Did the Ancient Danes Count by Tens, Twelves or Twenties? A vignette with a very slight mathematical content Bent E. Petersen - May 9, 1998

Here are the first 20 numerals in Danish:

If your native language is English you are unlikely to guess the correct pronunciation of any of these numerals except perhaps 18! Apart from the vowels differing from English, many of these words have tricky stops in the sound or have totally unexpected sounds. For example 16, "seksten" sounds vaguely like "sighs-den".

We see the same pattern here as in English. The first twelve numerals have individual names of great antiquity. The numerals from 13 through 19 are "teens" so to speak. For example, at one time "tretten" was probably   "tre og ti" (three and ten), just as "thirteen" in English.

The range of  English "teen" numerals is the same as in Danish and they are formed in the same way. In Danish this pattern is continued for larger numerals. Thus 28 is "otte og tyve" - usually spelled as "otteogtyve" - a word approached by great trepidation by the non-Danish speaker attempting to sort out the pronunciation. I imagine "fireoghalvfjersindstyve" (74) would inspire outright fear!  You may object that I have used an archaic spelling. Fair enough, but then be warned that the spelling of the ordinal 74^th (seventy fourth) is "fireoghalvfjersindstyvende" and this spelling is not archaic. We will analyze these imposing numeral names below.

One could try to make a case in Danish (or English) for a special status for 12. The distinct names for 11 and 12 suggest an ancient base 12 system. However it appears that "elleve" ("eleven") derives from old Teutonic "ainlif" which means one remaining and "tolv" ("twelve") derives from old Teutonic "twaliði" which means "two remaining". Apart from the subtractive aspect, as opposed to the additive aspect of the "teens" numerals, the basic idea is the same. These numerals are based on counting by tens. By the way, it interesting to note that the "liði" in "twaliði" derives from "laðan" or "laiðjan" which we still have with us as the verb "leave" in the sense "remain".

Let's move on to twenty. In English (and German) there seems little doubt that the origin of the name for 20 is "two tens". In Danish (and perhaps Icelandic)  the derivation is subtly different. The word "tyve" derives from an old plural form of  "ti" and so may simply be a lazy way of saying "two tens", but it is not actually "two tens". This can cause some confusion because in modern Danish "tyve" is used to mean 20 and also to mean 10 (as a plural noun) in compound words! Of course it doesn't confuse the Danes, who understand the complexities of making good beer, and so are unlikely to be troubled by the ambiguity of arcane names for numerals.

The names marked by question marks I heard as a very young child. It is likely that they were just affectations or jokes, and not correct usage. I do remember wondering what 5 had to do with 100!

In "fyrretyve" the suffix "tyve" is the plural of   "ti". We are looking at "four tens"  - not "four twenties"! Note the similarity to the English "forty" (obviously "four tens"). The other occurrences of "tyve" above however have the meaning "twenty" as we will see.

The name "tres" for 60 seems mysterious since it is obviously related to 3. The literary (and slightly archaic) form "tresindstyve" makes everything clearer! Here the suffix "tyve" means "twenty" so we obviously have 3 twenties. You might argue that it could be derived from "3 pairs of tens". Apart from suggesting that the ancient Danes used a combination binary and decimal number system, this interpretation has problems with the numeral 50.

The name "halvtres" or "halvtresindstyve" for 50 is not difficult to decipher. The prefix "halv" really means half. The name almost certainly means "halfway from two twenties to three twenties". It would not make any sense if interpreted in terms of pairs of tens. If that were the meaning then it would be simpler to use "5 tens", much as "4 tens" is the expression used for 40.

We can now untangle the fearsome 74 that I indicated above:

fireoghalvfjersindstyve

= fire og halv fjers inds tyve

= four plus three and one-half twenties

There's nothing to it - until you try to pronounce it, that is. Danish has a lot of tricks of speech, especially silent letters and letters replaced by stops. There's no help for you here.

I will point out that "fire" is pronounced very much as the English "fear". I admit it is a poor pun (all puns are), but at least it's a bilingual pun.

It seems clear to me that at one time the Danes used a base 20 number system, perhaps only for special purposes, and we are seeing a remnant of it. With the more modern shorter forms the evidence is less obvious. Eventually it may disappear.

It is interesting to note that the Icelanders who are quite conservative language-wise (and who can still read thousand year old sagas) use multiples of 10 for the numeral names:

Note the special case of  20. The word for the cardinal 2 is "tveir". Thus for 20 one expects a word on the pattern "tve....tíu" rather than "tuttugu". Unfortunately I do not know enough about Icelandic to comment on this special treatment of  20.

If we examine Norwegian numerals we find, with the exception of 20,   the Norwegians count  with clear, cold logic, consistently by tens. The numeral 20, however, is referred to as "tjue" or "tyve". Otherwise there is no trace of the Danish flirtation with base 20 in modern Norwegian. However Norwegian and Danish do share a peculiarity concerning fractions, a peculiarity which is related to the Danish counting-by-twenties:

Here "annen", "tredje", "fjerde" and "femte" are the ordinal forms of 2, 3, 4, and 5 respectively, whereas in the Danish counting-by-twenty usage the cardinal form of the numerals is employed. It pays to be careful in Danish! For example,

Not every word with the prefix "halv" is a number. For example, "halvflaske" is half a bottle of beer, which is really 3/8 of a liter, and so would pass as a whole bottle in some less enlightened parts of the world.

In summary I do believe the ancient Danes at one time used a base 20 number system, or a mixed base 20 and base 10 system. On reflecting on the English and American "four-score" and the French "quatre-vingts" I begin to believe that perhaps other people did as well.

November 23, 1998: Some time after writing the article above I came across an interesting book on ancient number systems:

Karl Menninger, Number Words and Number Symbols, A Cultutal History of Numbers, Dover Publications, 1992, reprint of  MIT Press edition of 1969, translation of Vandenhoeck & Ruprecht original German edition of 1957-1958.

Menniger makes the interesting suggestion that the French "quatre-vingts" was actually introduced by the Normans. There are many other suggestions, ideas, and of course, examples from around the world, throughout the book. It is well-worth getting a copy of it.

Disclaimer: This article is a vignette with a vaguely mathematical content. It is intended to amuse and to entice. It is not a scholarly paper. If you seriously want to know more or to question what is here, then search the expert literature.

Copyright © 1998 Bent E. Petersen. This document may be used, copied and distributed freely, entire and intact, for any purpose, but may not be altered. If you want to improve on it, which can certainly be done, then please write your own.

No claim of historical veracity is made, quite the contrary.

Last updated Thursday, April 10, 2003 11:28 PM

If you have corrections, suggestions or comments feel free to email me,

bent@alum.mit.edu

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