Elements of Algebra

70 ELEMENTS SECT. I. whatever fraction we assume on this condition, the power resulting from it will be always an irrational number, greater or less than $2$; and, consequently, the logarithm of $2$ cannot be accurately expressed by such a fraction: therefore we must content ourselves with determining the value of that logarithm by such an approximation as may render the error of little or no importance; for which purpose, we employ what are called *decimal fractions*, the nature and properties of which ought to be explained as clearly as possible. 243. It is well known that, in the ordinary way of writing numbers by means of the ten figures, or characters, $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, the first figure on the right alone has its natural signification; that the figures in the second place have ten times the value which they would have had in the first; that the figures in the third place have a hundred times the value; and those in the fourth a thousand times, and so on: so that as they advance towards the left, they acquire a value ten times greater than they had in the preceding rank. Thus, in the number $1765$, the figure $5$ is in the first place on the right, and is just equal to $5$; in the second place is $6$; but this figure, instead of $6$, represents $10 \times 6$, or $60$; the figure $7$ is in the third place, and represents $100 \times 7$, or $700$; and lastly, the $1$, which is in the fourth place, becomes $1000$; so that we read the given number thus: One thousand, seven hundred, and sixty-five. 244. As the value of figures becomes always ten times greater, as we go from the right towards the left, and as it consequently becomes continually ten times less as we go from the left towards the right; we may, in conformity with this law, advance still farther towards the right, and obtain figures whose value will continue to become ten times less than in the preceding place: but it must be observed, that the place where the figures have their natural value is marked by a point. So that if we meet, for example, with the number $36.54892$, it is to be understood in this manner: the figure $6$, in the first place, has its natural value; and the figure $3$, which is in the second place to the left, means $30$. But the figure $5$, which comes after the point, expresses only $\frac{5}{10}$; and the $4$ is equal only to $\frac{4}{100}$; the figure $8$ is equal to $\frac{8}{1000}$; the figure $9$ is equal to $\frac{9}{10000}$; and the figure $2$ is equal to $\frac{2}{100000}$. We see then, that the more those figures advance towards the right, the more their