Appendix:Knuth numeric system

The suffix -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, 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, and so on.

For a more extensive table, see Myriad system.

In Knuth's -yllion proposal:

• 1 to 99 have their usual names.
• 100 to 9999 are divided before the 2nd-last digit and named "blah hundred blah". (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 "blah myriad blah". Knuth also introduces at this level a grouping symbol (comma) for the numeral. So, 382,1902 is "3 hundred 82 myriad 19 hundred 2".
• 10⁸ to 10¹⁶-1 are divided before the 8th-last digit and named "blah myllion blah", and a semicolon separates the digits. So 1,0002;0003,0004 is "1 myriad 2 myllion, 3 myriad 4".
• 10¹⁶ to 10³²-1 are divided before the 16th-last digit and named "blah byllion blah", and a colon separates the digits. So 12:0003,0004;0506,7089 is "12 byllion, 3 myriad 4 myllion, 5 hundred 6 myriad 70 hundred 89".
• 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 nearly forty-three myllion (4300 million) digits. (By contrast, a conventional "trigintillion" has merely 94 digits — not even a hundred, let alone a hundred million!)

• 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.

• Donald E. Knuth. Supernatural Numbers in The Mathematical Gardner (edited by D. A. Klarner). Wadsworth, Belmont, CA, 1981. 310—325. https://doi.org/10.1007/978-1-4684-6686-7_28
• Robert P. Munafo. Large Numbers, 1996-2008.