-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 Ancient 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	Ancient Chinese	Pīnyīn (Mandarin)	Jyutping (Cantonese)	Pe̍h-ōe-jī (Hokkien)
100	One	1	一	yī	jat1	it/chit
101	Ten	10	十	shí	sap6	si̍p/tsa̍p
102	Hundred	100	百	bǎi	baak3	pah
103	Ten hundred	1000	千	qiān	cin1	chhian
104	Myriad	1,0000	萬, 万	wàn	maan6	bān
105	Ten myriad	10,0000	十萬, 十万	shíwàn	sap6 maan6	si̍p/tsa̍p bān
106	Hundred myriad	100,0000	百萬, 百万	bǎiwàn	baak3 maan6	pah bān
107	Ten hundred myriad	1000,0000	千萬, 千万	qiānwàn	cin1 maan6	chhian bān
108	Myllion	1;0000,0000	億, 亿	yì	jik1	ik
1012	Myriad myllion	1,0000;0000,0000	萬億, 万亿	wànyì	maan6 jik1	bān ik
1016	Byllion	1:0000,0000;0000,0000	兆	zhào	siu6	tiāu
1024	Myllion byllion	1;0000,0000:0000,0000;0000,0000	億兆, 亿兆	yìzhào	jik1 siu6	ik tiāu
1032	Tryllion	1'0000,0000;0000,0000:0000,0000;0000,0000	京	jīng	ging1	kiann
1064	Quadryllion		垓	gāi	goi1	gai
10128	Quintyllion		秭	zǐ	zi2	tsi
10256	Sextyllion		穰	ráng	joeng4	liōng
10512	Septyllion		溝, 沟	gōu	kau1	kau
101024	Octyllion		澗, 涧	jiàn	gaan3	kán
102048	Nonyllion		正	zhēng	zing3	tsiànn
104096	Decyllion		載, 载	zài	zoi3	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.