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-yllion (pronounced/aɪljən/)^[1] is a proposal fromDonald Knuth for the terminology and symbols of an alternatedecimal superbase^{[clarification needed]} system. In it, he adapts the familiar English terms forlarge numbers to provide a systematic set ofnames for much larger numbers. In addition to providing an extended range,-yllion also dodges thelong and short scale ambiguity of -illion.

Knuth's digit grouping isexponential 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-unusedChinese numeral systems, in which units stand for 10⁴, 10⁸, 10¹⁶, 10³², ..., 10²ⁿ, and so on (with an exception that the -yllion proposal does not use a word forthousand which the original Chinese numeral system has). Today the corresponding Chinese characters are used for 10⁴, 10⁸, 10¹², 10¹⁶, and so on.

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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 hundredbar." (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 "foomyriadbar". 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 myllionbar", 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 byllionbar", 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, "onen-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 correspondingChinese "long scale" numerals are given, with thetraditional form listed before thesimplified 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 inJapanese and inKorean. For a more extensive table, seeMyriad system.

In order to construct names of the formn-yllion for large values ofn, Knuth appends the prefix "latin-" to the name ofn without spaces and uses that as the prefix forn. For example, the number "latintwohundredyllion" corresponds ton = 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 amyriadth.${\displaystyle {10}^{-16777216}}$ is avigintyllionth.

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

• Donald E. Knuth.Supernatural Numbers inThe Mathematical Gardener (edited byDavid A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325.
• Robert P. Munafo.The Knuth -yllion Notation (Archived 2012-02-13 at theWayback Machine 2012-02-25), 1996–2012.

• Scientific suffixes
• Numerals
• Mathematical notation
• Large integers
• Donald Knuth

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