-yllion

Look up -yllion in Wiktionary, the free dictionary.

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 âˆ’ 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 âˆ’ 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 âˆ’ 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 ${\displaystyle 10^{2^{n+2}}}$. "One trigintyllion" (${\displaystyle 10^{2^{32}}}$) 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" (${\displaystyle 10^{2^{102}}}$) 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 10³⁺ⁿ)), "myriad scale" (new number name every 10⁴ⁿ), and "mid scale" (new number name every 10⁸ⁿ). 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.

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 ${\displaystyle 10^{2^{202}}}$.

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

For instance, ${\displaystyle 10^{-4}}$ is a myriadth. ${\displaystyle 10^{-16777216}}$ is a vigintyllionth.

• Nicolas Chuquet â€“ French mathematician (c.1445–c.1455 – c.1488–1500)
• Jacques Pelletier du Mans â€“ French humanist, poet, mathematician (1517–1582)
• Knuth's up-arrow notation â€“ Method of notation of very large integers
• The Sand Reckoner â€“ Work by Archimedes