An examination of each of these examples, will show that the process consists in reducing the quantities to a common denominator, and then dividing the numerator of the dividend, by the numerator of the divisor. But, as the common denominator of the fraction is not used in performing the division, the result will be the same as if we invert the divisor, and proceed as in multiplication. Hence, the RULE, FOR DIVIDING AN INTEGRAL OR FRACTIONAL QUANTITY BY A FRACTION. Reduce both dividend and divisor to the form of a fraction; then invert the terms of the divisor, and multiply the numerators together for a new numerator, and the denominators together for a new denominator. NOTE. After inverting the divisor, the work may be abbreviated, by canceling all the factors common to both terms of the result. REVIEW. 142. How do you divide an integral or fractional quantity by a fraction? Explain the reason of this rule, by analyzing an example. When, and how, can the work be abbreviated? ART. 143.—To reduce a complex fraction to a simple one. This may be regarded as a case of division, in which the dividend and the divisor are either fractions or mixed quantities. 7 7 7.2 2 ( a + c ) + ( m + ; ) n ac+b mr+n ac+b, go acr+br m2 = ac с `mr+n cmr+cn In the same manner, let the following examples be solved. A complex fraction may also be reduced to a simple one, by multiplying both terms by the least common multiple of the denominators of the fractional parts of each term. Thus, we may 41 reduce to a simple fraction, by multiplying both terms by 6, 5 the least common multiple of 2 and 3; the result is. In some cases this is a shorter method, than by division. Either method may be used. ART. 144.-Resolution of fractions into series. An infinite series consists of an unlimited number of terms, which observe the same law. The law of a series is a relation existing between its terms, so that when some of them are known, the succeeding terms may be easily derived. REVIEW.-143. How do you reduce a complex fraction to a simple one, by division? How, by multiplication? RESOLUTION OF FRACTIONS INTO SERIES. 109 Thus, in the infinite series, 1—ax+a2x2—a3x3+a*x*, &c., any term may be found, by multiplying the preceding term by―ax. Any proper algebraic fraction, whose denominator is a polyno mial, can, by division, be resolved into an infinite series; for, the numerator is a dividend, and the denominator a divisor, so related to each other, that the process of division never can terminate, and the quotient will, therefore, be an infinite series. After a few of the terms of the quotient are found, the law of the series will, in general, be easily seen, so that the succeeding terms may be found without continuing the division. nite series, 1+x+x2+x3+x2+, &c. In a similar manner, let each of the following fractions be resolved into an infinite series, by division. · 1−x+x2—x3+x*-, &c., to infinity. 1 2. 1+x a-x REVIEW.-144. What is an infinite series? What is the law of a series? Give an example. Why can any proper algebraic fraction, whose denominator is a polynomial, be resolved into an infinite series, by division? CHAPTER IV. EQUATIONS OF THE FIRST DEGREE. DEFINITIONS AND ELEMENTARY PRINCIPLES. ART. 145.—The most useful part of Algebra, is that which relates to the solution of problems. This is performed by means of equations. An equation is an Algebraic expression, stating the equality between two quantities. Thus, x-3=4, is an equation, stating, that if 3 be subtracted from x, the remainder will be equal to 4. ART. 146.-Every question is composed of two parts, separated from each other by the sign of equality. The quantity on the left of the sign of equality, is called the first member, or side of the equation. The quantity on the right, is called the second member, or side. The members or quantities are each composed of one or more terms. ART. 147.—There are generally two classes of quantities in an equation, the known and the unknown. The known quantities are represented either by numbers, or the first letters of the alphabet, as a, b, c, &c.; and the unknown quantities by the last letters of the alphabet, as x, y, z, &c. ART. 148.-Equations are divided into degrees, called first, second, third, and so on. The degree of an equation, depends on the highest power of the unknown quantity which it contains. An equation which contains no power of the unknown quantity higher than the first, is called an equation of the first degree. Thus, 2x+5=9, and ax+b=c, are equations of the first degree. Equations of the first degree are usually called Simple Equations. An equation in which the highest power of the unknown quantity is of the second degree, that is, a square, is called an equution of the second degree, or a quadratic equation. REVIEW.-145. What is an equation? Give an example. 146. Of how many parts is every equation composed? How are they separated? What is the quantity on the left of the sign of equality called? On the right? Of what is each member composed? 147. How many classes of quantities are there in an equation? How are the known quantities represented? How are the unknown quantities represented? 148. How are equations divided? On what does the degree of an equation depend? What is an equation of the first degree? Give an example. What are equations of the first degree usually called? What is an equation of the second degree? Give an example. What are equations of the second degree usually called. |