15. The value of the exports from the United States in 1841, was $104691534. If an equal amount had been exported each day of the year excepting Sundays, what would it have been? OF THE PROPERTIES OF THE 9's. 83. Besides the methods already explained of proving the operations in figures, there is yet another called the method by casting out the 9's. That method we will now explain. -84. An excess of units over exact 9's, is the remainder after the number has been divided by 9: hence, any number less than 9 must be treated as an excess over exact 9's. Let us write down the numbers to be added, as at the right. Now, if we divide each number by 9, and place the quotients to the right, and the remainders in the column still to the right, we shall have, in the middle column, the exact number of 9's contained in each number, and in the column at the right, the excesses over exact 9's. By adding these columns, we find 1,5 OPERATION. Quotients after di- Remainders, or ex 3870 ... 430 ...0 2708 ... 300. 304. cesses over 9. 8 33 ... 7 9)6882 764 764-6 6 in the column of remainders, which is equal to one 9 and 6 over: hence, there are 764 exact 9's and 6 over. But it is evident that the sum of all the numbers, viz., 6882, must contain exactly the same number of 9's and the same excess over exact 9's, as are found in the numbers taken separately, since a whole is equal to the sum of all its parts any way taken: therefore, in the sum of any numbers whatever, the number of exact 9's and the excess over 9's are equal, respectively, to the aggregate of exact 9's and the excess of 9's in the numbers taken separately. QUEST.-83. What other methods of proof are there for arithmetical operations, besides those already explained? 84. What is an excess of 9's? How do the exact number of 9's and the excess of 9's in any sum compare with the exact 9's and the excess of 9's in the several numbers? 85. We will now explain a short process of finding the excess over an exact number of 9's in any number whatever; and to do this, we must look a little into the formation of numbers. In any number, written with a single significant figure, as 4, 50, 600, 8000, &c., the excess over exact 9's will always be equal to the number of units expressed by the significant figure; for, in any such number we shall always have 4 4 Also, &c. &c. 50 (9 +1)x5 600 (99 +1)×6 8000 = (999+1)×8 &c. Each of the numbers 9, 99, 999, &c., expresses an exact number of 9's; and hence, when multiplied by 5, 6, 8, &c., the several products will each contain an exact number of 9's; therefore, the excess over exact 9's, in each number, will be expressed by 4, 5, 6, 8, &c. If, then, we write any number whatever, as 6253, we may read it 6 thousand 2 hundred 50 and 3. Now, Now, the excess of 9's in the 6-thousand is 6; in 2 hundred it is 2; in 50 it is 5; and in 3 it is 3: hence, in them all, it is 16, which makes one 9 and 7 over: therefore, 7 is the excess over exact 9's in the number 6253. Hence, the excess over exact 9's in any number whatever, may be found by adding together the significant figures, and rejecting the exact 9's from the sum. NOTE. It is best to reject or drop the 9 as soon as it occurs : thus we say, 3 and 5 are 8 and 2 are 10; then dropping the 9, we say, 1 to 6 is 7, which is the excess; and the same for all similar operations. 1. What is the excess of 9's in 48701? In 67498? 2. What is the excess of 9's in 9472021? In 2704962 ? 3. What is the excess of 9's in 87049612? In 4987051 ? QUEST.-85. What will be the excess over exact 9's in any number expressed by a single significant figure? How may the excess over exact 9's be found in any number whatever? PROOF OF ADDITION BY CASTING OUT THE 9's. OPERATION. Excess of 9's. 94874 . 5 46073 2 50498 3674 341 8 2 8 7 195460-7 86.-1. In the first of these numbers we find the excess of 9's to be 5; in the second 2; in the third 8; in the fourth 2; and in the fifth 8: hence, in them all it is 25, which leaves 7 for the excess over exact 9's. We also find 7 to be the excess over exact 9's in the sum 195460: hence the work is supposed to be right. Notwithstanding this proof, it is possible, after all, that the work may be erroneous. For example, if either figure in the sum is too large by one or more units, and any other figure is too small by the same number of units, the excess over exact 9's will not be affected. But as it would seldom happen that one error would be exactly balanced by another, the work when proved may be relied on as correct. Similar sources of error exist a the proof of all arithmetical operations. 2. Add together, 8754608, 4908721, 6027983, 89704543, 2142367, and ̄28949760, and prove the result by rejecting the 9's. 3. Add together 40799903, 874162, 32704931, 6704192, 2146748, 94004169, and prove the result by casting out the 9's. OPERATION. 874136. 2 45302 5 828834 6 PROOF OF SUBTRACTION BY CASTING OUT THE 9's. 87.-1. Since the sums of the remainder and subtrahend must be equal to the minuend, it follows that the excess of 9's in these two numbers must be equal to the excess of 9's in the minuend: hence, to the excess of 9's in the remainder add the excess of 9's in the subtrahend, and the excess of 9's in the sum will be equal to the excess of 9's in the minuend. QUEST.-86. Explain the proof of addition by casting out the 9's. In what is the proof defective? 87. Explain the proof of subtraction by casting out the 9's. 2. From 874096 take 370494, and prove the work by rejecting the 9's. 3. From 47096702 take 1104967, and prove the work by rejecting the 9's. PROOF OF MULTIPLICATION BY CASTING OUT THE 9's. 88. We will first remark, that if any number containing an exact number of 9's be multiplied by another whole number, the product will also contain an exact number of 9's. Let it be required to multiply any two numbers together, as 641 and 232. We first find the excess over exact 9's in both factors, and then separate each factor into two parts, one of which shall contain exact 9's, and the other the excess, and unite the two together by the sign plus. It is now required to take 639 +2 = 641, is many times as there are units in 225 +7 232. Beginning with the 7, we have 14 641 OPERATION. 639 + 2 232 225+ 7 4473 +14 450 3195 1278 1278 148698 +14 for the product of 2 by 7, and 4473 for the product of 639 by 7 ; and this last contains an exact number of 9's. We then take 2, 225 times, which gives 450, which also contains an exact number of 9's. We next multiply 639 by the figures of 225, and each of the several products contains an exact number of 9's, since 639 contains an exact number. Hence, the entire sum 148698 contains an exact number of 9's, to which if we add the one 9 from the 14, we shall find the excess of 9's in the product to be 5; and as the same may be shown for any numbers, we conclude that, the excess of 9's in any product must arise from the product of the excess of 9's in the factors. QUEST.—88. Explain the proof of multiplication by casting out the 9's. What does the excess of 9's in any product arise from? But since the product of two numbers found in the ordinary way must contain the same number of 9's, and the same excess of 9's as a product found above, it follows that, if the excesses of 9's in any number of factors be multiplied together, the excess of 9's in such product will be equal to the excess of 9's in the product of the factors. When the excess of 9's in any factor is 0, the excess of 9's in the product is always 0. PROOF OF DIVISION BY CASTING OUT THE 9's. 89. Since the divisor multiplied by the quotient must produce the dividend, it follows that if the excess of 9's in the divisor be multiplied by the excess of the 9's in the quotient, the excess of 9's in the product will be equal to the excess of 9's in the dividend. 1. The dividend is 8162315040, the divisor 61835720, and the quotient 132: is the work right? 2. The dividend is 10264849920, the divisor 1440, and the quotient 7128368: is the work right? 3. The dividend is 74855092410, the quotient 78795, and the divisor 949998: is the work right? Let the pupils apply the property of the 9's to other examples. QUEST.-If the excess of 9's in any number of factors be multiplied together, what will the excess of 9's in the product be equal to? 89. How do you prove division by casting out the 9's? |