Who gave Denmark its insanely complex numbering system?

It's perhaps a depressing thing to hear for those wrestling with learning Danish numbers. But they didn't use to be this way. Before about 1300, they were, well, normal. And the Jutlanders may be to blame.

Who gave Denmark its insanely complex numbering system?
Photo: Morebyless/Flickr

As brilliantly described in this article, the Danish numbering system for the multiples of ten from 50 to 90 is brain-bendingly perverse. 

Starting with the word for fifty, halvtreds.

It is constructed of halvtredje, meaning “two and a half”, and sindstyve, meaning “times 20”, so 2½ times 20 equals 50.

And from there it only gets worse. 

  • 60tres, is the more simple tre (three) and sindstyve: 3 times 20 equals 60.
  • 70halvfjerds, is constructed of halvfjerde and sindstyve: 3½ times 20 equals 70.
  • 80firs, is the more simple fire (four) and sindstyve: 4 times 20 equals 80.
  • 90halvfems, is constructed of halvfemte and sindstyve: 4½ times 20 equals 90.

Leaving aside the fact that the suffix halv means something different in each of the three times that it is used, why on earth do the Danes suddenly switch from multiplying by ten to multiplying by 20? 

The answer is that they didn’t use to. 

According to the famed Danish language researcher Johannes Brøndum-Nielsen, up until about 1300, Danish used the forms siutyugh, “six tens” for 60, following the same system right up to 100. 

So what happened? 

No one knows for sure, but the earliest use of multiples of 20 found in a Danish text is in the municipal law for the city of Flensborg, from about 1300, which included the forms fiyrsin tiughæ (4 x 20) and half fæmpt sin tiygh (4½ times 20 equals 90). 

Flensborg is now in Germany, but was then at that point in very south of Jutland. By 1400, the use of these strange forms had spread as far east as Lund in Skåne, then also part of Denmark. 

“It can be stated with a considerable degree of certainty that the process of adopting the vigesimal system began in Western Denmark and spread eastwards,” concludes Błażej Garczyński, a PHD student at Adam Mickiewicz University, in a research paper on Danish numbers

He also points out that the numbers from 50 to 100 are generally more prone to developing varied forms, as they are used much less frequently than 10-50.

The question is where the early medieval burghers of Flensborg got the strange idea of multiplying 20s by unusual fractions? 

As anyone who has learned French — with its quatre-vingt (80) and quatre-vingt-dix (90) — will know, Danish is not alone in having twenty-based counting. 

Such numeral forms are described as “vigesimal”, and according to the Swedish language historian Stig Eliasson, the consensus is that the forerunners of Indo-European languages such as Danish did not use them. 

“Proto Indo-European is considered to have been thoroughly decimal,” he explains in his comparison of vigesimal counting in Old Danish and Basque

“Nevertheless,” he continues, “twenty-based counting shows up in quite a few of its daughter languages, the best known western case being perhaps that of Old French. In the Celtic languages, vigesimal counting is found in the Gaelic – Modern Irish, Scottish Gaelic, and Manx – as well as the Brittonic branch – Welsh, Cornish, and Breton.” 

The form is also common in Basque, the Northern version of which uses a vigesimal system all the way from 20 to 180. 

As Celtic languages are also Indo-European, some researchers have argued that Basque is the only European language in which the vigesimal system is original. 

Eliasson told The Local that it was possible, however, that the system originated in languages spoken among the people in northwest Europe before the Celts arrived, of which Basque is the only survivor, and that those languages then influenced Celtic languages, such as Gaulic, which then in turn influenced Old French, Gaelic, and perhaps even Danish.

“It is conceivable that Basque might somehow have been part of a kind of Pre-Celtic numerical Sprachbund [linguistic area] that might have been at the roots of West European vigesimal counting. But we know nothing about such a Sprachbund if it actually existed,” he said. 

Some argue that rather than the French inheriting their vigesimal habits from the Gauls, the Normans picked them up from the Celts. 

None of this, however, explains the sheer weirdness of the Danish practice of multiplying 20 by 2½, 3½ and 4½. Most vigesimal languages, taking the same pattern as the French quatre-vingt-dix, just add a ten to the closest multiple of 20. 

“There are no obvious details that link these two numeral systems in such a way as to suggest direct linguistic (semantic) copying from Basque to Danish,” Eliasson told The Local. 

But this does not mean that the Danish system was not influenced by other vigesimal counting systems. 

“I believe that there may be a connection between the various vigesimal systems in Western Europe and that an important role has been played by cultural contact, at least in the Danish case,” he said.

“Vigesimal counting may have been practiced in trade and hence triggered the development of the Danish vigesimal numerals.The vigesimal numerals in Danish might have been created in response to vigesimal counting practices in contact with speakers of languages with vigesimal numerals structured perhaps in partly different ways than what was to be the case in Danish.” 

