ARITHMETIC. CHAPTER I. 1. ARITHMETIC is the art of expressing numbers, and the mode of computing by them. 2. NUMBER signifies a unit, or more units than one. 3. All numbers are expressed by the following ten characters or figures, either singly or in combination, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, read, nought or cypher, one, two, three, four, five, six, seven, eight, nine. 4. When any one of these figures stands by itself, it has what is called its simple value, which never varies. 5. When two or more figures are used to express a number, all except that in the units' place (to the right of the other figures), have what is termed a local value, which varies according to the distance from the units' place. Thus, if a figure stands one place to the left, its local value is ten times its simple value; if two places to the left, its local value is one hundred times its simple value, and so on. 6. 1, 3, 5, 7, &c., are called odd numbers: 2, 4, 6, 8, &c., are called even numbers. 7. NOTATION is the art of expressing numbers by characters or figures: numeration, the art of reading them when written. B 8. For the convenience of reading and writing numbers, we divide them into periods, each consisting of three places of figures, as in the following TABLE. This TABLE has three periods. Units, Thousands, Millions, consisting of three figures each, which, commencing at the right in every period, are invariably read, units, tens, hundreds. In this Table, to show the sameness of the places which form cach period, the units' place is marked by one dot, the tens' by two, and the hundreds' by three dots. The periods are all alike as regards the places which form them. It follows, that if we can write numbers consisting of three places of figures, we can write any number, however great. We must be careful to write the numbers in their proper periods. For instance, if we require to write six, we place it in the first place to the right; because it is 6 ones or units, we must place it in the “units' period." Again, if we wish to write six thousand, we have merely to remove the 6 from its position in the units' period to a similar one in the thousands' period; thus, 6,000, that is, we pass over the first period, the places of which must be filled up with cyphers; the great use of the cypher being to fill up the places which would otherwise be vacant. In the same manner, if we have to write three hundred and six units, three hundred and six thousand, and three hundred and six millions; we have, respectively, 306; 306,000; 306,000,000, each number being placed in its proper period. 9. The next period to the left, after millions, is called billions, next trillions, and so on, to an indefinite extent. The number of periods is indefinite; the number of places only three. 10. TO READ ANY NUMBER. RULE.-10. Commencing at the right, divide it into periods, allowing three places to each. 2o. Then read each period from the left, adding the name of the period to the number which its figures express. To read 43506403; by dividing it into periods, we have, 43,506,403, which we read forty-three, five hundred and six, four hundred and three. Now, by adding the names of the periods, we have, forty-three millions, five hundred and six thousand, four hundred and three (ones or units), 11. Ours being a decimal system of Notation, the figures increase to the left in a ten-fold ratio: in other words, ten units make one ten, that is, one unit in the tens' place; ten of these units in the tens' place make one hundred, and so on. 12. From the nature of NOTATION, it is evident, that if we remove a figure one place to the left, we make its value ten times greater. Thus 60 is equal to ten times 6. If we remove it two places to the left, we make its value one hundred times greater, and so on; thus 600 is equal to one hundred times 6. On the contrary, if we remove a figure one place to the right, we decrease its value ten-fold; if we remove it two places to the right, we decrease its value one hundred-fold. In the one case, the units become tens, the tens, hundreds, &c.: in the other the hundreds become tens, the tens, units. 13. A NOUGHT placed to the left of a figure or number has no effect; since it does not alter its position with respect to the units' place, it cannot change the local value of the figure (5). The following exercises have been drawn up carefully, and, like all others in this work, are formed on the progressive principle, beginning with the most simple examples. Express in figures the following numbers :1. Nine, twenty, thirty-seven, forty. 2. Forty-six, fifty-nine, sixty seven. 3. One hundred and sixty-five. 4. Two hundred and seventy-eight. 5. Five hundred and ninety. 6. Nine hundred and eleven. 7. Four hundred and nine. ANS. 9, 20, 37, 40 46, 59, 67 165 278 590 911 409 12. Four thousand five hundred and seventy. 4570 13. Five thousand nine hundred and twenty-one. 5921 14. Eight thousand and nine. 8009 15. Three thousand and seventy-six. 3076 16. Nine thousand and ninety-seven. 9097 17. Twelve thousand three hundred and ten. ANS. 12310 13097 14704 16412 19809 20074 24069 39502 49072 54089 27. Sixty-seven thousand eight huudred and four. 67804 28. Ninety thousand six hundred and seven. 29. Ninety-three thousand and eighty-five. 30. Ninety-nine thousand and ninety-nine. 90607 93085 99099 31. One hundred and four thousand and sixty-nine. 104069 32. Two hundred and six thousand seven hundred and one. 206701 33. One hundred and nine thousand and seventy. 109070 34. Three hundred and ninety-six thousand and forty four. 396044 35. Four hundred and six thousand nine hundred and ten. 406910 36. Seven hundred and nine thousand seven hundred and nine. 709709 37. Seven hundred and two thousand and seventy. 38. Five hundred and nine thousand and forty-one. 39. Nine hundred thousand and ninety-eight. 702070 509041 900098 40. Nine hundred and six thousand and eighty-five. 906085 Additional exercises will be found in Addition and all the succeeding rules.-See Notes, at the end of the volume. ADDITION. 14. ADDITION teaches to add two or more numbers or quantities of the same kind. The result or answer is called the sum. The numbers to be added are called addends. 15. The addition of abstract numbers or of applicate numbers of one denomination is called SIMPLE ADDITION. When a number is not applied to any particular object it is called an abstract number. |