[Let the above exercises be repeated and varied, till each pupil can perform them correctly with rapidity and ease.] THE FIRST PRINCIPLES OF DECIMAL ARITHMETIC. [To be committed to memory by the Pupil.] When figures are written horizontally, or side by side, I. Every figure is ten times greater than the same figure immediately on its right, and ten times less than the same figure immediately on its left. II. Every figure becomes tenfold greater by being removed one rank or place toward the left; tenfold less by being removed one rank or place toward the right. III. Ten units of any one place make one unit of the next place to the left; and one unit of any one place makes ten units of the next place to the right. IV. When there is a separatrix, the place of units is immediately on its left; when there is none, the right hand figure occupies the unit's place. V. The cipher is superfluous, unless it occupies the place of units, or intervenes between a significant figure and the place of units. Questions to be put by the Teacher.—Which are the two most important kinds of numerals? In what cases are the Roman numerals chiefly used ? What is their probable origin? What does the capital I represent? The V? The X? What is the origin of the C and M? Of the L and D? On what principle are the Roman numerals horizontally arranged ? Ans. When a smaller one, &c. What is the probable origin of the Arabic numerals? Of how many characters does the Arabic notation consist ? Why are the first nine called significant figures? Why called digits? What is the tenth character called ? What is its use? What is meant by the simple value of a figure? By its local value ? Why is our system of computation called Decimal Arithmetic ? What change is made in the value of a figure by removing it one place or rank to the left? One place to the right? Three places to the left? Two to the right? How is a number increased tenfold, a hundred fold, &c., or multiplied by 10, 100, 1000, &c. ? How is a number decreased tenfold, a hundred fold, &c., or divided by 10, 100, 1000, &c. ? What is the meaning of the word unit? Does the word apply to more than one rank of figures ? What is the use of the reversed comma, or separatrix? What is the objection to the use of the comma or the period as a separatrix? Where is the place of units when there is a separatrix ? When there is none ? How many ranks or orders of figures make a period? Why should large numbers be separated into periods? Repeat the names of the first five periods. By what character should the periods be separated ? Are the names of the orders the same or different in different periods ? Repeat their names. What does the word fraction signify? On what principle does the notation of decimal fractions proceed? Ans. On the same principle as that of whole numbers. [See p. 119.] What is the principle? In dividing a number into periods, with what rank should we always commence? What is the first principle of arithmetic? The second ? The third ? The fourth? The fifth ? [This, and indeed every chapter, should be studied till the pupil can answer all the questions at the close correctly and without hesitation.] CHAPTER II. THE SHORTENED PROCESSES OF INCREASE AND DECREASE OF INTEGERS AND DECIMAL FRACTIONS. SECTION I.--Addition. [The following exercises should be transferred to the slate, and practised till the sum of each set of two, three, or more figures can be rapidly read off without spelling ; (that is, without naming the individual figures), horizontally* as well as vertically ; irregularly as well as regularly, without taking into view the local value of the figures. Every figure down to * Every pupil should learn to add horizontally as well as vertically. In ledgers of country merchants, &c., much of this kind of work is necessary. the range of stars, * ****, should be considered as of the denomination of units, the sign of addition, t, being omitted, except in the first range of exercises, merely to save room. The class should be daily exercised on the black-board with these or similar combinations. For an explanation of the signs used in Addition see Oral Arithmetic, p. 57. 