By practice the pupil will soon find it unnecessary to divide into periods any lines of figures, except those of considerable magnitude. Exercises in converting the Expression of Numbers from Figures to Words. Write down in words, or name, the numbers signified by the following expressions: — RULE II. To express numbers by figures: (1.) Make a sufficient number of ciphers or dots, and divide them into periods of three each. (2.) Then, commencing at the left, place, in their proper positions, beneath the dots or ciphers, the value figures necessary for expressing the proposed number. (3.) If any places remain unoccupied, let them be filled with ciphers. Thus, the method of expressing the number, two hundred and five millions, twenty thousand, seven hundred, and nine, will be found in the following and thence, by filling the unoccupied places, 205,020,709. By practice the learner will soon be able to dispense with the use of the dots or ciphers. Exercises in converting the Expression of Numbers from Words to Figures. Express the following numbers in figures: Exer. 1. Fifty-two. 2. Three hundred. 3. Five hundred and four. 4. One thousand and twenty-four. 5. Two thousand. 6. One thousand, eight hundred, and fifteen. Exer. 7. Seven thousand, eight hundred, and fifty-four. 8. Three thousand and eight. 9. Five thousand and seventy. 10. Four thousand, five hundred, and four. 12. Six hundred and fifty thousand, and ninety. 15. Eleven millions, two thousand. 16. One hundred and ten millions, and twenty thousand. 17. One million, and fifty thousand. 18. One billion, two hundred thousand. 19. Seventy billions, ten thousand, and eightyeight. 20. Nine hundred billions, sixty-eight millions, and twenty. 21. The following numbers express the distances of the principal primary planets from the sun, in miles; express them in figures:-Mercury, thirty-five millions, five hundred thousand; Venus, sixty-six millions, three hundred thousand; the Earth, ninety-one millions, six hundred thousand; Mars, one hundred and forty millions; Jupiter, four hundred and seventy-six millions; Saturn, eight hundred and seventy-four millions Uranus, one thousand, seven hundred and sixty millions; Neptune, two thousand, seven hundred and fifty millions.* Such is the facility with which large numbers are expressed, both by figures and in language, that we have generally a very imperfect conception of their real magnitudes. The following considerations will assist in enlarging the ideas of the pupil on this subject :-To count a million, at the rate of one in each second of time, would require between twenty-three and twenty-four days of twelve hours each. The seconds in six thousand years are less than one fifth of a trillion.† *The distance of Uranus might be stated as one billion, seven hundred and sixty millions; and that of Neptune as two billions, seven hundred and fifty millions of miles: but the mode of naming these distances used above is well suited for conveying the information in this case, because here the million of miles may conveniently be considered as a unit of distance, and the distances of these planets may be thought of as so many millions of miles, instead of being thought of as so many miles. In like manner large sums of money are often spoken of in millions of pounds, instead of in pounds. Thus people would often prefer to say two and a half millions of pounds, rather than two millions, five hundred thousand pounds. This one-fifth of a trillion of the new numeration would be designated as one-fifth of a billion, in the common or old numeration. A quadrillion* of leaves of paper, each the two-hundredth part of an inch in thickness, would form a pile, the height of which would be three hundred and thirty times the moon's distance from the earth. Let it be remembered, also, that a million is equal to a thousand repeated a thousand times; and a billion t is equal to a million repeated a thousand times. In the ancient Roman notation, I. signified one, V. five, X. ten, L. fifty, and C. one hundred. To these characters were added, at a later period, D., signifying five hundred, and M., one thousand. When any character was followed by another of equal, or of less value, the compound value was equal to the simple values of both taken together; but when a character preceded one of greater value, both together expressed a value equal to the difference of their simple values. Thus, II. expressed two; XI., eleven, and IX., nine; CX., one hundred and ten, and XC., ninety. We find also I. put for 500; and by every such annexed, the value is made ten times as great. Thus, I. signifies 5000; 10., 50,000, &c. CIO. was also used to express 1000, and the prefixing of C and the annexing of. increased the value ten times. Thus, CCIO. signified 10,000; CCCIɔɔɔ, 100,000, &c. A line drawn over a letter, made it signify as many thousands as the letter itself expressed units. Thus, V. expressed 5000; C., 100,000, &c. The following table, together with the preceding observations, will give an adequate idea of the Roman notation : THE object of ADDITION, considered generally, is to find the number or quantity which is equivalent to two or more given numbers or quantities taken together. The number or quantity which is equivalent to two or more given numbers or quantities taken together is called their SUM. When simply numbers in the proper sense of the term-that is to