Reciprocals
A special case of division to be considered is the determination of the reciprocal of a number n, or 1 n. Following the ordinary rule for division, it is evident that setting n on C to 1 on D, gives 1 n on D under 1 on C. It is more important to observe that by inverting the operation—setting 1 (or 10) on C to n on D—we can read 1 n on C over 1 (or 10) on D. Hence whenever a result is read on D under an index of C, we can also read its reciprocal on C over whichever index of D is available.
The Number of Digits in a Reciprocal is obvious when n = 10, 100, or any power (p) of 10. Thus 1
10 = 0·1; 1
100 = 0·01; 1
10p = 1 preceded by p − 1 cyphers. For all other cases we have the rule:—Subtract from 1 the number of digits in the number.
Ex.—1
339 = 0·00295.
There are 3 digits in the number; hence, there are 1 − 3 = −2 digits in the answer.
Ex.—1
0·0000156 = 64,100.
There are −4 digits in the number; hence, there are 1 − (−4) = 5 digits in the result.