The Rise and Fall of the Decimal PointAuthor(s): Chris WeeksSource: Mathematics in School, Vol. 32, No. 1, History of Mathematics, Codes andCrytography (Jan., 2003), pp. 35-36Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212233 .

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by chris week

Introduction

The use of a decimal (or denary) counting system, together with positional notation and a separator between the integer part and fractional part of the number, has a history that stretches back to the sexagesimal counting system of the Old Babylonian period (c. 1900-c. 1600 BC). The use of a mark as a separator between the integer and fractional part of a number came into use at the end of the 16th century and Napier, in his work on logarithms Rabdologiae (1617), used a comma but the convention was not consistently applied. Wingate's Arithmetick, published in London in 1696 has this to say in Chapter XXII, entitled Decimal Fractions (pp. 168f.):

III. A Decimal fraction may be express'd without the Denominator, by prefixing a point or comma before (to wit, on the left hand of) the Numerator, so may be written thus,. 5, or thus,, 5.

[the punctuation is exactly as was printed]. However, by the 19th century, in Britain the convention of using the midpoint as a 'half-high' decimal point, both in printed works and in manuscript, became firmly established whereas the convention in continental Europe was to use the comma and in the United States to use a point on the line. Printers use the terms 'period' for the point on the line and 'midpoint' or 'prick' for the half-high dot.

From the beginning of the 20th century, and with the evolution of mass produced school mathematics textbooks we find in Britain the use of the midpoint for the decimal separator, as in '2-54', firmly established and this convention remained more or less fixed until about the middle of the 20th century. Changes in convention began to be seen from about 1960, influenced mainly by the decimalization of British currency in 1971, the half-hearted adoption of metric systems of measurement and associated SI conventions, and electronic technology (calculators) and, perhaps, a decline in British imperial influence. These changes have had a little- noticed effect on some mathematical conventions that brings us into disharmony with standard practice in Europe and the rest of the world.

Printing and Typing

In Britain, when (mathematical) manuscripts were hand- written and it was the job of the printer to type-set the work, the midpoint was always used for a decimal point. With the wider use of the typewriter, particularly after the introduction of light portable models, we see the appearance of a period for the decimal point in typed documents. Also, in the 1960s most secondary schools acquired bulk printing machines and the mathematics teacher was now able to produce typed examination papers where 2.54 would appear rather than '2.54', though by careful physical manipulation of the platten it was possible to produce the 'correct' decimal

point. Even so, school secretaries were rarely patient enough to achieve this effect.

Electric typewriters (which adjusted letter widths) became more common from the beginning of the 1970s, and machines such as the IBM 'golf-ball' provided a large range of fonts, including the midpoint. Secretaries, by this time, had become so used to using the period for a decimal point that, in my experience, this new facility was rarely used.

Decimal Currency

The use of the period for a decimal point would have only been observed by those using scientific works. The vast majority would have continued to use the midpoint, if indeed they ever used decimal notation at all. Decimal notation is not necessary for the imperial measurement system and all parts and sub-parts of imperial units are found by fractions which do not (usually) involve ten. For schools, a change came with the (half-hearted) adoption of a metric measurement system and the use of, say, '2-5 cm' (typed as '2.5 cm') instead of the more common '2 cm 5 mm'.

The major change in Britain in the convention of using the midpoint came in 1971 with the decimalization of our currency. Overnight, an amount such as A6 3s. 6d. was converted to the rather incongruous A6-171/2. In fact, with inflation, it was not long before the 1/2p coin disappeared, sparing the mathematics teacher from having to deal with that particular nonsense. Whereas before decimalization, written amounts of money would always show the units (pounds, shillings, pence), after decimalization we were advised to write all currency amounts using a dash: A6-17. But typed amounts, and also printed statements of accounts, adopted the period: A6.17.

The other important technological change that took place in the 1970s was the advent of hand-held calculators. Within ten years of their first appearance they were, by the beginning of the 1980s, available for use in mathematics classrooms, and ten years later had become so cheap that everybody used them. The display window of an electronic calculator is relatively simple and the decimal point always appears on the line. It would have been an uphill task to have tried to encourage the young learners of mathematics to use the midpoint when the evidence of their own eyes shows a period.

Despite the arrival of the calculator and the widespread adoption of the period for currency, printed mathematical works, including school textbooks, continued to use the midpoint for many years. Some publishers continued with the midpoint for longer than others, and it may well have continued in school textbooks (encouraged by the authors) long after the practice had ceased elsewhere. (The latest example I can find is an A level text published by Oxford

Mathematics in School, January 2003 The MA web site www.m-a.org.uk 35

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University Press in 1987.) From the late 1980s, however, it is safe to say that the printed use of the midpoint had completely disappeared in Britain. The National Curriculum makes no mention of any convention for decimal parts, except to use a period in its own text. Over a period of thirty years, from 1960 to 1990, the decimal point fell and its (unremarked) decline was complete. (But not quite, it seems. Since writing this, a colleague has pointed out that you will still find midpoints in the Daily Telegraph, that stout defender of all things British.)