say, numbers without fractions-are to be added, the operation is called SIMPLE ADDITION. When quantities expressed simply by numbers of units, the units being all of one magnitude, or, as it may be said, all of one denomination and all whole units, as *This quadrillion of the new numeration is the same as a thousand billions in the old numeration. But a billion of the old numeration is equal to a million repeated a million of times. all yards, or all gallons, are to be added, the arithmetical operation consists simply in the adding of the mere numbers which express the quantities; and is, in like manner, called SIMPLE ADDITION. When quantities expressed numerically by units of one denomination together with fractional parts of the same units, or else by various fractional parts of the same units without whole units, are to be added (for instance, if a length of 3 inches and of an inch, and a length of 1 inch and of an inch are to be added together; or else if of an inch and of an inch, and of an inch are to be added together), the operation is not simple, but is somewhat complicated, and is called FRACTIONAL ADDITION, or ADDITION OF FRACTIONS. When quantities numerically expressed in different units, or, what means the same, when quantities in the expression of which units of different denominations occur, are to be added, the operation is called COMPOUND ADDITION. The quantities to be added must be all of the same kind, though the magnitudes and denominations of the units in which they are expressed may be different. Thus the quantities to be added may be all money, and so all of the same kind, while they may be expressed by numbers of pounds, numbers of shillings, and numbers of pence; or they may be all time, and so all of one kind, while expressed in numbers of hours, numbers of minutes, and numbers of seconds. Quantities of different kinds do not admit of being added together in arithmetic. Thus we cannot add together five inches, four gallons, and three minutes, to find either how many, or how much, of any kind of thing there would be. The sign+, called plus, is employed in arithmetic and other parts of mathematics, to signify that the numbers or quantities between which it is placed, are to be added together; and the sign =, called the sign of equality, is used to denote, that the numbers or quantities between which it stands, are equal to one another. Thus, the expression, 12 + 9 = 21, which is read, 12 plus 9 (that is, 12 more by 9) equal to 21, means that 12 and 9, added together, amount to 21; or that the sum of 12 and 9 is 21. RULE FOR SIMPLE ADDITION. To add numbers or to add quantities expressed by numbers of units all of one magnitude or denomination: (1.) Place the numbers so that units may stand under units, tens under tens, &c. (2.) Find the sum of the column of units, set down the last figure of it below that column, and carry to the next the number expressed by the remaining figure, or figures, if there be any. (3.) Proceed as before with the remaining columns, and at the last column set down its entire amount. Exam. 1. 9468 Thus, to add together 9468, 2956, and 79, let them be set as in the margin, and proceed thus:-9 and 6 are 15, and 8 are 23; set down 3, and carry 2 to the column of tens. Then 2 and 7 are 9, and 5 are 14, and 6 are 20; set down 0, and carry 2: 2 and 9 are 11, and 4 are 15; set down 5, and carry 1: 1 and 2 are 3, and 9 are 12; set down 12 and the sum, or answer required, is twelve thousand, five hundred, and three. 2956 79 12503, sum. It is to be particularly observed, however, that after the acquirement of a little experience in arithmetic, a pupil, or a person doing addition in actual business, ought to use far fewer words, aloud or mentally, than those which, for the purpose of explanation, are stated in this example. For instance, in adding the second column, counted from right towards left, he ought not to say, "Two and seven are nine, and five are fourteen, and six are twenty," but ought only to say quickly, aloud or mentally, the following words, or even fewer still:-Carry two, nine, fourteen, twenty." He should then set down 0 and go on with carrying two to the next column. Advice of the same kind might be given in reference to many other examples for explaining arithmetical processes in what follows throughout this book; but the present explanation may suffice for guarding the pupil against a practice which sometimes interferes with attainment of quickness and ease in arithmetical work. Reason of the Rule, The rule for performing addition depends on the nature of notation, and on the obvious principle, that the whole is equal to the sum of all its parts. By placing units under units, tens under tens, &c., we are enabled the more easily to add together the figures of the corresponding local values; and one is carried for every ten, because, by the nature of notation, ten in any column is equivalent only to one in the column immediately to the left of it. We commence with the units merely for the convenience of carrying to the next columns. Thus, in the preceding example, the sum of the column of units is 23; and therefore, after setting down 3, we have 20 remaining. But, by the nature of notation, 2 in the next column is equal to 20 in this; and therefore we carry only 2. Some teachers may, perhaps, consider it proper to make pupils commit the following table to memory : |