32 MATHEMATICAL THEORY OF INVESTMENT 13. The characteristic and the mantissa for a system of loga- rithms with base 10. A logarithm which is not an integer, like any other similar number, consists of two parts, one a whole number and the other a number which may be expressed, exactly or approximately, by a decimal fraction. Moreover, any loga- rithm, or for that matter any negative number, may be written in such form that the decimal part is positive.' For example, -.6090=-!+.3010, which may be written 1.3010. Again, -1.2351 =-2+.7649 =2.7649. The change is accomplished in every case by adding 1 to the decimal part and subtracting 1 from the integral part. "When a logarithm is written in such form that the decimal part is positive, the decimal part is called the mantissa and the integral part the characteristic. In the common system the mantissa is the same for all numbers having the same sequence of figures. For moving the decimal point is equivalent to multiplying or dividing by an integral power of 10. Since 10 is the base, its logarithm is 1 and the logarithm of any integral power of 10 is an integer. By Theorems III and IV, multiplying or dividing a number by an integral power of 10 will increase or diminish the logarithm of the number by an integer, and the characteristic alone will be changed. The mantissa of a logarithm is found from a table of loga- rithms prepared in advance. The characteristic depends upon the position of the decimal point. It is easily found by means of a table showing the loga- rithms of powers of 10. Such a table may be written in the following form: lO-^.OOOl lO-^.OOl lO-^.Ol lO-^.l 10° =1 101 =10 102 =100 108 =1000 104 =10000 LOGARITHMS 33 A number with one figure to the left of the decimal point is either 1 or a number between 1 and 10, and, from the table, its logarithm must lie between 0 and 1. Such a logarithm will have 0 for its characteristic. Similarly, the logarithm of a number with two figures to the left of the decimal point is either 1 (the logarithm of 10) or it lies between 1 and 2. In either case the characteristic is 1. In general, the characteristic of the logarithm of a number greater than 1 is positive, or zero, and is-one less than the number of figures to the left of the decimal point. This statement may be turned about to give the RULE FOR POINTING A NUMBER WITH A GIVEN POSITIVE, OR ZERO, CHARACTERISTIC: The number of places to the left of the decimal point, in a number whose logarithm has a positive, or zero, characteristic, is one greater than the number of units in the characteristic. Similarly, by inspecting that part of the table which gives the negative powers of 10, we see that the characteristic of the loga- rithm of a pure decimal is negative and is numerically one greater than the number of zeros immediately following the decimal point. For example, the number .0035 lies between .001 and .01. Its logarithm, therefore, lies between — 3 and — 2 and so must be — 3 plus a positive mantissa. From the foregoing statement concerning the characteristic of the logarithm of a pure decimal we obtain the RULE FOR POINTING A NUMBER WHOSE LOGARITHM HAS A NEGATIVE CHARACTERISTIC : To point off a number whose loga- rithm has a negative characteristic, place as many zeros between the first significant figure on the left and the decimal point as there are units in the negative characteristic, less one. Negative characteristics are usually written with the negative sign above the characteristic instead of in front of it; thus, 2.3010. A still better method is to express the negative characteristic as the difference between a positive integer and 10, writing the —10 at the end. Thus, we would write 8.3010 —10 rather than 2.3010. To facilitate division of logarithms, we may increase both the positive and the negative parts of the characteristic by such a