When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.
Of especial importance is Sidney Young, Fractional Distillation (1903).
A rigorous demonstration was wanting for many years, Leonhard Euler's proof for negative and fractional values being faulty, and was finally given by Niels Heinrik Abel.
The logarithms to base io of the first twelve numbers to 7 places of decimals are log 1 =0.0000000 log 5 log 2 = 0.3010300 log 6 log 3 =0.477 121 3 log 7 log 4 =0.6020600 log 8 The meaning of these results is that The integral part of a logarithm is called the index or characteristic, and the fractional part the mantissa.
Fractional silver coin is not legal tender above 20 pesos, and bronze and nickel coins not above 1 peso, but the government maintains conversion offices where such coins can be converted into silver pesos without loss.