12. Terms used.—The representation of numbers by spoken sounds is called numeration; their representation by written signs is called notation. The systems adopted for numeration and for notation do not always agree with one another; nor do they always correspond with the idea which the numbers subjectively present. This latter presentation may, in the absence of any accepted term, be called number-ideation; this word covering not only the perception or recognition of particular numbers, but also the formation of a number-concept.
13. Notation of Numbers.—The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system. The essential features which distinguish this from other systems are (1) the limitation of the number of different symbols, only ten being used, however large the number to be represented may be; (2) the use of the zero to indicate the absence of number; and (3) the principle of local value, by which a symbol in effect represents different numbers, according to its position. The symbols denoting a number are called its digits.
A brief account of the development of the system will be found under [Numeral]. Here we are concerned with the principle, the explanation of which is different according as we proceed on the grouping or the counting system.
(i) On the grouping system we may in the first instance consider that we have separate symbols for numbers from “one” to “nine,” but that when we reach ten objects we put them in a group and denote this group by the symbol used for “one,” but printed in a different type or written of a different size or (in teaching) of a different colour. Similarly when we get to ten tens we denote them by a new representation of the figure denoting one. Thus we may have:
| ones | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| tens | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| hundreds, | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| &c. | &c. | &c. |
On this principle 24 would represent twenty-four, 24 two hundred and forty, and 24 two hundred and four. To prevent confusion the zero or “nought” is introduced, so that the successive figures, beginning from the right, may represent ones, tens, hundreds, ... We then have, e.g., 240 to denote two hundreds and four tens; and we may now adopt a uniform type for all the figures, writing this 240.
(ii) On the counting system we may consider that we have a series of objects (represented in the adjoining diagram by dots), and that we attach to these objects in succession the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, repeating this series indefinitely. There is as yet no distinction between the first object marked 1 and the second object marked 1. We can, however, attach to the 0’s the same symbols, 1, 2, ... 0 in succession, in a separate column, repeating the series indefinitely; then do the same with every 0 of this new series; and so on. Any particular object is then defined completely by the combination of the symbols last written down in each series; and this combination of symbols can equally be used to denote the number of objects up to and including the last one (§ 10).
In writing down a number in excess of 1000 it is (except where the number represents a particular year) usual in England and America to group the figures in sets of three, starting from the right, and to mark off the sets by commas. On the continent of Europe the figures are taken in sets of three, but are merely spaced, the comma being used at the end of a number to denote the commencement of a decimal.