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The Welsh Vigesimal Number System

Date: 02/17/2003 at 11:47:33
From: Andy Spitzley
Subject: The Welsh Vigesimal Number System

What can you tell me about the Welsh version of the Vigesimal Number 

Date: 02/17/2003 at 15:52:27
From: Doctor Jodi
Subject: Re: The Welsh Vigesimal Number System

Hi Andy,

Thanks for your question! I didn't know that the Welsh had such an 
interesting number system. It turns out that they have two sets of 
counting words: decimal and vigesimal.

I searched for "Vigesimal Welsh" and "Vigesimal Wales" at Google. 
There's not much out there. The best hope I have for you is two books 
I found recommendations of (at the end). Welsh uses vigesimal and 
decimal numbers. I didn't find any evidence of a vigesimal number 
system per se, just the continued use of counting numbers 

From Michael Morgan at 


"In Welsh, for example, 30 is ten-on-twenty, 31 is eleven-on-twenty, 
40 is two-twenties, 60 is three-twenties, 80 is four-twenties (pedwar 
ugain, cf. French quatre-vingts).

"... Welsh teens are fairly complicated: 11 is one-on-ten, 12 is 
two-ten, 13 is three-on-ten, 14 is four-on-ten, 15 is five-ten, 16 
is one-on-five-ten, 17 is two-on-five-ten, 18 is two-nines, 19 is 
four-on-five-ten. These numbers combine with the units of twenties 
mentioned above, so that a number like 97 for example, is: dau ar 
bymtheg ar bedwar ugain (two-on-five-ten-on-four-twenties). Simple 
really, and I can't understand why this system hasn't been more widely 

You can find a list of all the numbers at

   clwb malu cachu - numerals 

According to this postscript paper, 

the use of base twenty is not uncommon: "Languages that at least 
partly use this vigesimal system are: Albanian, Basque, Breton, 
Danish, Faroese, French, Irish, Scottish Gaelic and Welsh."

Someone on the archives recommended

Menninger, Karl. 1970 Number Words and Number Symbols. Cambridge, 
Mass.: MIT Press.
A bookseller includes this in its description:

"All cultures have evolved or inherited number systems of some sort 
and the comparison yields significant clues as to the universals of 
language and culture and a measure of their actual divergence."  recommends
Jan Gullberg. Mathematics: From the Birth of Numbers, W.W. Norton & 
Company, New York, 1997. 

"A history of mathematics. Not mathematicians, mathematics. Early 
chapters are on the etymology of counting systems, including systems 
of enumeration in about 20 or 30 languages (Icelandic, Irish, Old 
English, Old Norse,....) There is quite a variety in counting systems. 
The most unusual was Welsh, which is vigesimal (20 based). Thirty is 
literally "second half times twenty" where 1/2 is the first half, 
1 1/2 is the second half, 2 1/2 is the third half, etc. 

"Later chapters include number theory, logic, topology, calculus, 
and differential equations. An excellent overview of many fields of 
mathematics, and a good starting point for exploring more. "

I hope this helps. Please let us know what you find out.

- Doctor Jodi, The Math Forum 

Date: 02/17/2003 at 18:59:42
From: Andy Spitzley
Subject: Thank you (The Welsh Vigesimal Number System)

Thank you very much for your response. I will use your tips 
to further my research.

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