Elements of Algebra

CHAP. XXIII. OF ALGEBRA. 71 values diminish; and at last, those values become so small, that they may be considered as nothing *. 245. This is the kind of numbers which we call decimal fractions, and in this manner logarithms are represented in the Tables. The logarithm of 2, for example, is expressed by $0.3010300$; in which we see, 1st. That since there is $0$ before the point, this logarithm does not contain an integer; 2dly, that its value is $\frac{3}{10} + \frac{0}{100} + \frac{1}{1000} + \frac{0}{10000} + \frac{3}{100000} + \frac{0}{1000000} + \frac{0}{10000000}$. We might have left out the two last ciphers, but they serve to shew that the logarithm in question contains none of those parts, which have $1000000$ and $10000000$ for the denominator. It is however to be understood, that, by continuing the series, we might have found still smaller parts; but with regard to these, they are neglected, on account of their extreme minuteness. 246. The logarithm of 3 is expressed in the Table by $0.4771213$; we see, therefore, that it contains no integer, and that it is composed of the following fractions: $\frac{4}{10} + \frac{7}{100} + \frac{7}{1000} + \frac{1}{10000} + \frac{2}{100000} + \frac{1}{1000000} + \frac{3}{10000000}$. But we must not suppose that the logarithm is thus expressed with the utmost exactness; we are only certain that the error is less than $\frac{1}{10000000}$; which is certainly so small, that it may very well be neglected in most calculations. 247. According to this method of expressing logarithms, that of 1 must be represented by $0.0000000$, since it is really $= 0$: the logarithm of 10 is $1.0000000$, where it evidently is exactly $= 1$: the logarithm of 100 is $2.0000000$, or $2$. And hence we may conclude, that the logarithms of all numbers, which are included between $10$ and $100$, and * The operations of arithmetic are performed with decimal fractions in the same manner nearly, as with whole numbers; some precautions only are necessary, after the operation, to place the point properly, which separates the whole numbers from the decimals. On this subject, we may consult almost any of the treatises on arithmetic. In the multiplication of these fractions, when the multiplicand and multiplier contain a great number of decimals, the operation would become too long, and would give the result much more exact than is for the most part necessary; but it may be simplified by a method, which is not to be found in many authors, and which is pointed out by M. Marie in his edition of the mathematical lessons of M. de la Caille, where he likewise explains a similar method for the division of decimals. F. T. The method alluded to in this note is clearly explained in Bonnycastle's Arithmetic.