What rank does the cipher occupy? Why? Ans. Because there are no No. 3. - What is the value of the 3 ? The 4 ? The 2 ? What rank does the cipher occupy here? Why? No. 4. Why is there no cipher in this number? No. 5. What is the value of the first figure on the right? Why? The fourth from the right? Why thousands ? The third ? Why? How many times is the third greater than the second ? The third than the first ? The fourth than the second ? The fourth than the third ? The fourth than the first? How many times is the first contained in the second ? In the fourth ? In the third ? How many times is the second contained in the fourth? In the third ? How many times is the third contained in the fourth ? No. 6. — What is the use of the cipher here ? Why is there none in the place of thousands ? Ans. Because the cipher is useless, unless it stands, &c. [Show this principle by an example on the blackboard.] Does the cipher stand for any number? What would this number be, if the cipher were omitted ? If another cipher were placed beside the first, thus : [place one] what effect would it produce on the 2? Ans. Its value would be fold. What effect would be produced on the 3 ? If a cipher were placed after the 3 [place one), what effect would be produced on the number? Would both the 2 and 3 be increased tenfold ? No. 7. If another cipher were introduced between the two ls, what effect would be produced, that is, what figures would change their value ? Add a cipher after the 2, and then say What change is thus produced, on each figure severally, and on the whole number? No. 8. — What effect would a cipher produce on this number, if placed to the left of the 1? To the right of the 1 ? Between the 5 and 6 ? After the 6? No. 9. What effect would a cipher produce on this number, if placed to the left of the 1 ? On its right? Beside the other cipher? To the right of the 2? How many are 10 times 26 ? How many tens in 2050 ? How many hundreds in 2500 ? How many tens in 3700 ? How many hundreds in 2000 ? Tens in 540 ? Tens in 270 ? [While proceeding with the following sections, the class should still be exercised in notation and numeration, as above, varied till the subject is perfectly familiar.] SECTION XVI. - Multiplication by Higher Numbers. 1. How many are 12ty? Ans. A hundred and twenty. Why? Because 10ty are a hundred, and 2ty are twenty. How many are 15ty, then? How many are 13ty? 16ty ? 18ty ? 17ty ? 14ty ? 19ty ? 2. How many are 20ty ? Ans. Two hundred. Why? Because each of the 10ty make a hundred. How many are 24ty? 27ty? 23ty? 36ty? 11ty? 45ty? 72ty? 69ty? 37ty? 84ty ? [Continue and extend similar questions till sufficiently familiar.] 3. How many are 100ty? Ans. A thousand. Why? Because ty means tens, and ten times 100 are a thousand. How many are 160ty ? 140ty ? 170ty? 240ty? 110ty ? 520ty? 370ty? [Continue and extend till familiar.] 4. How many are 124ty? Why? 356ty? Why? 247ty? 563ty? 116ty? 218ty? 311ty ? &c. [In reviewing, these questions should be varied by asking, How many are 10 times 12, 16, 84, 270, &c., in place of 12ty, 16ty, 84ty, 270ty, &c.] 5. How many are 2 times 20 ? Why? Because, as 2 times 2 are 4, 2 times 2ty are 4ty. How many are 2 times 30 ? 50 ? 40 ? 70? Why? Because, as 2 times 7 are 14, 2 times 7ty must be 14ty. 60 ? 90 ? 80 ? 6. How many are 3 times 20 ? 40 ? Ans. 12ty or 120. 50? 30 ?70? 90 ? 60 ? 80 ? 7. How many are 4 times 20 ? 50? 30 ? 60 ? 40 ? 90 ? 70 ? 80 ? 8. How many are 5 times 20 ? 90 ? 30? 70? 60 ? 40 ? 80 ? 50 ? 9. How many are 6 times 20 ? 40 ? 70 ? 30? 90 ? 50 ? 80 ? 60 ? 10. How many are 7 times 20 ? 80 ? 60 ? 40 ? 50 ? 30 ? 90? 70 ? 11. How many are 8 times 20? 30? 90? 70? 50 ? 80 ? 60 ? 40 ? 12. How many are 9 times 20 ? 90 ? 60? 70 ? 40 ? 80 ? 50 ? 30 ? 13. How many are 2 times 13? How do you know? Ans. Because 2 times 10 are 20 and 2 times 3 are 6. [In oral arithmetic, the higher order should always be multiplied first, because the figures are thus taken in their natural order, but chiefly because in practice it is found more easy and convenient.] 2 times 14? Why? [Repeat why after the questions that follow, till the reasoning is perfectly familiar.] 2 times 15 ? 16 ? 24 ? 27 ? 34 ? 45 ? 17 ? 47 ?. 28 ? 19 ? 39 ? 14. How many are 3 times 13? 15 ? 14? 16 ? 19 ? 17 ? 21 ? 18 ? 24 ? 37 ? Why? Because 3 times 30 are 9ty, and 3 times 7 are 2ty one; together 11ty one, or a hundred and eleven. 3 times 54 ? Why? 72? Why? 87, &c. ? 96 ? 38 ? 34 ? 37 ? 15. How many are 4 times 13? 19 ? 26 ? 54? 18? 72 ? 87 ? 96 ? 38 ? 34 ? 37 ? 53 ? 62 ? 99 ? 79 ? 88 ? 56 ? 49 ? 16. How many are 5 times 15 ? 13 ? 17 ? 26 ? 22 ? 23? 32? 47 ? 73 ? 31 ? 54? 27? 85 ? 96 ? 74 ? 17. How many are 6 times 13? 18? 15 ? 17 ? 19? 54? 36 ? 28? 72 ? 69 ? 93 ? 77 ? 65 ? 59? 48? 18. How many are 7 times 18? 13? 15 ? 19? 14? 27 ? 94 ? 36 ? 52 ? 73 ? 87 ? 76? 84 ? 55 ? 29 ? 19. How many are 8 times 13? 19 ? 16? 14? 17 ? 94 ? 85 ? 22 ? 73 ? 87 ? 54 ? 45 ? 95 ? 17 ? 57 ? 20. How many are 9 times 27 ? 35 ? 13?18? 72 ? 81 ? 58 ? 62 ? 73 ? 95 ? 46? 32 ? 17 ? 29 ? 55 ? 84 ? 21. How many are two times 126? 2 times 524 ? 2 times 346 ? 725 ? 274 ? 373 ? 644? 375 ? 863? 588? 453 ? 22. How many are 3 times 132 ? 144 ? 365 ? 427 ? 629 ? 863? 275 ? 529 ? 246 ? 