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526
ARITHMETIC

two halfpennies). There are, therefore, four denominations, the bases for conversion of one denomination into the next being successively four (or two), twelve and twenty. Within each denomination, however, the denary notation is employed exclusively, e.g. “twelve shillings” is denoted by 12s.

The diversity of scales appears to be due mainly to four causes: (i) the tendency to group into scores (§ 20); (ii) the tendency to subdivide into twelve; (iii) the tendency to subdivide into two or four, with repetitions, making subdivision into sixteen or sixty-four; and (iv) the independent adoption of different units for measuring the same kind of magnitude.

Where there is a division into sixteen parts, a binary scale may be formed by dividing into groups of two, four or eight. Thus the weights ordinarily in use for measuring from ¼ oz. up to 2 ℔ give the basis for a binary scale up to not more than eight figures, only 0 and 1 being used. The points of the compass might similarly be expressed by numbers in a binary scale; but the numbers would be ordinal, and the expressions would be analogous to those of decimals rather than to those of whole numbers.

In order to apply arithmetical processes to a quantity expressed in two or more denominations, we must first express it in terms of a single denomination by means of a varying scale of notation. Thus £254, 13S. 6d. may be written each of the numbers in brackets indicating the number of units in one denomination that go to form a unit in the next higher denomination. To express the quantity in terms of £, it ought to be written this would mean £254 (13 6⁄12)/20 or £(254 + 13⁄20 + 6⁄20·12), and therefore would involve a fractional number.

A quantity expressed in two or more denominations is usually called a compound number or compound quantity. The former term is obviously incorrect, since a quantity is not a number; and the latter is not very suggestive. For agreement with the terminology of fractional numbers (§ 62) we shall describe such a quantity as a mixed quantity. The letters or symbols descriptive of each denomination are visually placed after or (in actual calculations) above the figures denoting the numbers of the corresponding units; but in a few cases, e.g. in the case of £, the symbol is placed before the figures. There would be great convenience in a general adoption of this latter method; the combination of the two methods in such an expression as £123, 16s. 4½d. is especially awkward.

18. Numeration.—The names of numbers are almost wholly based on the denary scale; thus eighteen means eight and ten, and twenty-four means twice ten and four. The words eleven and twelve have been supposed to suggest etymologically a denary basis (see, however, Numeral).

Two exceptions, however, may be noted.

(i) The use of dozen, gross (= dozen dozen), and great gross (= dozen gross) indicates an attempt at a duodenary basis. But the system has never spread; and the word “dozen” itself is based on the denary scale.

(ii) The score (twenty) has been used as a basis, but to an even more limited extent. There is no essential difference, however, between this and the denary basis. As the latter is due to finger-reckoning, so the use of the fingers and the toes produced a vigesimal scale. Examples of this are given in § 20; it is worthy of notice that the vigesimal (or, rather, quinary-quaternary) system was used by the Mayas of Yucatan, and also, in a more perfect form, by the Nahuatl (Aztecs) of Mexico.

The number ten having been taken as the basis of numeration, there are various methods that might consistently be adopted for naming large numbers.

(i) We might merely name the figures contained in the number. This method is often adopted in practical life, even as regards mixed quantities; thus £57,593, 16s. 4d. would be read as five seven, five nine three, sixteen and four pence.

(ii) The word ten might be introduced, e.g. 593 would be five ten ten ninety (= nine ten) and three.

(iii) Names might be given to the successive powers of ten, up to the point to which numeration of ones is likely to go. Partial applications of this method are found in many languages.

(iv) A compromise between the last two methods would be to have names for the series of numbers, beginning with ten, each of which is the “square” of the preceding one. This would in effect be analysing numbers into components of the form a. 10b where a is less than 10, and the index b is expressed in the binary scale, e.g. 7,000,000 would be 7·104·102, and 700,000 would be 7·104·101.

The British method is a mixture of the last two, but with an index-scale which is partly ternary and partly binary. There are separate names for ten, ten times ten (= hundred), and ten times ten times ten (= thousand); but the next single name is million, representing a thousand times a thousand. The next name is billion, which in Great Britain properly means a million million, and in the United States (as in France) a thousand million.

19. Discrepancies between Numeration and Notation.—Although numeration and notation are both ostensibly on the denary system, they are not always exactly parallel. The following are a few of the discrepancies.

(i) A set of written symbols is sometimes read in more than one way, while on the other hand two different sets of symbols (at any rate if denoting numerical quantities) may be read in the same way. Thus 1820 might be read as one thousand eight hundred and twenty if it represented a number of men, but it would be read as eighteen hundred and twenty if it represented a year of the Christian era; while 1s. 6d. and 18d. might both be read as eighteenpence. As regards the first of these two examples, however, it would be more correct to write 1,820 for the former of the two meanings (cf. § 13).

(ii) The symbols 11 and 12 are read as eleven and twelve, not (except in elementary teaching) as ten-one and ten-two.

(iii) The names of the numbers next following these, up to 19 inclusive, only faintly suggest a ten. This difficulty is not always recognized by teachers, who forget that they themselves had to be told that eighteen means eight-and-ten.

(iv) Even beyond twenty, up to a hundred, the word ten is not used in numeration, e.g. we say thirty-four, not three ten four.

(v) The rule that the greater number comes first is not universally observed in numeration. It is not observed, for instance, in the names of numbers from 13 to 19; nor was it in the names from which eleven and twelve are derived. Beyond twenty it is usually, but not always, observed; we sometimes instead of twenty-four say four and twenty. (This latter is the universal system in German, up to 100, and for any portion of 100 in numbers beyond 100.)

20. Other Methods of Numeration and Notation.—It is only possible here to make a brief mention of systems other than those now ordinarily in use.

(i) Vigesimal Scale.—The system of counting by twenties instead of by tens has existed in many countries; and, though there is no corresponding notation, it still exhibits itself in the names of numbers. This is the case, for instance, in the Celtic languages; and the Breton or Gaulish names have affected the Latin system, so that the French names for some numbers are on the vigesimal system. This system also appears in the Danish numerals. In English the use of the word score to represent twenty—e.g. in “threescore and ten” for seventy—is superimposed on the denary system, and has never formed an essential part of the language. The word, like dozen and couple, is still in use, but rather in a vague than in a precise sense.

(ii) Roman System.—The Roman notation has been explained above (§ 15). Though convenient for exhibiting the composition of any particular number, it was inconvenient for purposes of calculation; and in fact calculation was entirely (or almost entirely) performed by means of the abacus (q.v.). The numeration was in the denary scale, so that it did not agree absolutely with the notation. The principle of subtraction from a higher number, which appeared in notation, also appeared in numeration, but not for exactly the same numbers or in exactly the same way; thus XVIII was two-from-twenty, and the next number was one-from-twenty, but it was written XIX, not IXX. 