Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar English terms for
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...
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
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
ambiguity of -illion.
Knuth's digit grouping is
exponential
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
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, ..., 102''n'', and so on (with an exception that the -yllion proposal does not use a word for
thousand
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
A group of one thousand th ...
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
The terms foobar (), foo, bar, baz, and others are used as metasyntactic variables and placeholder names in computer programming or computer-related documentation. - Etymology of "Foo" They have been used to name entities such as variables, f ...
'' 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 . "One trigintyllion" () 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" () 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
Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...
traditional
A tradition is a belief or behavior (folk custom) passed down within a group or society with symbolic meaning or special significance with origins in the past. A component of cultural expressions and folklore, common examples include holidays or ...
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 104''n''), and "mid scale" (new number name every 108''n''). Today these numerals are still in use, but are used in their "myriad scale" values, which is also used in
Japanese
Japanese may refer to:
* Something from or related to Japan, an island country in East Asia
* Japanese language, spoken mainly in Japan
* Japanese people, the ethnic group that identifies with Japan through ancestry or culture
** Japanese diaspor ...
and in
Korean
Korean may refer to:
People and culture
* Koreans, ethnic group originating in the Korean Peninsula
* Korean cuisine
* Korean culture
* Korean language
**Korean alphabet, known as Hangul or Chosŏn'gŭl
**Korean dialects and the Jeju language
** ...
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 .
Negative powers
To refer to small quantities with this system, the suffix ''-th'' is used.
For instance, is a ''myriadth.''