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In this introductory treatise on Arithmetic, the utmost simplicity of language has been studied in expressing the rules; and in every instance their meaning is shewn by examples wrought out at length before the eye of the pupil, accompanied with detailed explanations of the mode of working the questions.
In the present edition, the exercises and explanatory notes have been greatly extended, and many new rules have been added, that were not given in the former edition ; such as Equation, Partnership, the Square and Cube Roots, Duodecimals, &c. The work will be found to contain all the ordinary rules taught at school.
An account of the proposed Decimal System of reckoning money, with examples and exercises, has been given as an Appendix.
INTRODUCTION TO ARITHMETIC.
ARITHMETIC is the art of counting or reckoning by means of Numbers.
Number expresses either a unit--that is, one of anything, as one man, one table-or a collection of units of the same kind, as two horsemen, five books, a thousand feet.
NOTATION AND NUMERATION. NOTATION is the method of expressing numbers by means of certain signs or figures; thus—1, 2, 3.
NUMERATION is the art of numbering, that is, of reading or expressing numbers in words; thus-1, one; 2, two.
THE FIGURES used to express numbers are the following :one, two, three, four, five, sis, seven, eight, nine, nothing or nought 1 2 3 4 5 6 7 8 9 0
The first nine of these figures, when standing separately or singly, thus—1, 2, 3, 9, represent the numbers from one to nine. The last, 0, called a nothing or nought, expresses no number, and has no value in itself; but it is used to affect the value of the other figures when annexed to them, as will presently be explained.
It is by means of these ten figures and their combinations that all numbers are expressed.
Each of these figures, besides the simple value which it has when standing alone, as 1, one, 2, two, &c., has also, when standing in connection with other figures, a local value—that is, a value depending on the place it occupies in the number.
Thus, 4 by itself means simply four, namely, four units; but if it become the second figure from the right, by a nothing or any other figure being placed after it-tlius, 40, or 45—the 4 has ten times its former value, and means 4 tens, or forty; and if it become the third from the right, by another figure being annexed --thus, 400, or 456–the 4 is again increased tenfold in value, and means 4 hundreds; and so on, the addition of each new figure increasing tenfold the value of those before it.
It is in this way that the nothing, though it has no value in itself, affects the value of the other figures, by altering their place or rank. Each of the other figures has a siinilar effect on the rest when annexed to them.