The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa. When the base is 10, the logarithms of all numbers in which the digits are the same, no matter where the decimal point may be, have the same mantissa; thus, for example, log 2.5613 = 0.4084604, log 25.613 = 1.4084604, log 2561300 = 6.4084604, &c. In the case of fractional numbers (i.e. numbers in which the integral part is 0) the mantissa is still kept positive, so that, for example, _ _ log .25613 = 1.4084604, log .0025613 = 3.4084604, &c. the minus sign being usually written over the characteristic, and not before it, to indicate that the characteristic only, and not the whole expression, is negative; thus _ 1.4084604 stands for -1 + .4084604. The fact that when the base is 10 the mantissa of the logarithm is independent of the position of the decimal point in the number affords the chief reason for the choice of 10 as base. The explanation of this property of the base 10 is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact. It should be mentioned that in most tables of trigonometrical functions, the number 10 is added to all the logarithms in the table in order to avoid the use of negative characteristics, so that the characteristic 9 denotes in reality ~1, 8 denotes ~2, 10 denotes 0, &c. Logarithms thus increased are frequently referred to for the sake of distinction as _tabular logarithms_, so that the tabular logarithm = the true logarithm + 10. In tables of logarithms of numbers to base 10 the mantissa only is in general tabulated, as the characteristic of the logarithm of a number can always be written down at sight, the rule being that, if the number is greater than unity, the characteristic is less by unity than the number of digits in the integral portion of it, and that if the number is less than unity the characteristic is negative, and is greater by unity than the number of ciphers between the decimal point and the first significant figure. It follows very simply from the definition of a logarithm that log(a) b × log(b) a = 1, log(b) m = log(a) m × (1/log(a) b). The second of these relations is an important one, as it shows that from a table of logarithms to base a, the corresponding table of logarithms to base b may be deduced by multiplying all the logarithms in the former by the constant multiplier 1/log(a)b, which is called the _modulus_ of the system whose base is b with respect to the system whose base is a.