Fractions

Art. 131. EXPRESSIONS in the form of fractions occur more frequently in Algebra than in arithmetic. Most instances in division belong to this class. Indeed the numerator of every fraction may be considered as a dividend, of which the denominator is a divisor.

According to the common definition in arithmetic, the denominator shows into what parts an integral unit is supposed to be divided; and the numerator shows how many of these parts belong to the fraction. But it makes no difference, whether the whole of the numerator is divided by the denominator; or only one of the integral units is divided, arfd then the quotient taken as many times as the number of units in the numerator. Thus $\frac{3}{4}$ is the same as $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$. A fourth part of three dollars, is equal to three fourths of one dollar.

132. The value of a fraction, is the quotient of the numerator divided by tho denominator.

Thus the value of $\frac{6}{2}$ is $3$. The value of $\frac{ab}{b}$ is $a$.

From this it is evident, that whatever changes are made in the terms of a fraction ; if the quotient is not altered, the value remains the same. For any fraction, therefore, we* may substitute any other fraction which will give the same quotient.

Thus $\frac{4}{2}=\frac{10}{5}=\frac{4ba}{2ba}=\frac{8drx}{4srx}=\frac{6+2}{3+1}$. For the quotient in each of these instances is $2$.

133. As the value of a fraction is the quotient of the numerator divided by the denominator, it is evident from Art. 125, that when the numerator is equal to the denominator, the value of the fraction is a unit; when the numerator is less than the denominator, the value is less than a unit; and when the numerator is greater than the denominator, the value is greater than a unit.

The calculations in fractions depend on a few general principles, which will here be stated in connexion with each other.

134. If the denominator of a fraction remains the same, multiplying the numerator by any quantity, is multiplying the value by that quantity; and dividing the numerator, is dividing the value. For the numerator aud denominator are a dividend and divisor, of which the value of the fraction is the quotient. And by Art. 127 and 128, multiplying the dividend is in effect multiplying the quotient, and dividing the dividend is dividing the quotient.

Thus in the fractions $\frac{ab}{a}, \frac{3ab}{a}, \frac{7abd}{a},c$.

The quotients or values are $b, 3b, 7bd, c$.

Here it will be seen that, while the denominator is not altered, the value of the fraction is multiplied or divided by the same quantity as the numerator.

Cor. With a given denominator, the greater the numerator, the greater will be the value of the fraction ; and, on the other hand, the greater the value, the greater the numerator.

135. If the numerator remains the same, multiplying the denominator by any quantity, is dividing the value by that quantity; and dividing the denominator, is multiplying the value. For multiplying the divisor is dividing the quotient; and dividing the divisor is multiplying the quotient. (Art. 129, 130.)

In the fractions $\frac{24ab}{6b}, \frac{24ab}{12b}, \frac{24ab}{3b}, \frac{24ab}{b}, c$.

The values are $4a, 2a, 8a, 24a, c$.

Cor. With a given numerator, the greater the denominator, the less will be the value of the fraction ; and the less the value, the greater the denominator.

136. From the last two articles it follows, that dividing the numerator by any quantity, will have the same effect on the value of the fraction, as multiplying the denominator by that quantity; and multiplying the numerator will have the same effect, as dividing the denominator.

137. It is also evident from the preceding articles, that if THE NUMERATOR AND DENOMINATOR BE BOTH MULTIPLIED, OR BOTH DIVIDED, BY THE SAME QUANTITY, THE VALUE OF THE FRACTION WILL NOT BE ALTERED.

138. Any integral quantity may, without altering its value, be thrown into the form oh a fraction, by multiplying the quantity into the proposed denominator, and taking the product for a numerator.
Thus $a = \frac{a}{1}=\frac{ab}{b}=\frac{ad+ah}{d+h}= \frac{6adh}{6dh}, \& c$. For the quotient of each of these is $a$.

139. There is nothing, perhaps, in the calculation of algebraic fractions, which occasions niore perplexity to a learner, than the positive and negative signs. The changes in these are so frequent, that it is necessary to become familiar with the principles on which they are made. The use of the sign which is prefixed to the dividing line, is to show whether the value of the whole fraction is to be added to, or subtracted from, the other quantities with which it is connected. (Art. 43.) This sign, therefore, has an influence on the several terms taken collectively. But in the numerator and denominator, each sign affects only the single term to which it is applied.

The value of $\frac{ab}{b}$ is $a$. (Art. 132.) But this will become negative, if the sign - be prefixed to the fraction.

Thus $y+\frac{ab}{b} = y + a$. But $y - \frac{ab}{b} = y - a$.

So that changing the sign which is before the whole fraction, has the effect of changing the value from positive to negative, or from negative to positive.

Next, suppose the sign or signs of the numerator to be changed.

By Art. 120, $\frac{ab}{b}=+a$. But $\frac{-ab}{b} = -a$.

And $\frac{ab - bc}{b} = +a - c$. But $\frac{-ab + bc}{b} = -a + c$.

That is, by changing all the signs of the numerator, the value of the fraction is changed from positive to negative, or the contrary.

Again, suppose the sign of the denominator to be changed.

As before $\frac{ab}{b} = +a$. But $\frac{ab}{-b} = -a$.

140. We have then, this general proposition; If the sign prefixed to a fraction, or all the signs of the numerator, or all the signs of the denominator be changed; the value of the fraction will be changed, from positive to negative, or from negative to positive.

From this is derived another important principle. As each of the changes mentioned here is from positive to negative, or the contrary ; if any two of them be made at the same time, they will balance each other.

Thus by changing the sign of the numerator,
$\frac{ab}{b}= +a$ becomes $\frac{-ab}{b}= -a$.

But, by changing both the numerator and denominator, it becomes $\frac{-ab}{-b}=+a$, where the positive value is restored.

By changing the sign before the fraction, $y + \frac{ab}{b} = y + a$ becomes $y -\frac{ab}{b}= y - a$.

But by changing the sign of the numerator also, it becomes $y - \frac{(-ab)}{b}$ where the quotient $-a$ is to be subtracted from $y$, or which is the same thing, (Art. 78) $+a$ is to be added, making $y + a$ as at first. Hence,

141. If all the signs both of the numerator and denominator, or the signs of one of these with the sign prefixed to the whole fraction, be changed at the same time, the value of the fraction will not be altered.

Thus $\frac{6}{2} = \frac{-6}{-2} = -\left(\frac{-6}{2}\right) = -\frac{6}{-2}= +3$.

Hence the quotient in division may be set down in different ways. Thus $\frac{a-c}{b}$ is either $\frac{a}{b} + \left(\frac{-c}{b}\right)$, or $\frac{a}{b} - \frac{c}{b}$.

The latter,method is the most common. See the exam pies in Art. 124.


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