Elements of Algebra

CHAP. VII. OF ALGEBRA. 21 value as 1, or one integer; for the same reason, all the following fractions, $2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 8/8$, &c. are equal to one another, each being equal to 1, or one integer. 73. We have seen that a fraction whose numerator is equal to the denominator, is equal to unity. All fractions therefore whose numerators are less than the denominators, have a value less than unity: for if I have a number to divide by another, which is greater than itself, the result must necessarily be less than 1. If we cut a line, for example, two feet long, into three equal parts, one of those parts will undoubtedly be shorter than a foot: it is evident then, that $2/3$ is less than 1, for the same reason; that is, the numerator 2 is less than the denominator 3. 74. If the numerator, on the contrary, be greater than the denominator, the value of the fraction is greater than unity. Thus $3/2$ is greater than 1, for $3/2$ is equal to $2/2$ together with $1/2$. Now $2/2$ is exactly 1; consequently $3/2$ is equal to $1 + 1/2$, that is, to an integer and a half. In the same manner, $4/3$ is equal to $1\ 1/3$, $5/3$ to $1\ 2/3$, and $7/3$ to $2\ 1/3$. And, in general, it is sufficient in such cases to divide the upper number by the lower, and to add to the quotient a fraction, having the remainder for the numerator, and the divisor for the denominator. If the given fraction, for example, were $43/12$, we should have for the quotient 3, and 7 for the remainder; whence we should conclude that $43/12$ is the same as $3\ 7/12$. 75. Thus we see how fractions, whose numerators are greater than the denominators, are resolved into two members; one of which is an integer, and the other a fractional number, having the numerator less than the denominator. Such fractions as contain one or more integers, are called improper fractions, to distinguish them from fractions properly so called, which having the numerator less than the denominator, are less than unity, or than an integer. 76. The nature of fractions is frequently considered in another way, which may throw additional light on the subject. If, for example, we consider the fraction $3/4$, it is evident that it is three times greater than $1/4$. Now, this fraction $1/4$ means, that if we divide 1 into 4 equal parts, this will be the value of one of those parts; it is obvious then, that by taking 3 of those parts we shall have the value of the fraction $3/4$. In the same manner we may consider every other fraction; for example, $7/12$; if we divide unity into 12 equal parts, 7 of those parts will be equal to the fraction proposed.