1996–2012.