INTRODUCTION. + VULGAR FRACTIONS. ART. 1. A FRACTION is a quantity which represents a part or parts of an integer or whole. 2. A simple fraction consists of two members or terms, the numerator and the denominator; the denominator shews into how many equal parts the whole, or unity, is divided; and the numerator the number of those parts taken. The numerator is usually placed over the denominator with a line. 2 between them. Thus (two thirds,) signifies that unity is 3' divided into three equal parts, and that two of those parts are taken. It must be observed, that we suppose every integer to be divisible into any number of equal parts at pleasure. 3. A proper fraction is one whose numerator is less 4. An improper fraction is one whose numerator is equal to, or greater than, its denominator, as 314 6 7 6' 5 5. A compound fraction is a fraction of a fraction, as 5 6 3 4 where is the whole quantity of which is to be 6. A quantity consisting of a whole number and a fraction is called a mixed number, as 7, which signifies 7 integers to 7. Every integer may be considered as a fraction whose denominator is 1; thus 5, or 5 units, is 5 1 8. A continued fraction is one whose denominator is continued by being itself a mixed number, and the denominator of the fractional part again continued as before, and so on: thus 9. To multiply a fraction by any whole number multiply the numerator by that number and retain the same denominator. and 7 times as many of those parts are taken in the latter case as in the former. 10. To divide a fraction by any whole number multiply the denominator by that number and retain the same numerator. the parts in the latter case is four times as great as in the former, and the same number of parts is taken in both cases; therefore the former fraction is one fourth of the latter. 11. A simple fraction may be considered as representing the quotient arising from the division of the numerator by the denominator. * To avoid repetition the Reader is referred to the first section of the Algebra for the explanation of the signs +, −, ×, =,, &c. 3 Thus the fraction represents the quotient of 3 divided 4 3 1 (Art. 7), and this divided by 4 is the fraction If the integer be supposed a pound, or twenty shillings, of £1, which is 15 shillings, is equal to 4 which is also 15 shillings. 12. If the numerator and denominator of a fraction be both multiplied by the same number, the value of the fraction is not altered. For, if the numerator be multiplied by any number, the fraction is multiplied by that number (Art. 9); and if the denominator be multiplied by the same number, the fraction is divided by it (Art. 10); and if a quantity be both multiplied and divided by the same number, its value is not altered. COR. Hence, if the numerator and denominator of a fraction be both divided by the same number, its value is not The operation by which a quantity is changed from one denomination to another, or by which a fraction has its terms diminished, without altering its value, is called Reduction. 13. To reduce a whole number to a fraction with a given denominator. Multiply the proposed number by the given denominatorand the product will be the numerator of the fraction re, quired. Ex. Reduce 5 to a fraction whose denominator is 6. fraction (Art. 7), the numerator and denominator of which 1 are multiplied by 6, therefore its value is not altered. (Art. 12.) 14. To reduce a mixed number to an improper fraction. Multiply the integral by the denominator of the fractional part, to this product add the numerator of the fractional part, and make its denominator the denominator of the sum. Reduce 7 to an improper fraction. = Ex. 1. 39 5 15. To reduce an improper fraction to a mixed number. Divide the numerator by the denominator for the integral part, and make the remainder the numerator of the fractional part, and the divisor its denominator. Ex. Reduce to a mixed number. The fraction 39 5 39 5 = 74; because the unit being divided into 5 parts, 39 such parts are to be taken, that is, 7 units and 4 such parts. 16. To reduce a compound fraction to a simple one. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. therefore two thirds of the same quantity, which must be Mixed numbers must be reduced to improper fractions, before the rule can be applied. Thus 17. To reduce a continued fraction to a simple one. Apply the rule (Art. 14) for reducing a mixed number to an improper fraction, commencing at the lowest extremity of the continued fraction, and proceeding gradually upwards until the whole is reduced to a simple fraction. But as this operation requires the use of a rule not yet proved, the example is deferred to Art. 38. Whenever the numerator and denominator of a fraction have a common measure, that is, a number which divides each of them without remainder, greater than unity, the fraction may be reduced to lower terms by dividing both the numerator and denominator by this common measure. Ex. 105 is reduced to 21 24 by dividing both the nume 7 21 rator and denominator by 5; and is again reduced to 24 8 by dividing its numerator and denominator by 3. That the value of the fraction is not altered appears from Art. 12. |