When the dividend is 1, the divisor and quotient are reciprocals of each other. Hence, Multiplying by the reciprocal of a number is equivalent to dividing by the number. Dividing by the reciprocal of a number is equivalent to multiplying by the number. The quotient of a power by another power of the same number may be expressed by writing the number with an exponent equal to the exponent of the dividend diminished by the exponent of the divisor. Any number with a minus exponent is the reciprocal of the number with an equal plus exponent. The quotient of two equal powers of the same number may be expressed by unity, or by the number with zero for an exponent. That is, The zeroth power of any number is equal to 1. To test the accuracy of the work in Division, multiply the divisor and quotient together. The product should. be equal to the dividend. When a parenthesis includes two or more numbers, the included numbers must first be reduced to a single number, and the result put in place of the parenthesis. In contracted division of decimals: First, determine the number of significant figures required in the quotient. Secondly, begin to cut off figures from the right of the divisor, when the number of figures still required in the quotient is two less than the number of digits in the divisor. Thirdly, in multiplying the divisor by each quotient figure, multiply the figure of the divisor cut off, and carry the nearest ten. Prove it. Thus (10+1) (10 −1) = 152-1 11x9= 102-1 1179+1=102 CHAPTER VIII. MISCELLANEOUS EXERCISES. BEFORE giving any more examples, we shall give a method of testing the accuracy of results in the four operations by what is called CASTING OUT NINES. 170. A product of two integral factors is called a multiple of either of its factors. Every power of 10 is one more than some multiple of 9. Thus, 10=9+1; 102=11×9+1; 103=111 × 9+1, etc. Every multiple of a power of 10 by a single digit is, therefore, some multiple of 9, plus that digit. For example, 500 = 5 × 11 × 9+5; 7000 777 x 9+7, etc. But as every number consists of the sum of such multiples of powers of 10, every number is a multiple of 9, plus the sum of its own digits. Thus, 24,573 is a multiple of 9 plus 2+4+5+7+3. If a number, therefore, be divided by 9, the remainder will be the same as that arising from dividing the sum of its digits by 9. In finding the remainder from dividing the sum of the digits by 9, we may, of course, omit the nines, or any two or three digits which we see at a glance will make 9. Thus, to find the remainder on dividing 1,926,754 by 9, we see at once that 1, 2, 6, and 5, 4, make nines, and the single 7 will be the remainder. So in 254,786, we reject 5, 4, and 2, 7, and only add 8+6, from the sum of which reject 9, and there is left 5. 171. This truth may be applied to test the accuracy of results obtained in addition, subtraction, multiplication, From the several numbers to be added, the nines are cast out; then, from the sum of the several remainders, a final remainder of 6 is obtained, which corresponds with remainder obtained by casting out the nines from the sum. Therefore, the work may be presumed to be correct. The nines are cast out from the minuend and subtrahend, and the remainder of the subtrahend is subtracted from that of the minuend, giving a final remainder, 5, which corresponds with the remainder obtained by casting out the nines from the difference. It is obvious, that a number of nines + a remainder, subtracted from a number of nines + a remainder, will give an exact number of nines + the difference of the remainders. 1 (3) 51,786,531 = a number of nines +0 nines+Subtract. +8 Since the remainder from casting out nines from the minuend is less than that of the subtrahend, it is necessary to add a nine to the remainder of the minuend. =45 x 54+ 2 x 54 + 45 × 7+2×7. 45 × 54, 2 × 54, and 45 × 7 are a number of nines. That is, the entire product is a number of nines +2 ×7; or, since 2×7=9+5, the entire product is a number of nines +5. And the product of 47 × 61, namely, 2867, is a number of nines+5; therefore, the work is presumed to be correct. The work may be arranged as follows: It is important to observe, that if there be an error which this test does not indicate, the error must be a multiple of nine: |