The figures of this number are divided by commas, into portions of three figures each, beginning at the right; but the last division on the left, which in the present instance has but two figures, may sometimes have but one. Each of these divisions corresponds to the collections designated by the words unit, thousand, million, billion, trillion, and their figures express successively the units, tens, and hundreds of each. (Thus, the expression †The following are the denominations generally adopted in English and American books on arithmetic : It will be seen that the two methods agree for the nine first places, reckoning from the left; that beyond this the same written number, or figure, is read differently, or bears different names, in in words of a proposed number is formed, by reading each division of figures as if it stood alone, and adding, after its units, the name belonging to this division.) The above example is read, twenty-four trillions, eight hundred and ninety-seven billions, three hundred and twenty-one millions, five hundred and eighty thousands, three hundred and forty-six units. 9 Numbers admit of being considered in two ways; one is, when no particular denomination is mentioned, to which their units belong, and they are then called abstract numbers; the other, when the denomination of their units is specified, as when we say, two men, five years, three hours, &c. these are called concrete numbers. It is evident, that the formation of numbers, by the successive union of units, is independent of the nature of these units, and that this must also be the case with the properties resulting from this formation; by which properties we are enabled to compound and decompound numbers, which is called calculation. We shall now explain the principal rules for the calculation of numbers, without regard to the nature of their units. Addition. 10. THIS operation, which has for its object the uniting of several numbers in one, is only an abbreviation of the formation. of numbers by the successive union of units. If, for instance, it were required to add five to seven, it would be necessary, in the the two systems. Seven billions, for instance, in French is the same as seven thousands of millions in English. The difference, however, is only in the name; the written character, and the fundamental principles of numeration, are the same in both languages. The French begin with a new name after every three figures, the English after every six. These collections of six, however, are subdivided into portions of three; so that the French method is, on the whole, the more simple, and on this account more easily retained in the memory, while at the same time the names of the higher orders of numbers are much shorter. series of the names of numbers, one, two, three, four, five, six, seven, &c., to ascend five places above seven, and we should then come to the word twelve, which is consequently the amount of seven units added to five. It is upon this process that the addition of all small numbers depends, the results of which are committed to memory; its immediate application to larger numbers would be impossible, but in this case, we suppose these numbers divided into the different collections of units contained in them, and we may add together those of the same name. For instance, to add 27 to 32, we add the 7 units of the first number to the 2 of the second, making 9; then the 2 tens of the first with the 3 of the second, making 5 tens. The two results, taken together, form a total of 5 tens and 9 units, or 59, which is the sum of the numbers proposed. What is here said applies to all numbers, however large, that are to be added together; but it is necessary to observe, that the partial sums, resulting from the addition of two numbers, each expressed by a single figure, often contain tens, or units of the next higher collection, and these ought consequently to be joined to their proper collection. In the addition of the numbers 49 and 78, the sum of the units 9 and 8 is 17, of which we should reserve 10, or ten, to be added to the sum of the tens in the given numbers; next we say that 4 and 7 make 11, and joining to this the ten we reserved, we have 12 for the number of tens contained in the sum of the given numbers; which sum, therefore, contains 1 hundred, 2 tens, and 7 units, that is, 127. 11. By proceeding on these principles, a method has been devised of placing numbers that are to be added, which facilitates the uniting of their collections of units, and a rule has been formed, which the following example will illustrate. Let the numbers be 527, 2519, 9812, 73, and 8; in order to add them together, we begin by writing them under each other, placing the units of the same order in the same column; then we draw a line to separate them from the result, which is to be written underneath it. We at first find the sum of the numbers contained in the colum of units to be 29, we write down only the nine units, and reserve the 2 tens, to be joined to those which are contained in the next column, which, thus increased, contains 13 units of its own order; we write down here only the three units, and carry the ten to the next column. Proceeding with this column as with the others, we find its sum to be 19; we write down the 9 units and carry the ten to the next column, the sum of which we then find to be 12; we write down the 2 units under this column and place the ten on the left of it; that is, we write down the sum of this column, as it is found. By this means we obtain 12939 for the sum of the given numbers. 12. The rule for performing this operation may be given thus, Write the numbers to be added under each other, so that all the units of the same kind may stand in the same column, and draw a line under them. Beginning at the right, add up successively the numbers in each column; if the sum does not exceed 9, write it beneath its column, as it is found; if it contains one or more tens, carry them to the next column; lastly, under the last column write the whole of its sum.t †The best method of proving addition is by means of subtraction. The learner may, however, in general, satisfy himself of the correctness of his work by beginning at the top of each column and adding down, or by separating the upper line of figures and adding up the rest, and then adding this sum to the upper line. Examples for practice. Add together 8635, 2194, 7421, 5063, 2196 and 1225. Ans. 26734. Add together 84371, 6250, 10, 3842, and 631 Ans. 95104. Ans. 19366. Add together 7861, 345, 8023. Ans. 16229. Ans. 246373. Add together 66947, 46742, and 132684. Subtraction. 13. AFTER having learned to compose a number by the addition of several others, the first question that presents itself is, how to take one number from another that is greater, or, which amounts to the same thing, to separate this last into two parts, one of which shall be the given number. If, for instance, we have the number 9, and we wish to take 4 from it, we should, by doing this, separate it into two parts, which by addition would be the same again. To take one number from another, when they are not large, it is necessary to pursue a course opposite to that prescribed in the beginning of article 10, for finding their sum; that is, in the series of the names of numbers, we ought to begin from the greatest of the numbers in question, and descend as many places as there are units in the smallest, and we shall come to the name given to the difference required. Thus, in descending four places below the number nine we come to five, which expresses the number that must be added to 4 to make 9, or which shows how much 9 is greater than 4. In this last point of view, 5 is the excess of 9 above 4. If we only wished to show the inequality of the members 9 and 4, without fixing our attention on the order of their values, we should say that their difference was 5. Lastly, if we were to go through the operation of taking 4 from 9,, we should say that the remainder is 5. Thus we see that, although the words, excess, remainder, and difference, are synonymous, each answers to a particular manner of considering the separation of the number 9 into the parts 4 and 5, which operation is always designated by the name subtraction. ction.) |