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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What are the best websites and apps to learn Danish?

Most language schools and Danish language courses recommend students immerse themselves in the language as much as possible and educational websites and apps are a great way to do that.

What are the best websites and apps to learn Danish?

People moving to a foreign country face a number of challenges. From adapting to a new culture and prospecting for job opportunities to creating a new social circle, the list goes on and on. 

Relocating to Denmark is no exception.However, if you’re able to pick up the language quickly, all of these processes will become much more manageable.

READ MORE: The seven stages of learning Danish every foreigner goes through

While many people start this journey by joining a language course, many language students heed the advice of their teachers and pair these classes with websites and apps that enhance their language learning experience.

By regularly using such digital platforms, you’ll likely notice improvements in both pronunciation and your vocabulary – not to mention that these options are convenient, interactive, and often free of charge.

1. DR, DRTV, and online channels

You can find a number of free-to-access TV programs in Danish – a lot of them are subtitled – both as part of the standard broadcasting and online.

Some language schools recommend beginners start with DR Ramasjang and Ultra, Danish TV aimed at children and teens that can be easier to follow than some adult programmes. Starting with UltraNyt, a children’s news programme, might be the best option for beginners.

If you’re looking for more advanced news and information on what’s happening in the country, consider checking out Danish channels with online programs, such as TV 2 News, TV 3, and DR1.

Note that you can also use the DRTV app to access all of their programmes.

2. Language learning apps

The interactivity and ease of access that comes with digital devices such as smartphones, tablets, and computers has led to a significant increase in the popularity of language learning apps in recent years.

Conventional language courses cannot compare to the convenience of language learning apps. These digital solutions offer students the flexibility to access them at their preferred time and location without any limitations.

The heavy hitters in the industry include Duolingo, Babbel, and Memrise, all of which have quite extensive Danish courses. Most offer a lot of free content (usually, the first lesson or two), with Duolingo coming out on top as it’s free to use, but ad-supported (you have the option to pay to increase functionalities and remove ads).

When it comes to depth, Babbel can perhaps offer more than Duolingo.

3. Podcasts

Offering an immersive language learning experience, podcasts can help regular listeners expand their vocabulary and improve their pronunciation and listening skills.

The fact that you can access them virtually whenever and wherever – whether you’re commuting to work or waiting in line at the store – is also a significant draw for many students.

The Sara & Monopolet podcast from DR is a great podcast for students at the intermediate and advanced levels. In it, the host Sara Bro hosts different well-known guests on a weekly basis, with the goal of helping listeners solve a wide range of dilemmas. Note that it’s pretty long (roughly 1 hour and 40 minutes per episode).

In the 30-minute Baglandet (“back country” in Danish) podcasts, you’ll dive into hard-hitting stories from people of diverse backgrounds and experiences living in Denmark. The objective of Baglandet is to bring attention to individuals who have faced the repercussions of specific political choices.

While not technically a podcast, the Danish 101 language course is a highly developed and popular language learning program (covering levels from beginners to advanced) in podcast format, so we’re including it in this section. One of the advantages of Danish 101 is its approach – the text is initially spoken, followed by a slower repetition, and then reinforced with a translation.

Other interesting podcasts include Det Forenede Kongerige (covering the British music scene), Hvem er… (deep dives on individuals behind some of Denmark’s most renowned music), and you can also use the news subscription app Zetland to access a news podcast about the most important news of the day.


If combined with other resources like textbooks and language apps, podcasts can help students on their road to fluency in Danish. Photo by Will Francis on Unsplash

4. YouTube resources

You can find many Danish language schools, teachers, and tutors dedicated to spreading both knowledge and the love of the Danish language on YouTube.

Channels such as Danish Mastery, Learn Danish with, and David Jørgensen’s videos have several lessons most students can stand to benefit from, as long as they take the time to follow along regularly and by paying close attention to the classes.

5. Online textbook, grammar, dictionary, and vocabulary resources

While YouTube lessons are quite convenient, if you’re looking for more structured exercises and lessons, you can also find several Danish language textbooks and grammar guides online.

Grammar Explorer has both text and audio materials, and you can access this Danish grammar book online as long as you have Adobe Flash Player installed on your device. If you’re struggling with conjugation, you can use the Verbix conjugator to improve your knowledge of Danish verbs. Learn Danish 101 also offers useful grammar guides online.

Need help with learning new words in Danish? You can always improve your vocabulary by using Quizlet, a website and app which help users develop their Danish vocabulary by using flashcards, or Cram, which has an extensive list of flashcards too.

There is also a number of dictionaries and encyclopedias available online, such as the Lexilogos Danish-English Dictionary, the FreeDict Danish-English Dictionary, the Danish Dictionary, and the Danish-English Dictionary from Oxford University – to mention just a few.

6. Danish Here and Now

The Danish Ministry for Refugee, Immigration and Integration Affairs developed a free online Danish course for beginners – Danish Here and Now.

It consists of 12 lessons, and each lesson has five chapters. Note that the course is intended to serve as an introduction to the language, and it will likely not be as useful to advanced users.

Still, it’s a great resource if you are looking for a place to start learning Danish.