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ARITHMETIC

In using an ordinal we direct our attention to a term of a series, while in using a cardinal we direct our attention to the interval between two terms. The total number in the series is the sum of the two cardinal numbers obtained by counting up to any interval from the beginning and from the end respectively; but if we take the ordinal numbers from the beginning and from the end we count one term twice over. Hence, if there are 365 days in a year, the 100th day from the beginning is the 266th, not the 265th, from the end.

8. Meaning of Names of Numbers.—What do we mean by any particular number, e.g. by seven, or by two hundred and fifty-three? We can define two as one and one, and three as one and one and one; but we obviously cannot continue this method for ever. For the definition of large numbers we may employ either of two methods, which will be called the grouping method and the counting method.

(i) Method of Grouping.—The first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition. Thus, on the denary system (§16) we can give independent definitions to the numbers up to ten, and then regard (e.g.) fifty-three as a composite number made up of five tens and three ones. Or, on the quinary-binary system, we need only give independent definitions to the numbers up to five; the numbers six, seven, ... can then be regarded as five and one, five and two, ..., a fresh series being started when we get to five and five or ten. The grouping method introduces multiplication into the definition of large numbers; but this, from the teacher’s point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary arithmetical processes.

(ii) Method of Counting.—The second method consists in taking a series of names or symbols for the first few numbers, and then repeating these according to a regular system for successive numbers, so that each number is defined by reference to the number immediately preceding it in the series. Thus two still means one and one, but three means two and one, not one and one and one. Similarly two hundred and fifty-three does not mean two hundreds, five tens and three ones, but one more than two hundred and fifty-two; and the number which is called one hundred is not defined as ten tens, but as one more than ninety-nine.

9. Concrete and Abstract Numbers.—Number is concrete or abstract according as it does or does not relate to particular objects. On the whole, the grouping method refers mainly to concrete numbers and the counting method to abstract numbers. If we sort objects into groups of ten, and find that there are five groups of ten with three over, we regard the five and the three as names for the actual sets of groups or of individuals. The three, for instance, are regarded as a whole when we name them three. If, however, we count these three as one, two, three, then the number of times we count is an abstract number. Thus number in the abstract is the number of times that the act of counting is performed in any particular case. This, however, is a description, not a definition, and we still want a definition for “number” in the phrase “number of times.”

10. Definition of “Number.”—Suppose we fix on a certain sequence of names “one,” “two,” “three,” ..., or symbols such as 1, 2, 3, ...; this sequence being always the same. If we take a set of concrete objects, and name them in succession “one,” “two,” “three,” ..., naming each once and once only, we shall not get beyond a certain name, e.g. “six.” Then, in saying that the number of objects is six, what we mean is that the name of the last object named is six. We therefore only require a definite law for the formation of the successive names or symbols. The symbols 1, 2, ... 9, 10, ..., for instance, are formed according to a definite law; and in giving 253 as the number of a set of objects we mean that if we attach to them the symbols 1, 2, 3, ... in succession, according to this law, the symbol attached to the last object will be 253. If we say that this act of attaching a symbol has been performed 253 times, then 253 is an abstract (or pure) number.

Underlying this definition is a certain assumption, viz. that if we take the objects in a different order, the last symbol attached will still be 253. This, in an elementary treatment of the subject, must be regarded as axiomatic; but it is really a simple case of mathematical induction. (See Algebra.) If we take two objects A and B, it is obvious that whether we take them as A, B, or as B, A, we shall in each case get the sequence 1, 2. Suppose this were true for, say, eight objects, marked 1 to 8. Then, if we introduce another object anywhere in the series, all those coming after it will be displaced so that each will have the mark formerly attached to the next following; and the last will therefore be 9 instead of 8. This is true, whatever the arrangement of the original objects may be, and wherever the new one is introduced; and therefore, if the theorem is true for 8, it is true for 9. But it is true for 2; therefore it is true for 3; therefore for 4, and so on.

11. Numerical Quantities.—If the term number is confined to number in the abstract, then number in the concrete may be described as numerical quantity. Thus £3 denotes £1 taken 3 times. The £1 is termed the unit. A numerical quantity, therefore, represents a certain unit, taken a certain number of times. If we take £3 twice, we get £6; and if we take 3s. twice, we get 6s., i.e. 6 times 1s. Thus arithmetical processes deal with numerical quantities by dealing with numbers, provided the unit is the same throughout. If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract. But in the latter case it must always be understood that there is some unit concerned, and the results have no meaning until the unit is reintroduced.

II. Notation, Numeration and Number-Ideation

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. 