(ii.) Approximate Calculation.
111. Multiplication.—When we have to multiply two numbers, and the product is only required, or can only be approximately correct, to a certain number of significant figures, we need only work to two or three more figures (§ 83), and then correct the final figure in the result by means of the superfluous figures.
A common method is to reverse the digits in one of the numbers; but this is only appropriate to the old-fashioned method of writing down products from the right. A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals between .1 and 1.0. The method E of § 101 being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 101); and the multiplication is performed from the left, two extra figures being kept in. Thus, to multiply 27.343 by 3.1415927 to one decimal place, we require 2 + 1 + 1 = 4 figures in the product. The result is 085.9 = 85.9, the position of the decimal point being determined by counting the figures before the decimal points in the original numbers.
112. Division.—In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the partial product) at each stage. Thus, to divide 85.9 by 3.1415927 to two places of decimals, we in effect divide .0859 by .31415927 to four places of decimals. In the work, as here shown, a 0 is inserted in front of the 859, on the principle explained in § 106. The result of the division is 27.34.
113. Logarithms.—Multiplication, division, involution and evolution, when the results cannot be exact, are usually most simply performed, at any rate to a first approximation, by means of a table of logarithms. Thus, to find the square root of 2, we have log √2 = log (21/2) = ½ log 2. We take out log 2 from the table, halve it, and then find from the table the number of which this is the logarithm. (See [Logarithm].) The slide-rule (see [Calculating Machines]) is a simple apparatus for the mechanical application of the methods of logarithms.
When a first approximation has been obtained in this way, further approximations can be obtained in various ways. Thus, having found √2 = 1.414 approximately, we write √2 = 1.414 + θ, whence 2 = (1.414)2 + (2.818)θ + θ2. Since θ2 is less than ¼ of (.001)2, we can obtain three more figures approximately by dividing 2 − (1.414)2 by 2.818.
114. Binomial Theorem.—More generally, if we have obtained a as an approximate value for the pth root of N, the binomial theorem gives as an approximate formula p√N = a + θ, where N = ap + pap − 1θ.
115. Series.—A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations. A decimal is of course a series of this kind, e.g. 3.14159 ... means 3 + 1/10 + 4/102 + 1/103 + 5/104 + 9/105 + ... A series of aliquot parts is another kind, e.g. 3.1416 is a little less than 3 + 1⁄7 − 1⁄800.