23. How many are 4 times 132 ? 321 ? 126 ? 428 ? 125 ? 637 ? 528 ? 276 ? 677 ? 24. How many are 5 times 132 ? 234 ? 621 ? 532 ? 724 ? 452 ? 671 ? 346 ? 248? 25. How many are 6 times 132 ? 342 ? 254 ? 524? 362 ? 241 ? 526?728? 126 ? 26. How many are 7 times 132 ? 241 ? 324 ? 246 ? 542 ? 233 ? 621 ? 126? 272 ? 27. How many are 8 times 132 ? 244 ? 166 ? 342 ? 725 ? 637 ? 256 ? 428 ? 572 ? 28. How many are 9 times 132 ? 214? 326 ? 148 ? 522 ? 615 ? 926 ? 328? 218? SECTION XVII. - Definitions. . 1. WHEN two or more unequal numbers are joined together into one, the process is called addition, and the whole number is called the sum, or amount. Thus, joining 2 and 4 to make 6, or 3, 4, and 5, to make 12, is called adding those numbers, and 6 is called the sum, or the amount, of 2 and 4, and 12 the sum or the amount of 3, 4, and 5. 2. When two or more equal numbers are joined into one, the process is called multiplication. The number which is to be repeated is called the multiplicand, and the number which shows how many times the multiplicand is to be repeated is called the multiplier, and the increased number, or the multiplicand repeated as often as is required, is called the product. Thus, the process 4 times 5 are 20, is multiplication. The 4, which shows the number of times that 5 is to be taken, is the multiplier, 5 is the multiplicand, and 20 is the product. A more convenient name for the multiplicand and multiplier, as it applies equally to both, is that of factor. It is evident that both may always be called by the same name, since 4 times 5 is the same as 5 times 4, a remark applicable to any two numbers whatever. The word factor, in this connection, signifies maker ; product signifies the number made, or produced. Multiplication, then, is nothing but a short way of performing addition, when the numbers to be added are equal. For, to say 4 times 5 are 20, is precisely the same as to say 5 and 5 and 5 and 5 are 20. 3. When one number is to be taken away once from another number, the process is called subtraction. The number to be diminished is called the minuend, the number to be taken away the subtrahend, and the number remaining after the subtrahend is taken away is called the difference or remainder. Thus, if we take 5 from 8, 3 will remain. Here 8 is the minuend, or number to be diminished; 5 the subtrahend, or number to be subtracted, or taken away; and 3 the difference, or remainder. 4. When many subtractions of the same number are to be performed, or when we wish to find how many times one number can be taken from another, the process is called division. This is, evidently, nothing more than a short way of performing subtraction, since it comes to precisely the same thing, whether we find, at once, that 5 is contained in 20 4 times, which is called division, or produced by the slower method called subtraction, taking 5 from 20 as many times as possible, thus changing the 20 to 15, to 10, to 5, and to 0. The number to be divided is called the dividend ; the number by which we divide is called the divisor ; and the result of the division is called the quotient. Thus, if it be required to find how many times 4 is contained in 20, 4 is the divisor, 20 the dividend, and 5, the number of times that 4 is contained in 20, is the quotient. Sometimes the divisor is not contained an exact number of times in the dividend, and, consequenily, there will be a remainder at the close of the operation. Thus, if it be required to find how many times 5 is contained in 22, we find it to be 4 times, and 2 over. The 2 is the remainder, and it forms an undivided part of the dividend. 5. It is evident that the dividend is a product of the divisor and quotient, since, if 4 be contained 5 times in 20, it is plain that 4 times 5 will make 20, and so of any numbers whatever. As the divisor and quotient, then, may be considered factors of the dividend, division may be defined the process for finding one factor when the product and the other factor are given. When a remainder occurs, as this remainder is an undivided portion of the dividend, it must be added in if the divisor and quotient are multiplied to reproduce the dividend. Thus, if there be 4 fives in 23, and 3 over, the dividend evidently consists of 3 more than the 4 fives. [Show this on the blackboard.] 6. The termination end, ent, or and, in several of these terms, is derived from a Latin word signifying being, or thing. In this connection it stands for number. Hence, multiplicand signifies the number to be multiplied; minuend, the number to be diminished; subtrahend, the number to be subtracted; dividend, the number to be divided, and quotient, the number showing how many (the quota) times the divisor is contained in the dividend. The termination er, or or, signifies a man, or thing, that works, as in the words baker, miller, printer, farmer, &c. Hence, the multiplier, the factor, and the divisor, are the numbers by which the work is performed, whether in multiplication or division. 7. Signs, or characters, have been invented to express these different processes. Thus, a vertical cross, t, is the sign of addition, and an inclined cross, in the shape of the letter X, X, is the sign of multiplication, or contracted addition. Thus, |