1. What is the sum of 1+5 1+3 1+9 1+ 2+5 4+2 3+2 5+9 3+1 2+1 4+1 6+1 8+1 2+2 276 277 2+2 3+3 38 39 30 48 49 40 58 59 60 69 67 53 63 73 54 64 74 55 65 75 66 77 67 89 89 94 79 89 67 45 16 76 85 72 95 88 80 27 58 43 25 84 95 52 36 42 69 53 2. What is the sum of (take three figures at once, first horizontally, then vertically] 123 538 281 817 405 142 539 293 518 752 456 246 523 352 263 874 218 536 364 134 324 143 162 461 847 635 406 921 246 947 389 283 719 613 328 152 357 731 916 527 346 334 724 896 398 416 128 237 634 812 486 176 510 276 938 654 948 372 415 926 3. What is the sum of [four figures, first horizontally, then vertically. At first these may be thrown together in pairs ; afterwards all four should be caught at a glance, just as we do the letters of a word.] 3267 3948 7913 2896 5276 2468 8276 1928 9418 1234 3546 3754 3894 1357 1358 6734 7826 5678 2687 2137 1728 9123 9425 3695 8512 9123 1504 9862 9264 4567 1072 2739 1235 4372 2864 2816 4321 3142 9817 5942 6347 1893 9351 3743 1672 6417 3246 7637 9165 4715 4234 5214 3514 2738 7234 8424 2738 2637 6712 3746 6328 9516 1832 3056 [In the first range of exercises that follow, two figures in each column form 10. In the second range, it takes three figures. In the first four exercises of each range the figures that form the combination are together. In the others, they are not. The object of such an exercise is to make the pupil quick at observing the tens. The figures should be transferred to the slate, and studied till the class is prepared to read the sums on the blackboard as rapidly as words.] 4. What is the sum of (horizontally as well as vertically] 2512 4792 1741 1934 2531 1568 2541 7512 3282 6318 2235 9176 3427 5725 6138 1262 6375 1241 3624 2483 4362 4213 3421 3598 4735 5618 7486 1514 8579 5342 4972 4321 5283 2365 2756 7354 1362 2314 1485 1314 6251 1613 1613 7637 5478 2312 3252 2514 2143 2143 4352 2233 9674 6544 7387 5768 5768 5416 6563 7313 3514 6789 4334 3262 5. In any series of three figures that regularly increase or decrease by the common difference 1, such as 4, 5, 6; or 7, 6, 5; what will be the effect if the common difference, 1, be taken from the largest figure and added to the smallest ? Will the same equality be produced by such an operation when the common difference is 2, 3, or any other number, such as 2, 5, 8; or 9, 6, 3; &c. [Show this principle on the black-board, thus : 4, 5, 6 become 5, 5, 5, by carrying 1 from the 6 to the 4; and 5, 7, 9, become 7, 7, 7, by carrying two from the 9 to the 5. The same thing occurs, of course, when the numbers decrease regularly. Thus, 9, 8, 7, become 8, 8, 8; and 15, 12, 9, become 12, 12, 12. [The following exercises are designed to aid the pupil in rapidly discovering numbers with a common difference, three of which, of course, are equal to three times the mean number. Towards the beginning, the series are placed in regular order; afterwards they are arranged irregularly, many having the fourth figure interposed. In the last four exercises, the figure out of the regular series in each column is either 3, 6, or 9, which has the effect of increasing each figure in the series, by 1, 2, or 3. Thus, 5, 6, 7, and 3=3X7, and not 3X6; because each of the figures in the series, namely, 5, 6, 7, is increased 1 by the 3; while 5, 6, 7, and 6=3X8, each of the series being increased 2 by the 6; and, for a like reason, 5, 6, 7, and 9=3X9. In these exercises the numbers should be added vertically only. 5982 2947 8479 4738 3234 1394 2134 1252 3458 6171 7568 8293 5456 4676 5466 7393 2347 7282 6657 6174 7678 7958 8798 2934 1236 8393 2925 7382 9128 8213 1251 3418 2392 1237 1948 6323 3693 4921 2139 4741 4748 2982 4663 5578 5471 5843 5451 3693 1857 3454 7351 7466 6582 6765 8763 5832 6966 5676 8242 1857 7693 3969 3675 6929 Carrying the Tens. [Each figure should have its local, as well as its simple, value in the exercises that follow. In order that the pupil may thoroughly realize this, he should read the first exercise as follows: two thousand six hundred and fifty-four. Seven thousand nine hundred, &c. As some of the exercises contain decimal fractions, it will be proper occasionally to remind the pupil that numbers of different denominations, and consequently belonging to different ranks, cannot be added together. 6. Add togther the following numbers, placed vertically for the sake of convenience : The long process. The shortened process. 2654 2654 7972 7972 3814 3814 2483 2483 16923 sum total. 14 sum of the thousands. 16923 sum of the whole. Addition by these two processes amounts to precisely the same thing. The only difference is this: in the longer process the sum of each different rank is set down separately, and then added into one whole; whereas, in the shortened process, |