(3.) Any fraction whose numerator is not less than its de238 nominator is an improper fraction. Thus, 25 are improper fractions. (4.) Any quantity which is not of a fractional form is said to be of an integral form. Thus, 2, 3, a, b are integral forms. (5.) Any quantity whose expression is partly integral and partly fractional is called a mixed number or form. (6.) Any single expression which denotes any number of parts of the quantity which is expressed by 1, is called a 3 5 a с d single or simple fraction. Thus, ' ' ' ab are single 4' 9' fractions. (7.) A fraction of a fraction is called a compound fraction. 3 7 a C Thus, 4 of and a 8 d are compound fractions. (8.) Because the value of a quantity is not affected by dividing it by 1, we may express any integer or integral quantity in the form of a fraction, by writing 1 under it. 3 a Thus, for 3 and a, we may write 1'1 (9.) It is evident, from the rule of signs in Division, if the entire numerator and denominator of a fraction have like signs, that the fraction is positive; but if they have unlike signs, the fraction is negative. And it is clear that we may change the signs of the entire numerator and denominator of a fraction without affecting its sign. (10.) If any term, either of the numerator or denominator of a fraction has no sign expressed, the sign + is to be understood. Also, if no sign is prefixed to the line that separates the numerator from the denominator of a fraction, the sign + must be understood. (11.) If the sign - is prefixed to the line that separates the numerator from the denominator of a fraction, the meaning is that the fraction is to be subtracted, or that the signs of its entire numerator or denominator must be changed. requires the sign of cd or of e-f to be a changed, and is equivalent to %+ (12.) Fractions are said to be reduced, when they are changed from one form of expression to any other equivalent Erase all the factors that are common to the numerator and denominator; or, which comes to the same thing, divide both the numerator and denominator by their greatest common divisor, and the fraction will be reduced as required. According to what was shown in Division, the reduced fraction will be equivalent to the given fraction. a3 + b3 a2 ab + b2 Also, a2 a-b by dividing the numerator and denominator by their greatest common divisor a + b. To reduce a mixed quantity to a fraction, usually called an improper fraction. RULE. Multiply the integral part by the denominator of the fraction; then add the product and the numerator of the fraction, according to their signs; and the result being placed over the denominator, with a line drawn between them, will be the fraction required. Thus, a ac b с с = ac±b = ; for since a== с ac we get a ± ac ± b с = ; for clearly the sum of the quotients of ac divided by c, and ±b divided by c, must equal the quotient 7a2-1162 11. Reduce 3a2 5b2+ to a fraction. 14a-196 Ans. 42a3 - (57b — 7)a2 — 70b3a — 1162 + 95b3 14a 196 CASE III. To reduce an improper fraction to an integral or mixed quantity. RULE. Divide the numerator by the denominator for the integral part, and if there is a remainder write it over the denominator, with its proper sign, for the fractional part. This case is the reverse of the last, and the reason of the rule requires no explanation. |