Math or Magic? Decoding the Danish Number System

Imagine you are standing in a bakery in Copenhagen. You have successfully ordered a wienerbrød (pastry) and a coffee. You feel confident. You feel like a local. Then, the cashier tells you the total price: “Det bliver treoghalvtreds kroner.”

Your brain freezes. You know tre is three. You know treog… means “three and…” But what on earth is a halvtreds? If you speak German, you might panic—is it related to hundreds? If you speak English, you are completely lost. You hand over a 100-kroner note and hope for the best.

Welcome to the beautiful, bewildering world of Danish numbers. While the rest of Scandinavia (and indeed, most of the Germanic-speaking world) stuck to a decimal (base-10) system, the Danes decided to take a detour into history and mathematics. They adopted a vigesimal system—a way of counting based on the number twenty.

But Danish doesn’t just multiply by twenty; it combines fractions, archaic multiplication terms, and a reverse word order to create what might be the most fascinating counting system in Europe. Is it math? Is it magic? Let’s decode it.

The German Connection: The “One-and-Twenty” Switch

Before we dive into the deep end of the linguistics pool, let’s look at the shallow end. Numbers 1 through 20 in Danish are fairly standard for a Germanic language.
En, to, tre, fire, fem… sound distinctly related to English (one, two, three, four, five) and very close to German (eins, zwei, drei, vier, fünf).

However, once you hit twenty-one, Danish adopts the “Germanic inversion.” In English, we say the tens first, then the units (Twenty-one). In Danish, like in German, you say the units first, then the tens.

• English: Twenty-two
• German: Zweiundzwanzig (Two-and-twenty)
• Danish: Toogtyve (Two-and-twenty)

So far, this is manageable. It requires a bit of mental rewiring to say the last number first, but the logic is sound. It is when we reach the number 50 that the logic seemingly jumps out the window.

The Vigesimal Curveball: Counting in Scores

To understand why Danish numbers are unique, we first have to understand the number twenty. In English, we have a specific word for twenty called a “score” (famously used by Abraham Lincoln: “Four score and seven years ago”). A score is simply a set of twenty items.

French uses this logic famously for the number 80: quatre-vingts, or “four-twenties.” Danish takes this concept, injects it with steroids, and applies it to almost every multiple of ten between 50 and 90.

The archaic Danish word for a result of multiplication is sinds (times), and the word for twenty is tyve. In the old days, you would count by multiplying a number by twenty (tyve). However, over centuries of rapid speech—a phenomenon Danes are famous for—the “times twenty” part got swallowed.

Today, the suffix -sindstyve has been shortened simply to -s. This renders the math invisible to the naked ear, turning a logical equation into an abstract sound.

The Fractional Logic: “Half-Third”

Here is the missing key to the cipher: Danish uses an old way of describing fractions that exists in other languages but has largely died out in English.

In modern English, if you say “two and a half”, you mean 2 + 0.5.
However, historically, you could describe 2.5 as “half of the third number.” You have completed two whole numbers, and you are halfway through the third one.

• 1.5 = “Half-second” (Halvanden in Danish).
• 2.5 = “Half-third” (Halvtredje – archaic but essential for understanding 50).
• 3.5 = “Half-fourth” (Halvfjerde).
• 4.5 = “Half-fifth” (Halvfemte).

German speakers will recognize this time-telling logic immediately (halb drei means 2:30, or “half way to three”). Danish applies this logic to multiplication.

The Big Reveal: Decoding 50, 70, and 90

Now that we have the ingredients—the base-20 system and the “half-way” fractional counting—we can finally translate the numbers that baffle learners.

50: Halvtreds

The word halvtreds is actually a contraction of the old phrase halvtredje-sinds-tyve.

• Halvtredje: 2.5 (Half-third)
• Sinds: Times
• Tyve: 20
• The Math: 2.5 x 20 = 50

Over time, “halvtredje-sinds-tyve” was mumbled down to “halvtreds.”

70: Halvfjerds

Following the same pattern, 70 is a contraction of halvfjerde-sinds-tyve.

• Halvfjerde: 3.5 (Half-fourth)
• Sinds: Times
• Tyve: 20
• The Math: 3.5 x 20 = 70

90: Halvfems

Finally, we reach the nineties. Halvfems comes from halvfemte-sinds-tyve.

• Halvfemte: 4.5 (Half-fifth)
• Sinds: Times
• Tyve: 20
• The Math: 4.5 x 20 = 90

What About 60 and 80?

You’ll notice we skipped the even numbers. Thankfully, 60 and 80 are much easier because they deal in whole numbers, not fractions.

• 60 (Treds): Short for tre-sinds-tyve (Three times twenty). 3 x 20 = 60.
• 80 (Firs): Short for fire-sinds-tyve (Four times twenty). 4 x 20 = 80.

Putting It All Together: The Ultimate Example

Let’s go back to the bakery. You owe 53 kroner.

The cashier says: “Treoghalvtreds.”

1. Tre: 3
2. Og: And
3. Halvtreds: (2.5 x 20)

Literally translated, the cashier is saying: “Three and (halfway to the third times twenty).”

It is a linguistic equation that requires you to perform addition, logical fractional deduction, and multiplication, all while holding up the line for a pastry.

Why Did This Happen?

Linguists often ponder why Danish diverged so notably from its cousins, Swedish and Norwegian, which use straightforward decimal systems (femti, seksti, sytti).

The vigesimal system wasn’t always the standard. In the Middle Ages, Danish used a system similar to Swedish. However, counting by scores (20s) became fashionable in trade and counting money during the late medieval period, likely influenced by the Celtic and Basque regions or perhaps simply counting on fingers and toes (which totals 20).

While the long-form words (halvtredsindstyve) are visible on Danish bank notes, no one speaks them. The language evolved toward efficiency, clipping the ends of words until only the fractional prefixes remained. The result is a system that is historically rich but pedagogically terrifying.

A Survival Guide for Learners

If your head is spinning, don’t worry. You do not need to perform mental calculus every time you see a number. Here is the secret: Don’t do the math. Just memorize the sound.

Treat halvtreds (50) as a unique vocabulary word, just like “apple” or “house.” Dissociate it from the math. It equates to “fifty”, and that is all you need to know for daily survival. The etymology is a fascinating party trick, but trying to calculate 2.5 x 20 while listening to a phone number being recited at speed is a recipe for disaster.

However, understanding the logic does help in one specific area: Ordinal numbers.
When you want to say “the 50th”, Danish brings the full, archaic word back famously. “The 50th birthday” is den halvtredsindstyvende fødselsdag. Suddenly, the “times twenty” (sindstyve) reappears from the history books!

Conclusion

The Danish number system is a wonderful reminder that language is not always designed by engineers; it is grown by people, influenced by trade, history, and the messy evolution of culture. While it presents a steep learning curve, mastering it is a rite of passage.

So, the next time you are in Denmark and someone asks for femoghalvfems (95), take a breath. Smile. Remember that you aren’t just paying a bill; you are participating in a thousand-year-old tradition of counting by scores, halves, and wholes. And if all else fails? Just hand over a credit card.