-yllion

-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, ..., 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 characters are used for 10⁴, 10⁸, 10¹², 10¹⁶, 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")
• 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 ${\displaystyle 10^{2^{n+2}}}$. "One trigintyllion" (${\displaystyle 10^{2^{32}}}$) 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" (${\displaystyle 10^{2^{102}}}$) 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 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 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)	Chinese ("long scale")	Pīnyīn (Mandarin)	Jyutping (Cantonese)	Pe̍h-ōe-jī (Hokkien)
10^0	One	1	One	一	yī	jat^1	it/chit
10^1	Ten	10	Ten	十	shí	sap^6	si̍p/cha̍p
10^2	One hundred	100	One hundred	百	bǎi	baak^3	pah
10^3	Ten hundred	1000	One thousand	千	qiān	cin^1	chhian
10^4	One myriad	1,0000	Ten thousand	萬, 万	wàn	maan^6	bān
10^5	Ten myriad	10,0000	One hundred thousand	十萬, 十万	shíwàn	sap^6 maan^6	si̍p/cha̍p bān
10^6	One hundred myriad	100,0000	One million	百萬, 百万	bǎiwàn	baak^3 maan^6	pah bān
10^7	Ten hundred myriad	1000,0000	Ten million	千萬, 千万	qiānwàn	cin^1 maan^6	chhian bān
10^8	One myllion	1;0000,0000	One hundred million	億, 亿	yì	jik^1	ek
10^9	Ten myllion	10;0000,0000	One billion	十億, 十亿	shíyì	sap^6 jik^1	si̍p/cha̍p ek
10^12	One myriad myllion	1,0000;0000,0000	One trillion	萬億, 万亿	wànyì	maan^6 jik^1	bān ek
10^16	One byllion	1:0000,0000;0000,0000	Ten quadrillion	兆	zhào	siu^6	tiāu
10^24	One myllion byllion	1;0000,0000:0000,0000;0000,0000	One septillion	億兆, 亿兆	yìzhào	jik^1 siu^6	ek tiāu
10^32	One tryllion	1'0000,0000;0000,0000:0000,0000;0000,0000	One hundred nonillion	京	jīng	ging^1	kiaⁿ
10^64	One quadryllion		Ten vigintillion	垓	gāi	goi^1	kai
10^128	One quintyllion		One hundred unquadragintillion	秭	zǐ	zi^2	chi
10^256	One sextyllion		Ten quattuoroctogintillion	穰	ráng	joeng^4	liōng
10^512	One septyllion		One hundred novensexagintacentillion	溝, 沟	gōu	kau^1	kau
10^1024	One octyllion		Ten quadragintatrecentillion	澗, 涧	jiàn	gaan^3	kán
10^2048	One nonyllion		One hundred unoctogintasescentillion	正	zhēng	zing^3	chiàⁿ
10^4096	One decyllion		Ten milliquattuorsexagintatrecentillion	載, 载	zài	zoi^3	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 ${\displaystyle 10^{2^{202}}}$.

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 2012-02-25), 1996-2012.