-yllion

-yllion (pronounced ) 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, ..., 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

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 $10^$. "One trigintyllion" ($10^$) 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" ($10^$) 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 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 and in Korean. For a more extensive table, see Myriad system.

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 $10^$.

Negative powers

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

For instance, $10^$ is a ''myriadth.''
$10^$ is a ''vigintyllionth.''

Disadvantages

Knuth's system wouldn't be implemented well in Polish due to some numerals having the ''-ylion'' suffix in basic forms due to Polish-language rules, which change the syllables ''-ti-'', ''-ri-'', ''-ci-'' into ''-ty-'', ''-ry-'', ''-cy-'' in adapted loanwoards, present in all "-illions" above a billion (e.g. ''trylion'' as ''trillion'', ''kwadrylion'' as ''quadrillion'', ''kwintylion'' as ''quintillion'' etc; ''nonilion'' as ''nonnillion'' is the only exception, but also not always), causing ambiguity for numbers above 10³²-1.

See also

*
*
*
*

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
' ( 2012-02-25), 1996–2012.

{{DEFAULTSORT:Yllion
Scientific suffixes
Numerals
Mathematical notation
Large integers
Donald Knuth