FRACTIONS. DEFINITIONS AND NOTATION. 115. A FRACTIONAL UNIT is one of the equal parts into which a unit has been divided. Thus, , ±‚ / tional units. are frac 116. A FRACTION is a fractional unit, or a collection of fractional units. Thus,,, are fractions. a The denominator of a fraction indicates into how many parts the unit has been divided, and the numerator how many of those parts are taken. has been divided into b equal parts are taken. α Thus, indicates that 1 b parts, and that a such The numerator and denominator are called the terms of a fraction. 117. An ENTIRE QUANTITY is one which has no fractional part; as, ab, or a - b. 119. The VALUE of a fraction is the quotient of its numerator divided by its denominator. For, a fraction is an expression of division, the numerator answering to the dividend, and the denominator to the divisor. (Art. 18.) Thus, the value of the fraction ab b GENERAL PRINCIPLES OF FRACTIONS. 120. If the numerator be multiplied, or the denominator divided, by any quantity, the fraction is multiplied by the same quantity. Now, if we multiply its numerator by any quantity b, we have and, in like manner, if we divide its denominator by b, we obtain also a b. Hence, in both cases the value of the fraction has been multiplied by b. 121. If the denominator be multiplied, or the numerator divided, by any quantity, the fraction is divided by the same quantity. Now, if we multiply its denominator by any quantity b, we have and, in like manner, if we divide its numerator by b, we obtain also a. Hence, in both cases the value of the fraction has been divided by b. 122. If the numerator and denominator be both multiplied, or both divided, by the same quantity, the value of the fraction will not be changed. For, if any quantity be both multiplied and divided by the same quantity, the value of the former will not be changed. (Art. 46, Ax. 6.) 123. A fraction is POSITIVE when its numerator and denominator have the SAME sign, and NEGATIVE when they have DIFFERENT signs. For a fraction represents the quotient of its numerator divided by its denominator; consequently its proper sign must be determined as in division. (Art. 67.) 124. The SIGN of a fraction, or that prefixed to its dividing line, shows whether the fraction is to be added or subtracted. The sign written before the dividing line has been termed the apparent sign of the fraction, and that depending upon the value expressed by the fraction itself has been termed the real sign. a " b Thus, in + the apparent sign is +, and the real sign 125. If any one of the signs prefixed to the numerator, denominator, and dividing line of a fraction be changed, the value of the fraction will be changed accordingly. 126. Any two of the signs prefixed to the numerator, denominator, and dividing line of a fraction may be changed, without affecting the value of the fraction. 127. If all the signs prefixed to the terms and the dividing line of a fraction be changed, the value of the fraction will be changed accordingly. 128. REDUCTION OF FRACTIONS is the process of changing their forms without altering their values. CASE I. 129. To reduce a fraction to its lowest terms. A fraction is in its lowest terms, when its terms are prime to each other. Since dividing both numerator and denominator by the same quantity, or canceling equal factors in each, does not alter the value of the fraction (Art. 122), we have the following RULE. Resolve both terms of the fraction into their prime factors, and cancel all that are common to both. Or, Divide both terms by their greatest common divisor. x2 - bx = Ans. (xbx)(x+b) (x2+2bx+b23) ÷ (x + b) x + b' Here the reduction is performed by using the greatest common divisor of the terms of the fraction. |