ART. 118.-A fraction of a fraction, as called a compound fraction. a 1 2 m ART. 119. When a fraction has a fraction, either in its numerator, or in its denominator, or in both of them, it is called a complex ART. 120.-Algebraic fractions are represented in the same manner as common fractions in Arithmetic. The number or quantity below the line, is called the denominator, because it denominates, or shows the number of parts into which the unit is divided; and the number or quantity above the line, is called the numerator, because it numbers, or shows how many parts are taken. Thus, in the fraction, 3, the denominator, 4, shows, that the unit (for instance, 1 foot,) is divided into 4 equal parts, and the numerator, 3, shows, that 3 of these parts are taken. Again, in the fraction the denominator c, shows, that a unit is divided into c c' a equal parts, and a shows, that a of these parts are taken. The numerator and denominator, are called the terms of a fraction. ART. 121. In the preceding definitions of numerator and denominator, reference is had to a unit only. This is the simplest method of considering a fraction; but there is another point of view, in which it is proper to examine it. If it be required to divide 3 apples equally, between 4 boys, it can be effected, by dividing each of the 3 apples into 4 equal parts, and then giving to each boy 3 of those parts, expressed by 2. Now, the parts being equal to each other in size, it will be the same, for an individual to receive 3 parts from 1 apple, or 1 part from each of the 3 apples; that is, of one apple, is the same as of 3 apples; or, å of 1 unit, is the same as of 3 units. Thus, may be regarded as expressing two fifths of one thing, or one fifth of two things. REVIEW. 117. What is a simple fraction? Give an example. 118. What is a compound fraction? Give an example. 119. What is a complex fraction? Give an example. 120. In Algebraic Fractions, what is the quantity below the line called? Why? Above the line? Why? Givs an example. What do yo understand by the terms of a fraction! 1 is either the fraction of one unit taken m times, or it - n is the nth of m units. Hence, the numerator may be regarded, as showing the number of units to be divided; and the denominator, as showing the divisor, or what part is taken from each. NOTE TO TEACHERS.-Although it is important that the pupil should be perfectly familiar with the principles contained in the following propositions, the demonstrations may be omitted, especially by the younger class of pupils, until the book is reviewed. PROPOSITION I. ART. 122.-If we multiply the numerator of a fraction, without changing the denominator, the value of the fraction is increased as many times as there are units in the multiplier. If we multiply the numerator of the fraction by 3, without changing the denominator, we get . Thus: 2X3 6 Now, and have the same denominator, and, therefore express parts of the same size; but the second fraction, 9, has three times as large a numerator as the first,; it therefore expresses three times as many of those equal parts as the first, and is, consequently, three times as large. And the same may be shown of any fraction whatever. PROPOSITION II. ART. 123.—If we divide the numerator of a fraction, without changing the denominator, the value of the fraction is diminished, as many times as there are units in the divisor. If we take the fraction, and divide the numerator by 2, without changing the denominator, we get. Thus: 4÷2 2 Now, and have the same denominator, and, therefore, express parts of the same size; but the numerator of the second fraction,, is only one half as large as the numerator of the first, ; it therefore expresses only one half as many of those equal parts as the first, and is, consequently, only one half as large. And the same may be shown of other fractions. REVIEW.--121. In what two different points of view may every fraction be regarded? Give examples. 122. How is the value of a fraction affected. by multiplying the numerator only? How is this proposition proved? 123. How is the value of a fraction affected by dividing the numerator only. How is this proposition proved? PROPOSITION III. ART. 124.-If we multiply the denominator of a fraction, wit out changing the numerator, the value of the fraction is diminisher as many times as there are units in the multiplier. If we take the fraction, and multiply the denominator by 2, without changing the numerator, we get 3. Thus : 3 3 Now, each of the fractions, and §, have the same Lamerator, and, therefore, express the same number of parts; but, in the second, the parts are only one half the size of those in the first; consequently, the whole value of the second fraction, is only one half that of the first. And the same may be shown of any fraction whatever. PROPOSITION IV. ART. 125.-If we divide the denominator of a fraction, without changing the numerator, the value of the fraction is increased as many times as there are units in the divisor. If we take the fraction, and divide the denominator by 3, without changing the numerator, we get. Thus: 2 2 9÷33 Now, each of the fractions, and, have the same numerator, and, therefore, express the same number of parts; but, in the second, the parts are three times the size of those of the first; consequently, the whole value of the second fraction is three times that of the first. And the same may be shown of other fractions. PROPOSITION V. ART. 126.-Multiplying both terms of a fraction by the same, number or quantity, changes the form of the fraction, but does not alter its value. If we multiply the numerator of a fraction by any number, its value (by Prop. I.) is increased, as many times as there are units in the multiplier; and, if we multiply the denominator, the value (by Prop. III.) is decreased, as many times as there are units in the multiplier. Hence, if both terms of a fraction are multiplied by the same number, the increase from multiplying the numerator, REVIEW.-124. How is the value of a fraction affected by multiplying only the denominator? How is this proposition proved? 125. How is the value of a fraction affected by dividing the denominator only? How is this proposition proved? 126. How is the value of a fraction affected by mul tiplying both terms by the same quantity? Why? is equal to the decrease from multiplying the denominator; consequently, the value remains unchanged. PROPOSITION VI. ART. 127.-Dividing both terms of a fraction by the same num ber or quantity, changes the form of the fraction, but does not alter its value. If we divide the numerator of a fraction by any number, its value (by Prop. II.) is decreased, as many times as there are units in the divisor; and if we divide the denominator, the value (by Prop. IV.) is increased, as many times as there are units in the divisor. Hence, if both terms of a fraction are divided by the same number, the decrease from dividing the numerator is equal to the increase from dividing the denominator; consequently, the value remains unchanged. CASE I. TO REDUCE A FRACTION TO ITS LOWEST TERMS. ART. 128. Since the value of a fraction is not changed by dividing both terms by the same quantity (See Art. 127), we have the following RULE. Divide both terms by their greatest common divisor. Or, Resolve the numerator and denominator into their prime fac tors, and then cancel those factors common to both terms. REMARK. The last rule will be found most convenient, when one or both terms are monomials. REVIEW.-127. How is the value of a fraction affected by dividing both terms by the same quantity? Why? 128. How do you reduce a fraction to its lowest terms? NOTE. In the preceding examples, the greatest common divisor in each is a monomial; in those which follow, it is a polynomial; but, by separating the quantities into factors, or by the rule (Art. 106,) the greatest common divisor is readily found. This is equal to ART. 129.-Exercises in Division (See Art. 76,) in which the quotient is a fraction, and capable of being reduced to lower terms. In a similar manner, when one polynomial can not be exactly divided by another, the division may be indicated, and the result reduced to its most simple form. |