-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. Today the corresponding characters are used for 10⁴, 10⁸, 10¹², 10¹⁶, and so on.

Details and examples

For a more extensive table, see Myriad system. The corresponding Chinese numerals are given, with the traditional form listed before the simplified form. Today these numerals are still in use, but are used for different values.

Value	Name	Notation	Chinese	Pīnyīn (Mandarin)	Jyutping (Cantonese)	Pe̍h-ōe-jī (Hokkien)
10^0	One	1	一	yī	jat^1	it/chit
10^1	Ten	10	十	shí	sap^6	si̍p/tsa̍p
10^2	Hundred	100	百	bǎi	baak^3	pah
10^3	Ten hundred	1000	千	qiān	cin^1	chhian
10^4	Myriad	1,0000	萬, 万	wàn	maan^6	bān
10^5	Ten myriad	10,0000	十萬, 十万	shíwàn	sap^6 maan^6	si̍p/tsa̍p bān
10^6	Hundred myriad	100,0000	百萬, 百万	bǎiwàn	baak^3 maan^6	pah bān
10^7	Ten hundred myriad	1000,0000	千萬, 千万	qiānwàn	cin^1 maan^6	chhian bān
10^8	Myllion	1;0000,0000	億, 亿	yì	jik^1	ik
10^12	Myriad myllion	1,0000;0000,0000	萬億, 万亿	wànyì	maan^6 jik^1	bān ik
10^16	Byllion	1:0000,0000;0000,0000	兆	zhào	siu^6	tiāu
10^24	Myllion byllion	1;0000,0000:0000,0000;0000,0000	億兆, 亿兆	yìzhào	jik^1 siu^6	ik tiāu
10^32	Tryllion	1'0000,0000;0000,0000:0000,0000;0000,0000	京	jīng	ging^1	kiann
10^64	Quadryllion		垓	gāi	goi^1	gai
10^128	Quintyllion		秭	zǐ	zi^2	tsi
10^256	Sextyllion		穰	ráng	joeng^4	liōng
10^512	Septyllion		溝, 沟	gōu	kau^1	kau
10^1024	Octyllion		澗, 涧	jiàn	gaan^3	kán
10^2048	Nonyllion		正	zhēng	zing^3	tsiànn
10^4096	Decyllion		載, 载	zài	zoi^3	tsài

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 $10^{{2^{{n+2}}}}$. "One trigintyllion" ($10^{{2^{{32}}}}$) 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 thousand million, and still 7 digits short of a googol).

See also

• Alternatives to Knuth's proposal that date back to the French Renaissance came from Nicolas Chuquet and Jacques Peletier du Mans.
• A related proposal by Knuth is his up-arrow notation.
• The Sand Reckoner

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.