# -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 10^{4}, 10^{8}, 10^{16}, 10^{32}, ..., 10^{2n}, and so on. Today the corresponding characters are used for 10^{4}, 10^{8}, 10^{12}, 10^{16}, and so on.

Numeral systems |
---|

Hindu–Arabic numeral system |

East Asian |

Alphabetic |

Former |

Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

## Details and examples[edit]

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.

Value | Name | Notation | Chinese | Pīnyīn (Mandarin) | Jyutping (Cantonese) | Pe̍h-ōe-jī (Hokkien) |
---|---|---|---|---|---|---|

10^{0} |
One | 1 | 一 | yī | jat^{1} |
it/chit |

10^{1} |
Ten | 10 | 十 | shí | sap^{6} |
si̍p/tsa̍p |

10^{2} |
Hundred | 100 | 百 | bǎi | baak^{3} |
pah |

10^{3} |
Ten hundred | 1000 | 千 | qiān | cin^{1} |
chhian |

10^{4} |
Myriad | 1,0000 | 萬, 万 | wàn | maan^{6} |
bān |

10^{5} |
Ten myriad | 10,0000 | 十萬, 十万 | shíwàn | sap^{6} maan^{6} |
si̍p/tsa̍p bān |

10^{6} |
Hundred myriad | 100,0000 | 百萬, 百万 | bǎiwàn | baak^{3} maan^{6} |
pah bān |

10^{7} |
Ten hundred myriad | 1000,0000 | 千萬, 千万 | qiānwàn | cin^{1} maan^{6} |
chhian bān |

10^{8} |
Myllion | 1;0000,0000 | 億, 亿 | yì | jik^{1} |
ik |

10^{12} |
Myriad myllion | 1,0000;0000,0000 | 萬億, 万亿 | wànyì | maan^{6} jik^{1} |
bān ik |

10^{16} |
Byllion | 1:0000,0000;0000,0000 | 兆 | zhào | siu^{6} |
tiāu |

10^{24} |
Myllion byllion | 1;0000,0000:0000,0000;0000,0000 | 億兆, 亿兆 | yìzhào | jik^{1} siu^{6} |
ik tiāu |

10^{32} |
Tryllion | 1'0000,0000;0000,0000:0000,0000;0000,0000 | 京 | jīng | ging^{1} |
kiann |

10^{64} |
Quadryllion | 垓 | gāi | goi^{1} |
gai | |

10^{128} |
Quintyllion | 秭 | zǐ | zi^{2} |
tsi | |

10^{256} |
Sextyllion | 穰 | ráng | joeng^{4} |
liōng | |

10^{512} |
Septyllion | 溝, 沟 | gōu | kau^{1} |
kau | |

10^{1024} |
Octyllion | 澗, 涧 | jiàn | gaan^{3} |
kán | |

10^{2048} |
Nonyllion | 正 | zhēng | zing^{3} |
tsiànn | |

10^{4096} |
Decyllion | 載, 载 | zài | zoi^{3} |
tsài |

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") - 10
^{4}to 10^{8}− 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." - 10
^{8}to 10^{16}− 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." - 10
^{16}to 10^{32}− 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[edit]

- 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[edit]

- 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.