Conway's zillion numbers

Written by Fish on July 25, 2021. Last update on February 17, 2023.

In this page, English names of -illion numbers with Conway-Wechsler system is defined and listed. Online converter is also available.

• Definition
  • Conway-Wechsler system
  • For N≥1000
  • Modified system
  • Reference
• Software
• List of 1,200 zillion numbers
• List of 10,000 zillion numbers (another page)

Definition

Conway-Wechsler system

By extending the standard -illion system for naming large numbers, million, billion, trillion, and so on, Conway and Guy published a system for naming N'th zillion numbers for N<1000. N-th zillion number is 10^3N+3 for short scale and 10^6N for long scale. See long and short scale for detail. Robert Munafo explains Conway-Wechsler system and examples in detail. Standard English name is used for N<10, and Conway wrote

You can find a name for any zillion from the 10th to the 999th by combining parts from the appropriate columns of the following table, and then replacing the final vowel by "illion":

^*Note: when it is immediately before a component marked with ^S or ^X, "tre" increases to "tres" and "se" to "ses" or "sex" as appropriate. Similarly "septe" and "nove" increase to "septem" and "novem" or "septen" and "noven" immediately before components marked with ^M or ^N.

^**quin is changed from original quinqua.

As Robert explains, the Conway-Wechsler system disagrees some dictionary name. Miakinen proposed that as quindecillion is a common name and Latin name of 15 is quindecim, not quinquadecim, quinqua should be replaced to quin, and a similar change to all the Conway-Wechsler names involving the quinqua- prefix should be updated. Robert adopted the suggestion, and call this modified version (quin) as the Conway-Wechsler system. I also follow this convention. Note that Miakinen also proposed to replace letters of e to é in some cases for French spelling, but it is irrelevant in English.

For N≥1000

Conway and Guy (1996) extended this system with Allan Wechsler for N≥1000 as follows.

With Allan Wechsler we propose to extend this system indefinitely by combining these according to the convention that "XilliYilliZillion" (say) denotes the (1000000X + 1000Y + Z)th zillion, using "nillion" for the zeroth "zillion" when this is needed as a placeholder. So for example the million-and-third zillion is a "millinillitrillion."

Modified system

Even after changing quinqua to quin, Conway-Wechsler names have some difference from the common names, for example, sedecillion and novendecillion are better known as sexdecillion and novemdecillion respectively. Many people simplified the Conway-Wechsler system, where Cookiefonster summarized as follows.

This way we don't need to memorize when to put a s, x, m, or n where, and it's faithful to the dictionary -illions.

It was pointed out that the modified system is invalid, because N=103 and N=300 have same name trecentillion. Therefore, using the modified system is discouraged.

Reference

• Conway and Guy (1995) "The book of Numbers" Copernicus

Software

Self-made software for Python was used to produce the definition tables and the list of numbers in this page, and therefore they should be consistent. The software is published here under the MIT license. It can be used to obtain arbitrary N'th zillion number by using the extended system, as Python can handle arbitrary large integer.

How to use: Download zillion.py and make it executable. Then run the program with

where N means N'th zillion number. When 2 Ns are given, all values between them are shown. When more than 2 Ns are given, each values are shown. -m option uses the modified system. For other options, use -h option for help. For installation, just rename the file to zillion and copy the file to your system path.

It was confirmed that the output of zillion.py N for N<1000 exactly matches the list by Steve Olsen.

This program was last update on March 3, 2022.

Online converter is also available.

List of 1,200 zillion numbers

N'th zillion number is 10^3N+3 for short scale and 10^6N for long scale.