Elements of Algebra

20 ELEMENTS SECT. I. CHAP. VII. Of Fractions in general. 68. When a number, as $7$, for instance, is said not to be divisible by another number, let us suppose by $3$, this only means, that the quotient cannot be expressed by an integer number; but it must not by any means be thought that it is impossible to form an idea of that quotient. Only imagine a line of $7$ feet in length; nobody can doubt the possibility of dividing this line into $3$ equal parts, and of forming a notion of the length of one of those parts. 69. Since therefore we may form a precise idea of the quotient obtained in similar cases, though that quotient may not be an integer number, this leads us to consider a particular species of numbers, called fractions, or broken numbers; of which the instance adduced furnishes an illustration. For if we have to divide $7$ by $3$, we easily conceive the quotient which should result, and express it by $\frac{7}{3}$; placing the divisor under the dividend, and separating the two numbers by a stroke, or line. 70. So, in general, when the number $a$ is to be divided by the number $b$, we represent the quotient by $\frac{a}{b}$, and call this form of expression a fraction. We cannot therefore give a better idea of a fraction $\frac{a}{b}$, than by saying that it expresses the quotient resulting from the division of the upper number by the lower. We must remember also, that in all fractions the lower number is called the denominator, and that above the line the numerator. 71. In the above fraction $\frac{7}{3}$, which we read seven thirds, $7$ is the numerator, and $3$ the denominator. We must also read $\frac{2}{3}$, two thirds; $\frac{3}{4}$, three fourths; $\frac{3}{8}$, three eighths; $\frac{12}{100}$, twelve hundredths; and $\frac{1}{2}$, one half, &c. 72. In order to obtain a more perfect knowledge of the nature of fractions, we shall begin by considering the case in which the numerator is equal to the denominator, as in $\frac{a}{a}$. Now, since this expresses the quotient obtained by dividing $a$ by $a$, it is evident that this quotient is exactly unity, and that consequently the fraction $\frac{a}{a}$ is of the same