74. In every division there are three signs to be considered, the sign of the dividend, of the divisor and of the quotient. Similarly, there are three signs to consider in every fraction, the sign of the numerator, of the denominator and of the fraction itself, that is, the sign before the fraction. It is evident that changing the sign of the dividend and of the divisor will have no effect on the quotient, hence, Changing the sign of the numerator and of the denominator of a fraction will not alter the value of the fraction. If the numerator and denominator are compound expressions, the signs of these expressions are changed by changing the sign of every term in the expressions. In a division, if we change the sign of the dividend or of the divisor, but not of both, the sign of the quotient is changed, hence, to restore the quotient to its previous value, the sign of the quotient must also be changed. Therefore, to change the sign of the numerator or of the denominator, but not of both, change the sign before the fraction. If an algebraic expression is the product of any number of factors, it is evident, from the law of signs, that the signs of an even number of factors may be changed without altering the value of the product, but if the signs of an odd number of factors are changed, the sign of the product is changed. Hence, changing the signs of an odd number of factors of either the numerator or denominator of a fraction is equivalent to changing the sign of the numerator or denominator and therefore requires a change in the sign before the fraction. These principles are applied in the following problems. 4xy (y2 — x2). 5(x2y - xy2) 1. Reduce to lowest terms Changing the If the signs of the factor y x are changed, numerator and denominator will contain the common factor x signs of an odd number of factors of the numerator changes the - y. sign of the numerator and hence requires changing the sign before the fraction. Dividing numerator and denominator by the H. C. F. xy(x − y, we have Here we change the signs of both factors of the denominator and, as the change in an even number of factors does not change the sign of the denominator, the sign before the fraction remains unchanged. The sign of the factor 3 x in the numerator is changed, hence the sign before the fraction is also changed. As the denominator contains a negative term, 4, the sign of the denominator may be changed and the sign of the fraction changed to plus. 75. A mixed expression, corresponding to a mixed number in arithmetic, consists of an integral expression In arithmetic, a mixed number is reduced to a fraction by multiplying the whole number by the denominator of the fraction, adding the numerator and placing the sum over the denominator. The value of the whole number is not changed, because it is multiplied and divided by the same number, that is, the denominator of the fraction. In algebra, the same rule is used. Multiply the integral expression by the denominator, add the numerator to this product and place the denominator under the sum. Note, that in this rule, addition is an algebraic addition and includes subtraction, that is, if the sign before the fraction is negative, the terms of the numerator must be subtracted by changing their signs. The integral expression, a + x, is multiplied by the denominator x, the numerator a2 - b2 is subtracted from this product and the result placed over the denominator x. The same rule is used, whether the integral expression precedes or follows the fraction. 76. An improper fraction, in arithmetic, is changed to a mixed number by dividing the numerator by the denominator, using the remainder as the numerator of the remaining fraction. Similarly in algebra, if the numerator of a fraction is of a higher degree than the denominator, it may be changed to a mixed expression by the method of division. To change a fraction to a mixed expression, divide the numerator by the denominator. The quotient will be the integral expression and the remainder becomes the numerator of the remaining fraction. Note that the remaining fraction must be connected with the integral expression by the sign +. By changing the sign before the fraction, the sign of the numerator may be changed to +. REDUCTION OF FRACTIONS TO EQUIVALENT FRACTIONS HAVING THE LOWEST COMMON DENOMINATOR 77. In order to be able to add fractions, they must have equal or common denominators. As in arithmetic, so in algebra, the least common multiple of all the denominators is used as the lowest common denominator. As the value of each fraction must remain unchanged, numerator and denominator of each fraction are multiplied by the factor necessary to change each denominator to the lowest common denominator. |