-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.

Numeral systems
Hindu–Arabic numeral system
Western Arabic Eastern Arabic Bengali Gurmukhi Indian Sinhala Tamil Balinese Burmese Dzongkha Javanese Khmer Lao Mongolian Thai
East Asian
Chinese Suzhou Hokkien Japanese Korean Vietnamese Counting rods
Alphabetic
Abjad Armenian Āryabhaṭa Cyrillic Ge'ez Georgian Greek Hebrew Roman
Former
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit Kharosthi Mayan Muisca Quipu Prehistoric
Positional systems by base
2 3 4 5 6 8 10 12 16 20 60
Non-standard positional numeral systems
Bijective numeration (1) Signed-digit representation (Balanced ternary) factorial negative Complex-base system (2i) Non-integer representation (φ) mixed
List of numeral systems

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.

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 . "One trigintyllion" () 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.