2. Find a 4th proportional to 768, 381 and 9780. 53. When one number is to be subtracted from another, it is often convenient, first to subtract it from 10, then to add the difference to the other number, and afterwards to reject the 10. Thus, instead of a-b, we may put 10-b+a-10. In the first of these expressions, b is subtracted from a. In the other, b is subtracted from 10, the difference is added to a, and 10 is afterwards taken from the sum. The two expressions are equivalent, because they consist of the same terms, with the addition, in one of them, of 10-10-0. The alteration is, in fact, nothing more than borrowing 10, for the sake of convenience, and then rejecting it in the result. Instead of 10, we may borrow, as occasion requires, 100, 1000, &c. Thus a-b-100-b+a-100-1000-b+a-1000, &c. 54. The DIFFERENCE between a given number and 10, or 100, or 1000, &c. is called the ARITHMETICAL COMPLEMENT of that number. The arithmetical complement of a number consisting of one integral figure, either with or without decimals, is found, by subtracting the number from 10. If there are two inte gral figures, they are subtracted from 100; if three, from 1000, &c. Thus the arithmetical compl't of 3.46 is 10-3.46=6,54 of 34.6 is 100-34.6=65.4 of 346. is 1000-346.=654. &c. According to the rule for subtraction in arithmetic, any number is subtracted from 10, 100, 1000, &c. by beginning on the right hand, and taking each figure from 10, after increasing all except the first, by carrying 1. The difference, or arith'l comp't is 2.36875, which is obtained, by taking 5 from 10, 3 from 10, 2 from 10, 4 from 10, 7 from 10, and 8 from 10. But, instead of taking each figure, increased by 1, from 10; we may take it without being increased, from 9. Thus 2 from 9 is the same as 3 from 10, 3 from 9, the same as 4 from 10, &c. Hence, 55. To obtain the ARITHMETICAL COMPLEMENT of a number, subtract the right hand significant figure from 10, and each of the other figures from 9. If, however, there are ciphers on the right hand of all the significant figures, they are to be set down without alteration. In taking the arithmetical complement of a logarithm, if the index is negative, it must be added to 9; for adding a negative quantity is the same as subtracting a positive one. (Alg. 81.) The difference between -3 and +9, is not 6, but 12. 56. The principal use of the arithmetical complement, is in working proportions by logarithms; where some of the terms are to be added, and one or more to be subtracted. In the Rule of Three or simple proportion, two terms are to be added, and from the sum, the first term is to be subtracted. But if, instead of the logarithm of the first term, we substitute its arithmetical .complement, this may be added to the sum of the other two, or more simply, all three may be added together, by one operation. After the index is diminished by 10, the result will be the same as by the common method. For subtracting a number is the same, as adding its arithmetical complement, and then rejecting 10, 100, or 1000, from the sum. (Art. 53.) In the following proportion, the calculation is made in both ways. If the profit on 2625 dollars employed in trade, is 831 dollars; what is the profit on 536 dollars? The second method here, after rejecting 10, gives the same result as the first. But it is unnecessary, first to add two of the terms, and then the arithmetical complement of the othThe three may be added at once; and it will generally be expedient, to place the terms in the same order, in which they are arranged in the statement of the proportion. er. 57. In compound, as in single proportion, the term required may be found by logarithms, if we substitute addition for multiplication, and subtraction for division. Ex. 1. If the interest of $365, for 3 years and 9 months, be $82.13; what will be the interest of $8940, for 2 years and 6 months? In common arithmetic, the statement of the question is made in this manner, S 8940 dollars { And the method of calculation is, to divide the product of the third, fourth, and fifth terms, by the product of the two first.* This, if logarithms are used, will be to subtract the sum of the logarithms of the two first terms, from the sum of the logarithms of the other three. Two first terms 365 log. 2.56229 3.75 0.57403 Sum of the logs. of the 3d, 4th, and 5th 6.26378 Do. Term required 1st and 2d 1341 *See Webber's Arithmetic. 3.13632 3.12746 58. The calculation will be more simple, if, instead of subtracting the logarithms of the two first terms, we add their arithmetical complements. But it must be observed, that each arithmetical complement increases the index of the logarithm by 10. If the arithmetical complement be introduced into two of the terms, the index of the sum of the logarithms will be 20 too great; if it be in three terms, the index will be 30 too great, &c. The result is the same as before, except that the index of the logarithm is 20 too great. Ex. 2. If the wages of 53 men for 42 days be 2200 dollars; what will be the wages of 87 men for 34 days? 59. In the same manner, if the product of any number of quantities, is to be divided, by the product of several others; we may add together the logarithms of the quantities to be divided, and the arithmetical complements of the logarithms of the divisors. Ex. If 29.67 × 346.2 be divided by 69.24 × 7.862 × 497; what will be the quotient? |