APPENDIX V.
ON CHARACTERISTICS.
When the student comes to use logarithms, he will find what follows very useful. In the mean while, I give it merely as furnishing a rapid rule for finding the place of a decimal point in the quotient before the division is commenced.
When a bar is written over a number, thus, 7 let the number be called negative, and let it be thus used: Let it be augmented by additions of its own species, and diminished by subtractions; thus, 7 and 2 give 9 , and let 7 with 2 subtracted give 5 . But let the addition of a number without the bar diminish the negative number, and the subtraction increase it. Thus, 7 and 4 are 3 , 7 and 12 make 5, 7 with 8 subtracted is 15 . In fact, consider 1, 2, 3, &c., as if they were gains, and 1 , 2 , 3 , as if they were losses: let the addition of a gain or the removal of a loss be equivalent things, and also the removal of a gain and the addition of a loss. Thus, when we say that 4 diminished by 11 gives 7, we say that a loss of 4 incurred at the moment when a loss of 11 is removed, is, on the whole, equivalent to a gain of 7; and saying that 4 diminished by 2 is 6 , we say that a loss of 4, accompanied by the removal of a gain of 2, is altogether a loss of 6.
By the characteristic of a number understand as follows: When there are places before the decimal point, it is one less than the number of such places. Thus, 3·214, 1·0083, 8 (which is 8·00 ...) 9·999, all have 0 for their characteristics. But 17·32, 48, 93·116, all have 1; 126·03 and 126 have 2; 11937264·666 has 7. But when there are no places before the decimal point, look at the first decimal place which is significant, and make the characteristic negative accordingly. Thus, ·612, ·121, ·9004, in all of which significance begins in the first decimal place, have the characteristic 1 ; but ·018 and ·099 have 2 ; ·00017 has 4 ; ·000000001 has 9 .
To find the characteristic of a quotient, subtract the characteristic of the divisor from that of the dividend, carrying one before subtraction if the first significant figures of the divisor are greater than those of the dividend. For instance, in dividing 146·08 by ·00279. The characteristics are 2 and 3 ; and 2 with 3 removed would be 5. But on looking, we see that the first significant figures of the divisor, 27, taken by themselves, and without reference to their local value, mean a larger number than 14, the first two figures of the dividend. Consequently, to 3 we carry 1 before subtracting, and it then becomes 2 , which, taken from 2, gives 4. And this 4 is the characteristic of the quotient, so that the quotient has 5 places before the decimal point. Or, if abcdef be the first figures of the quotient, the decimal point must be thus placed, abcde·f. But if it had been to divide ·00279 by 146·08, no carriage would have been required; and 3 diminished by 2 is 5 ; that is, the first significant figure of the quotient is in the 5th place. The quotient, then, has ·0000 before any significant figure. A few applications of this rule will make it easy to do it in the head, and thus to assign the meaning of the first figure of the quotient even before it is found.