Welsh numerals

The traditional counting system used in the Welsh language is vigesimal, i.e. based on twenties where numbers from 11 to 14 are "x on ten", 16–19 are "x on fifteen" (though 18 is more usually "two nines"); numbers from 21 to 39 are "1–19 on twenty", 40 is "two twenty", 60 is "three twenty", etc.

There is also a decimal counting system, where numbers are "x ten y" unit(s), e.g. thirty-five (35) in decimal is tri deg pump (three ten five) while in vigesimal it is pymtheg ar hugain (fifteen – itself "five-ten" – on twenty).

Numerals

Number	Vigesimal system	Decimal system
0	sero/dim
1	un
2	dau (m), dwy (f)
3	tri (m), tair (f)
4	pedwar (m), pedair (f)
5	pum(p)
6	chwe(ch)
7	saith
8	wyth
9	naw
10	deg/un deg, deng
11	un ar ddeg ("one on ten")	un deg un
12	deuddeg, deuddeng	un deg dau/dwy
13	tri/tair ar ddeg	un deg tri/tair
14	pedwar/pedair ar ddeg	un deg pedwar/pedair
15	pymtheg, pymtheng	un deg pump
16	un ar bymtheg ("one on fifteen")	un deg chwech
17	dau/dwy ar bymtheg	un deg saith
18	deunaw/tri/tair ar bymtheg ("two nines"/"three on fifteen")	un deg wyth
19	pedwar/pedair ar bymtheg	un deg naw
20	ugain	dau ddeg
21	un ar hugain ("one on twenty")	dau ddeg un
22	dau/dwy ar hugain	dau ddeg dau/dwy
23	tri/tair ar hugain	dau ddeg tri/tair
24	pedwar/pedair ar hugain	dau ddeg pedwar/pedair
25	pump ar hugain	dau ddeg pump
26	chwech ar hugain	dau ddeg chwech
27	saith ar hugain	dau ddeg saith
28	wyth ar hugain	dau ddeg wyth
29	naw ar hugain	dau ddeg naw
30	deg ar hugain ("ten on twenty")	tri deg
31	un ar ddeg ar hugain	tri deg un
32	deuddeg ar hugain	tri deg dau/dwy
etc.
40	deugain ("two twenty")	pedwar deg
41	deugain ac un ("two twenty and one")	pedwar deg un
50	deg a deugain	pum deg hanner cant ("half a hundred")
51	un ar ddeg a deugain	pum deg un hanner cant ac un
60	trigain ("three twenty")	chwe deg
61	trigain ac un	chwe deg un
70	deg a thrigain ("ten and three twenty")	saith deg
71	un ar ddeg a thrigain ("one on ten and three twenty")	saith deg un
80	pedwar ugain ("four twenty")	wyth deg
81	pedwar ugain ac un	wyth deg un
90	deg a phedwar ugain ("ten and four twenty")	naw deg
91	un ar ddeg a phedwar ugain ("one on ten and four twenty")	naw deg un
100	can(t)
200	dau gant
300	tri chant
400	pedwar cant
500	pum cant
600	chwe chant
700	saith cant
800	wyth cant
900	naw cant
1000	mil
2000	dau fil/dwy fil
1,000,000	miliwn
1,000,000,000	biliwn
1,000,000,000,000	triliwn

Variation in form

There is some syntactically and phonologically triggered variation in the form of numerals. There are, for example, masculine and feminine forms of the numbers "two" (dau and dwy), "three" (tri and tair) and "four" (pedwar and pedair), which must agree with the grammatical gender of the objects being counted. The numerals for "five", "six" and "hundred" (pump, chwech and cant) also have reduced forms (pum, chwe, can) when they precede the object they are counting. The words for "ten", "twelve", and "fifteen" (deg, un deg dau/deuddeg, un deg pump/pymtheg) have the alternative forms deng, deuddeng, pymtheng used before nasals (which may be the result of mutation) and, occasionally, vowels; these forms are becoming less common. Numerals change as expected according to normal rules of consonant mutation; some also trigger mutation in some following words (see below for details).

Use of the decimal system

The decimal system is widely used, but is rather uncommon for dates and ages. Larger numbers, however, tend to be expressed in this system e.g. 1,965 mil, naw cant chwe deg pump. In referring to years, on the other hand, the number of thousands is stated, followed by the individual digits, e.g. 1965 mil naw chwe(ch) pump. This system appears to have broken down for years after 2000, e.g. whereas 1905 is mil naw dim pump, 2005 is dwy fil a phump.

The Welsh decimal counting system was devised by 19th-century Patagonian Welsh businessmen in Argentina for accountancy purposes. It was recommended to teachers for use in the first Welsh language schools in Patagonia by Richard Jones Berwyn in a book published in 1878. The system was later adopted in Wales in the late 1940s with the beginning of Welsh-medium education.

Use with nouns

The singular form of the noun is used with numbers, but for larger numbers an alternative form is permitted, where o ("of") with the plural noun follows the number. Except where using this plural form, the noun is placed directly after the number but before any parts of the number that are added using ar ("on") in the traditional system.

Nouns are also mutated following many numbers. Un triggers the soft mutation (treiglad meddal) of feminine nouns, other than those beginning with "ll" and "rh", but not masculine nouns. Dau and dwy both trigger the soft mutation (ll and rh included). Tri (but not tair) and chwe trigger the aspirate mutation. Several higher numbers (pum, saith, wyth, deng, deuddeng, and pymtheng) trigger the nasal mutation when used with blynedd ("year(s)"). The part of the number immediately preceding the noun will determine any mutation of the noun. In the plural form with o, the soft mutation is used as is normal after o.

The following example illustrates several of these points:

English	Thirty-six dogs
Traditional system	Un One ci dog ar on bymtheg fifteen ar on hugain twenty Un ci ar bymtheg ar hugain One dog on fifteen on twenty
	Un One ar on bymtheg fifteen ar on hugain twenty o of gŵn dogs Un ar bymtheg ar hugain o gŵn One on fifteen on twenty of dogs
Decimal system	Tri Three deg ten chwe six chi dog Tri deg chwe chi Three ten six dog
	Tri Three deg ten chwech six o of gŵn dogs Tri deg chwech o gŵn Three ten six of dogs