Half of twenty less than five times twenty

Welsh has two counting systems. One is decimal (i.e. centred on a base of tens) and the other is vigesimal (i.e. centred on a base of twenties). When we talk about a “base of tens” what we mean is that the system uses building blocks of 10s to form numbers. In English, we have such a decimal system. “Seventy” is “7 lots of ten” (“-ty” from the Old English tig meaning ten). But lots of languages don’t form all their numbers in reference to 10.

In Welsh, the number 25 could appear in two different forms. Dau ddeg pump is the decimal version, literally “two ten five”. Pump ar hugain is the vigesimal version, literally “five on twenty”.

Welsh ended up with two number systems because (perhaps somewhat apocryphally) Patagonian Welsh speakers in Argentina created a decimal system for use in their Welsh medium schools. This number system was then imported across the Atlantic back to Wales. This leads to a bit of a muddled situation, which often confuses learners, where there are two versions of all numbers over 11.

We tend to use these different systems in different contexts. So we talk about someone being pedwar ugain for age (literally “four twenty” for 80) and driving trigain miles (literally “three twenty” for 60), but we’re more likely to talk about there being dau ddeg (“two ten”, 20) chairs or other things we’re counting. Weights, measurements, money, time and ages are still mostly referred to using the “old” (i.e. vigesimal) system. However, this is rapidly changing as more and more adult learners become part of the Welsh language community and as school children just see the old numbers as exactly that: old.

The ancient Babylonians apparently had a sexigesimal number system (with a base of 60)… That’s objectively weird. Welsh’s number system being vigesimal based isn’t weird. Loads of languages have a system which is (at least in part) based on this: French, Basque & Inuit languages to name but a few. What’s perhaps intriguing about Welsh’s “old” system is that’s it’s slightly mixed. 15 is an important pivot sometimes:

16

Un ar bymtheg

One on fifteen

19

Pedwar ar bymtheg

Four on fifteen

35

Pymtheg ar hugain

Fifteen on twenty

The old system is full of inconsistencies. The pattern in the teens, for example, is broken by the words pymtheg for 15 and deunaw for 18 (“two nine”). If 14 is pedwar ar ddeg (“four on ten”), then why isn’t 15 just pump ar ddeg (“five on ten”)? And why not have 18 as wyth ar ddeg (“eight on ten”)? The system isn’t really strictly about a base of 20 as opposed to 10 either- sometimes it’s both. 90 is deg a phedwair ugain (“ten and four twenty”).

Danish is a language that makes good use of the idea of using 20 as the base. Danish is a (largely) vigesimal system, but interestingly the 20 bit has been lost slightly. Sindstyve is the part of Danish numbers meaning x20 which isn’t really pronounced in modern spoken Danish anymore. So tredsindtyve for 60 is just pronounced tres nowadays.

Here are some numbers in Danish that make the Welsh system look completely boring.

50

halvtreds

~“half of 20 less than 3 times twenty” [(3-½) x 20]

60

~“3 times twenty” tres [3 x 20]

70

halvfjerds

~“half of twenty less than four times twenty” [(4-½) x 20]

80

firs

~“four times twenty” [4 x 20]

90

halvfems

~“half of twenty less than five times twenty” [(5-½) x 20]

“Half of twenty less than five times twenty” is much more exciting than Welsh’s version of 90. Instead of trying to crowbar the vigesimal theme throughout, Welsh just opts for “ten and four twenties” for 90 and combines a decimal and a vigesimal system, much like French. Danish, however, is loyal to the vigesimal cause and renders 10 by talking about it as half of twenty.

If you only speak English and you're learning a vigesimal number system, then this can cause a headache. But this headache is compounded by the fact that there are actually two different number systems to learn in Welsh. Another thing that causes my learners and tutees to despair is feminine number forms (dwy gath (two cats) but dau gi (two dogs). Yet another trauma is the placement of the noun in quantified noun phrases, e.g. dwy gath ar hugain ("two cat on twenty" for 22 cats) and pedair punt a deugain ("four pound on two twenty" for £44). Ordinal numbers seem to cause even more distress: y bedwaredd ar bymtheg a phedwar ugain  ("the fourth on fifteen and four twenty" for 99th). The learners that make the most progress are always those who look at this kind of linguistic irregularity and think "there's a challenge!". Human languages have created a seemingly infinite number of interesting and different systems for things like numbers. Getting our heads around something like Danish or Welsh numbers might be tricky, but it's a good bit of linguistic legwork and a means of realising our linguistic potential. English numbers are so boring! We can do so much better!