Multiplication

The changes briefly summarized above occurred as a result of typical British 'muddling through', but there are consequences for mathematical notation which would not have been anticipated, and which cause some unease for teachers of mathematics. The conventional sign for multiplication is the 'cross' x but this becomes tedious when many products have to be written. For elementary work this is not a problem because of the simple algebraic convention that xyz implies the multiplication of the three factors. When dealing with the multiplication of a sequence of integers, as in work with permutations and with the associated binomial coefficients, it was always the practice to abandon the x in favour of the period. Thus the use of the period for multiplication, with no apparent need to defend its use, appears on page 3 of Durell's Advanced Algebra Book I (1932).

[therefore] the total number of ways of arranging the n unlike things in a row is n(n - 1)(n - 2)(n - 3) ... 3 . 2 . 1 = n!.

But the use of the period for multiplication appears much earlier. Workman's Tutorial Arithmetic (1902), page 20, declares

The . is also used as the sign for multiplication, but it is not to be recommended at first. Thus we may write 6.7. Be careful not to write 6-7, which means something very different (see Decimals, Chap. XVI).

Presumably because of its possible confusion with the decimal point, the use of the period as a notation for multiplication has all but disappeared from British mathematical works, at least at an elementary level, so that we now have the rather cumbersome, and visibly challenging, form of repeated x signs. I shall use examples taken from the A level texts recently published by Cambridge University Press under the editorship of Hugh Neill (2000) but any modern British mathematics school text could be used for illustration. In Statistics 1, page 83, we read

[...] the number of permutations is

8! 8! 8x7x6x5x4x3x2xl

(8 - 6)! 2! 2 x 1

and this use of the x sign in extended multiplications continues throughout the mathematics and further mathematics books of the same publisher. When the signs x and + are printed with equal weight it becomes difficult to distinguish the separate terms of a series. In Pure Mathematics 4, we have

[...] prove that, for all positive integers n,

1 1x4 + 2 x 5 + 3 x6 + ... +n(n+ 3) =In(n + 1)(n + 5). 3

Compare this with the format

1 1-4 + 2-5 + 3-6 +... + n(n + 3) = n(n + l)(n + 5)

3

where the terms of the series are immediately evident. The failure to use a dot (whether a period or a midpoint) for products also leads to major notation problems for authors Neill and Quadling when it comes to handling the vector product in Pure Mathematics 6. After introducing the vector product and the x notation in the usual way,

The vector product, or cross product, of two vectors p and q is given by

p x q = -pJllq sin n fi

they go on to evaluate expressions such as i x j and write (page 177)

The magnitude of i x j is 1 x 1 x sin 1/2 71 = 1, and its direction is k. [...] Therefore i x j = k.

There is no warning about the different status accorded to the x signs, nor is there any cautionary note to accompany this evaluation, two pages later:

1 0 (-3)x(-2)-(-5)x(-1) 1

-3 x -1= (-5)x0-1x(-2) = 2

-5 -2 1x(-1)-(-3)x0 -1

where one of the x signs means something quite different from the other six!

A return to the use of a dot for the multiplication of numbers, which is anyway consistent with the scalar (or dot) product for vectors, would be a welcome relief. Now that the decimal point has dropped to the line, the way is open to adopt the European and American practice of the midpoint to represent multiplication. Hence 6-7 would be equal to 42 and 6.7 would have its now conventional meaning of 67/10. This is the exact reverse of the advice given by Workman in 1902 but, one hundred years later, we should now be able to agree to come into line with the rest of the world, at least as regards the symbol for multiplication.

Colon: a Resume of Current Practice

It is clear that different publishing houses follow slightly different conventions. All seem to be agreed that the decimal separator should normally be the period. For multiplication, The Royal Society recommends the use of a x b or a - b, without indicating a preference, while other authorities advise against the midpoint. The following table provides a summary of the three current conventions in use, not only in the particular given regions, but also in those parts of the world that come under their respective influence:

'Old' Continental US British Europe

3x2 3-2 3.2 3.2

31/5 3.2 3.2 3,2

3200 3,200 3,200 3.200

Keywords: Decimals; History.

Author Chris Weeks, Downeycroft, Virginstow, Beaworthy, Devon EX21 5EA.

36 Mathematics in School, January 2003 The MA web site www.m-a.org.uk

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