Most researchers have concluded, however, in Eliasson’s words, that rather than having “pre-medieval roots or a trigger in language-contact”, Denmark’s numbers are “a spontaneous language-internal innovation in the Middle Ages”.

In other words, you can blame the Jutlanders. 

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The seven stages of learning Danish every foreigner goes through

You've got your dansk ordbog, you've downloaded all the apps, you are ready and willing to learn Danish. Then you move to Denmark and reality hits. Optimism, overwhelm, delight and then over it: These are some of the emotions familiar to those of us trying to learn the language, writes Emma Firth.

The seven stages of learning Danish every foreigner goes through

Stage one: Optimism 

You’ve decided to move to Denmark. You’ve watched The Killing and Borgen and can pick out the words ‘tak’ and ‘hej hej’, so you’re sure that within a year or so of actually living in the Scandinavian country, you’ll be sounding like Sarah Lund herself. You can’t wait to get started.

Tip: Hold onto the optimism because you’re about to have the shock of your life.

Stage two: Overwhelm

You arrive in Denmark, you’re overwhelmed by the next level life admin and you do not understand a word, not a word, of what is going on around you. You start to recognise written words while you’re out and about; ‘s-tog’, ‘gade’, ‘rugbrød’, but when you say them out loud, oh dear. You soon realise that you can’t learn Danish by reading it in your head. This is a language that needs to be listened to, at slow-speed, then de-coded, put back together and practiced. But you’re too tired for that because you’ve just moved country.

Tip: Enrol in the government’s free Danish language course as soon as you can. It will give you structure and motivation for starting to learn some useful vocabulary and vowel sounds. Duolingo and Google Translate are also your friends.

Stage three: Quiet delight

You’ve passed your first module of your Danish language course. You had a little chat in Danish and explained which country you come from, where you now live with and how many siblings and/or pets you have. This is it. You are going to be fluent in 18 months’ time (after Module 5). There’s tangible progress in your language skills and you are on your way to deciphering Danish.

Tip: Remember this feeling of progression and how good it feels because you’re going to have to keep it going for quite some time. Speak the little Danish you know, over and over again to gain confidence in hearing yourself make the sounds.

READ MORE: Five tips that make it easier to learn Danish

Stage four: Incredulity

You’re now half way through the language school modules. You’ve put hours and hours into learning this language. You know enough vocabulary to use in everyday life – it’s there in your head – you even know how to spell and conjugate the word. So why, when you go to say the sentence to the person behind the check-out, do they look at you in bewilderment and after another failed attempt, switch to English?

You start to feel like the hard work has been a waste, or perhaps you’re terrible at languages, maybe you’ve actually got an undetected speech impediment. The truth is, Danish takes a lot of hard work and practice to get to conversational stage. The vowel sounds are subtle and plentiful; the only way to master them is to keep speaking Danish. 

Tip: Don’t give up – you know far more than you sound like. Keep talking Danish wherever you can and push past those awkward exchanges, which unfortunately have to happen in order to progress to the next level. Force Danish speakers to stick to Danish, even just for five or ten minutes, or mix up a bit of English into your Danish so you can keep to the general thread of Danish conversation.

READ MORE: The best podcasts for learning and perfecting your Danish

Stage five: Reinforcements

The reason you can’t be understood is not you, it’s Danish. You realise that the language course alone is not going to make you fluent. You need reinforcements. You sign up to a language cafe, force yourself to listen to some Danish podcasts, start to watch more Danish TV and read some children’s books.

Tip: If you haven’t got a Danish person living with you, go and find one who will help you practice. There are schemes where a Danish volunteer can sit with you and help you practice speaking, or you can volunteer yourself in a local charity shop. If you have a cheerleader who reassures you that you can and will be understood, then you will get over that barrier many face after language school finishes.

Stage six: Breakthrough

You are being understood more than you’re not, you can read posters, apartment notices, letters in your e-boks. You are not so embarrassed by the vowel sounds coming out of your mouth and people are impressed you can understand a Danish exchange. 

Tip: Don’t take your foot off the pedal just yet. Keep going with the podcasts, the TV and the reading because stage four can and will still happen, and it can knock you off your course.

Stage seven: Acceptance

Despite your breakthroughs and miles on the clock, you realise you no longer know what fluency feels like. You will never sound exactly like a Dane; there will always be new words or expressions to learn; there will always be someone who responds with a “hvad?” to what you’re saying. But what you now accept is that this is the case with any language and we are all learning every day.

Tip: The more you use the language, the more you’ll enjoy it. One day, you may even find yourself sounding like Sarah Lund, to the untrained ear.