The Danish words for 20, 30 and 40 denote, as ours do, 2, 3 and 4 tens. But 'tres' and 'firs' for 60 and 80 are shortened forms of 'tresindstyve' and 'firsindstyve' which mean 'three times twenty' and 'four times twenty'. "Halvtreds" for 50 is a shortened form of 'halvtredsindstyve' meaning 'half three times twenty'. Here "half three" means 2-1/2, its significance being 'half way to three". Similarly "halvfjerds' and 'halvfems' denote 3-1/2, and 4-1/2 twenties. (Man & Number, p.23)

There are, however, examples of the use of 12 as a base. We still count eggs in dozens, we have twelve inches in the foot and 12 pence in the shilling, there were 12 ounces in the old pound.

It may be that the words 'eleven' and twelve' - irregularities in our present system of number words - are due to an earlier duodecimal system (i.e. a system with base twelve). They and the equivalent words in some other Indo-European languages appear originally to have meant 'one left' and 'two left' - left, that is, to be counted after all the fingers had been used. ... There are obvious advantages in the use of 12 as a collective unit as it can be divided into halves, thirds, quarters and sixths without the use of fractions ... (Man & Number, p.24)

It is unlikely that anyone deliberately planning a number system would decide to use the base ten; a duodecimal system (i.e. base 12) would be a more probable choice. For such a system two more symbols would be needed, i.e. there would be eleven single code symbols... In writing numbers greater than eleven the second place would be used to denote the number of twelves, the third place the number of twelve twelves, and so on.

Thus twelve would be written 10 and the number of this page (112) 94 (nine twelves and four). As twelve is an exact multiple of 3, 4 and 6 as well as of 2, division by these numbers would be simplified; there would be less carrying and fewer fractional results, e.g. 10÷3 would equal 4 and 100÷6 would equal 20. Further, the fractions 1/2, 1/3, 1/4 and 1/6 would be expressed in the form 0·6, 0·4, 0·3 and 0·2 but 1/5 and one-tenth would be expressed in recurring form. The advantages of such a system would be the greater if all the systems of weights and measures were also duodecimal. (Ibid., page 112)

It would be an enormous boon to international commerce if the hundreds of money systems used by different countries could be simplified. At present, though the standard units of the various coinages (pound, dollar, rupee etc) are of the utmost variety in their values, there is a large amount of uniformity in the method of dividing these units into smaller ones, by using ten or 100. of course, this is the result of the whole civilised world being in agreement about basing its method of counting and calculating on the number ten. If human beings had twelve fingers and toes instead of ten the nations of the world world prcbably all be using twelve instead of ten, and many sums would be simpler than they are now, for twelve has twice as many factors as ten (not counting one). A few enthusiasts have advocated the scrapping of our system of counting entirely and suggested that the whole world should re-learn to count in dozens, gross (144), and great gross (1,728) instead of tens, hundreds, and thousands. (Romance in Arithmetic)

(The metric system to be introduced in Britain)...there is little doubt that it will eventually happen. Such a development would be welcomed by the business community who claim that they can increase trade with foreign countries when those countries are able to understand our quotations and prices more easily. However many people - and not least mathematicians - are strongly opposed to the change, and regard it as a retrograde step believing the decimal system to be only superficially simpler or more useful than our present one. (Man Learns to Measure)

- MAN AND NUMBER: Donald Smeltzer, Black, 1958
- ROMANCE IN ARITHMETIC: Margaret Bowman, Univ. London Press 1961
- MAN LEARNS TO MEASURE: Keith Irwin, Dobson 1960

UNHELPFUL PATTERNS I note from various "Word Books" that five is an important grouping. On studying my son's Disney "Wordbook" I found all the numbers above five illustrated with groups of five:

for example 6 as ***** *,

16 as ***** ***** ***** *.Surely they could have used more instructive patterns, or do they think 5 more important than the other numbers? Richard Scarry's Wordbook does the same, so I wonder who pinched the idea from whom, and why...

If we had twelve fingers instead of ten we should probably have acquired the habit of counting by twelves and of considering all large numbers as composed of so many twelves.Of all the possible methods of considering the number, that which breaks it up into tens is the most convenient merely because we are accustomed to it. Scientific Arithmetic, 1896.