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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READERS REVEAL: What learning Danish changes about your life in Denmark

We asked our readers in Denmark why they learned to speak Danish and what it has changed about their lives.

READERS REVEAL: What learning Danish changes about your life in Denmark

For the majority of foreigners who live in Denmark, learning the Danish language comes with the territory of relocating to the country. It’s not always an easy process, but it can be a rewarding one.

We asked our readers what learning Danish had changed for them. We received a lot of interesting answers and input – thanks to all who took the time to answer our survey.

Some said that learning Danish was a personal decision while for others, it was a requirement of immigration rules.

“It was my own decision to learn the language to be able to understand what is happening around me in daily life (on public transport, in the shop, on the street, what my colleagues are chatting about when they don’t use English), and with the hope that I can easier build up some social connections with locals! If I live in a foreign country, then it’s the minimum to speak and understand the language at some extent,” said Dorina.

“I’m still in module 1 [of the national language school programme, ed.], so no change (to my life) yet, but I can see that my colleagues are valuing my effort very much,” she added.

Pedro told us that “as a person who’s lived in a few countries since I was very young, I do understand the enormous value of completely emerging oneself and learning the language of your current home.”

“It opens up a whole new world in a sense and it helps you to be fully engaged into a new society. And I’ve felt that the locals truly appreciate it when someone knows their language,” he said.

“That’s especially true with Danish since relatively there are so few speakers in the world,” he added.

“Regardless of my own desire to learn I do need to learn to pass a few language exams to fulfil my visa requirements, for permanent residency and hopefully citizenship exams,” Pedro also said while adding that knowing Danish would likely broaden his career options while in Denmark.

Lizzie meanwhile said she had begun learning Danish “because I decided to stay in the country after graduation, so it made sense to learn Danish, so I can integrate easier.”

“I’m from outside the EU. It was compulsory for me to learn (Danish) on a reunification visa,” Barry said.

Being able to speak Danish had a range of impacts on the lives of our respondents.

“After 10 years I still work in a predominately English workplace,” Barry said, adding that he used the local language “when out shopping and (for) other simple everyday interactions.”

Learning Danish has “enabled me to engage and become involved in society, build a social circle independent of my wife’s social circle and become more eligible in my previous professional field,” wrote Lyle, who was required to learn the language to meet visa requirements.

“Danes hold you in a higher regard when you engage in Danish even if you attempt and you suck a bit,” he said.

Another reader, Iulian, said language classes were a good place to “meet people having the same obstacles and make new friends.”

“And it is free now,” he noted.

READ ALSO: More foreigners go to Danish language classes after fees scrapped

“I have a nice relation with my 70-plus year-old neighbour who speaks only Danish. He helped us with so many things so far, things I would have not known if he would not have told me. It was possible because I learned some Danish, enough to understand each other,” Iulian said.

“Finding work and internships I think has been easier” with Danish, wrote Lizzie, adding that even at international companies, Danish can help you feel more at home.

“Almost everyone speaks Danish in the breaks,” she said.

“It’s allowed me to communicate with others, especially at my son’s vuggestue [childcare] where many don’t speak English as much as the general population,” Pedro added.

Dorina told us that “A whole new world opens up by understanding what’s going on around me.”

“The biggest achievement of starting the language is that I can already catch some words from locals and be able to differentiate words within a sentence when they speak,” she said.