The examples and problems which have been introduced are very numerous, and, it is believed, will be found to comprise every kind of question which is likely to come in the way of the student of Elementary Arithmetic. A chapter upon the Metric System has also been introduced, as seemed advisable, now that the Act permitting its employment has been passed. T. P. H. January, 1866. A TREATISE ON ARITHMETIC. CHAPTER I. NOTATION AND NUMERATION. 1. ARITHMETIC is the science of Numbers and the art of performing calculations by them, and investigating their relations. Any single thing-as, for instance, a pen, a sheep, a house is called a unit: we say there is one such thing. If another single thing of the same kind be put with it, there are said to be two such things; if another, three; if another, four; if another, five; and so on. Each of these collections of things of which we have spoken is a number of things; and the terms one, two, three, four, five, &c., by which we express how many single things or units are under consideration, are the names of numbers. A number therefore is a collection of units. This is also sometimes called an integer, or whole number. It will be seen that the idea of number is quite independent of the particular kind of units, a collection of which is counted. Thus, if there are four pigs, the B number of pigs is the same as if there were four pens. We can thus abstract a number from any particular unit or thing, and talk of the number four, the number five, &c. Numbers thus abstracted from their reference to any particular unit or thing are called abstract numbers. When a collection of things or objects is indicated, it is called a concrete number. We shall treat first of abstract numbers. 2. NOTATION. The art of expressing numbers by symbols, or figures, is called Notation. In the system of notation which we are about to explain, all numbers can be expressed by means of ten symbols (figures, or digits,* as they are called), representing respectively the first nine numbers, and nothing, i.e., the absence of number. These are— nine o called a nought, a cipher, or zero. N.B. Ten times ten is called one hundred; ten times a hundred, a thousand. 3. Numbers are represented by giving to the figures employed what is called a local value-i.e., a value depending upon the positions in which they are placed. Let a number of columns be drawn as on page 3, that being called the first which is on the right, and reckoning the order of the columns from right to left. *Digits. So called from digitus, a " finger." This decimal notation clearly took its origin from these natural counting instruments. |