(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).