Fractions are either Proper, Improper, Simple, Compound, Mixed, or Complex. 29 A Proper Fraction, is when the numerator is less than the denominator; as, , or 3, or 3. An Improper Fraction, is when the numerator is equal to, or exceeds, the denominator; as, or {, or 7. In these cases the fraction is called improper, because it is equal to or exceeds unity. A Simple Fraction, is a single expression, denoting any number of parts of the integer; as, 3, or 1. A Compound Fraction, is the fraction of a fraction, or two or more fractions connected with the word of between them; as of, or of § of 3. A Mixed Number, is composed of a whole number and a fraction together; as, 3, or 12. A Complex Fraction, is one that has a fraction or a mixed number for its numerator, or its denominator, or both; A whole or integer number may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is, or 4 is 1. A fraction denotes division; and its value is equal to the quotient obtained by dividing the numerator by the denominator: so is equal to 3, and is equal to 4. Hence then, if the numerator be less than the denominator, the value of the fraction is less than 1. But if the numerator be the same as the denominator the fraction is just equal to 1. And if the numerator be greater than the denominator, the fraction is greater than 1. REDUCTION OF VULGAR FRACTIONS. REDUCTION of Vulgar Fractions, is the bringing them out of one form or denomination into another; commonly to prepare them for the operations of Addition, Subtraction, &c. ; of which there are several cases. PROBLEM. To find the greatest common measure of two or more numbers. The common measure of two or more numbers, is that number which will divide them all without remainder: so, 3 is a common measure of 18 and 24; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this is the greatest common measure: so 6 is the greatest common measure of 18 and 24; the quotient of the former being 3, and of the latter 4, which will not both divide further. RULES. If there be two numbers only, divide the greater by the less; then divide the divisor by the remainder; and so on, dividing always the last divisor by the last remainder, till nothing remains; so shall the last divisor of all be the greatest common measure sought. When there are more than two numbers, find the greatest common measure of two of them, as before; then do the same for that common measure and another of the numbers; and so on, through all the numbers; so will the greatest common measure last found be the answer. If it happen that the common measure thus found is 1; then the numbers are said to be incommensurable, or not to have any common measure, or they are said to be prime to each other *. Ex. 1. Find the greatest common measure of 3852 and 762896. * It is not absolutely necessary that our products should be less than the dividend. All that the principle requires is, that we should take the difference between the dividend and the nearest multiple of the divisor. The method given in the rule is that most usually employed: though when the next higher multiple would be nearer to the dividend than the next lower, the actual work is considerably lessened by the adoption of the higher multiple. Thus in the example in the text, had we taken the quotient 4 in the third division, it will be obvious that one division would have been saved. For a proof of this rule see the corresponding subject in algebra. + The several quotients in both processes are numbered by subscribed figures, as 14 or 26 showing that 1 is the 4th quotient, and 2 is the 6th. In the new method the remainder is considered and treated as the divisor of the previous quotient, without being placed (after the first step) in the usual place of the divisor in common operations. This can occasion no difficulty in any case, as the divisor is not more removed from the place of the successive products than in the old method. One advantage is, that it saves the repetition of the writing of the dividend figures. A more important one (and which is of great practical convenience in the corresponding algebraical operation) is the compactness of the work, and the small space it occupies. Both considerations concur in recommending it to general adoption. The 2nd, 4th, &c. quotients are placed on the left of the work; the 1st, 3d, &c. on the right. It will conduce to ready re-examination of the work to draw the horizontal lines above each final remainder through the side lines, as in the example. Ex. 2. To find the greatest common measure of 1908, 936, and 630. And 36 is the g. c. m of 1908 and 936; and 18 the g. c. m. of 36 and 630 is the g. c. m. of all the three numbers 1908, 936, and 630. 3. What is the greatest common measure of 246 and 372 ? 4. What is the greatest common measure of 324, 612, and 1032 ? CASE I. To abbreviate or reduce fractions to their lowest terms. Ans. 6. Ans. 12. * DIVIDE the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients again in the same manner; and so on, till it appears that there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms. *That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evident. And when these divisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must be the least possible. Note. 1. Any number ending with an even number, or a cipher, is divisible, or can be divided, by 2. 2. Any number ending with 5, or 0, is divisible by 5. 3. If the right-hand place of any number be 0, the whole is divisible by 10; if there be two ciphers, it is divisible by 100; if three ciphers, by 1000: and so on; which is only cutting off those ciphers. 4. If the two right-hand figures of any number be divisible by 4, the whole is divisible by 4. And if the three right-hand figures be divisible by 8, the whole is divisible by 8. And so on. 5. If the sum of the digits in any number be divisible by 3, or by 9, the whole is divisible by 3, or by 9. 6. If the right-hand digit be even, and the sum of all the digits be divisible by 6, then the whole is divisible by 6. 7. A number is divisible by 11, when the sum of the 1st, 3d, 5th, &c. or of all the odd places, is equal to the sum of the 2d, 4th, 6th, &c. or of all the even places of digits. 8. If a number cannot be divided by some quantity less than the square root of the same, that number is a prime, or cannot be divided by any number whatever. 9. All prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units; and all other numbers are composite, or can be divided. It is not, however, to be inferred that all numbers which end in 1, 3, 7, 9, are prime numbers. No method, indeed, is yet known by which prime numbers can be either immediately calculated, or assigned, or detected. The best practical method for numbers not very high, is the sieve of Eratosthenes (KOKKLvov), an account of which may be seen in the Phil. Trans. by Dr. Horsley, and in most works on the theory of numbers. 10. When numbers, with the sign of addition or subtraction between them, are to be divided 10+8-4 =5+ by any number, then each of those numbers must be divided by it. Thus, 4-2-7. 11. But if the numbers have the sign of multiplication between them, only one of them must 10×8×3_10×4×3_10×4×1_10×2×1_20_ be divided. Thus, = = 6x2 6x1 20. Or, Divide both the terms of the fraction by their greatest common measure at once, and the quotients will be the terms of the fraction required, of the same value as at first. To reduce a mixed number to its equivalent improper fraction. * MULTIPLY the integer or whole number by the denominator of the fraction, and to the product add the numerator; then set that sum above the denominator for the fraction required. 2. Reduce 123 to a fraction. 3. Reduce 14 to a fraction. 4. Reduce 183 to a fraction. CASE III. Ans. 15. Ans.. Ans. 3848. To reduce an improper fraction to its equivalent whole or mixed number. + DIVIDE the numerator by the denominator, and the quotient will be the whole or mixed number sought. EXAMPLES. 1. Reduce to its equivalent number. Here or 12 ÷ 3 = 4, the Answer. This is no more than first multiplying a quantity by some number, and then dividing the result back again by the same: which it is evident does not alter the value; for any fraction represents a division of the numerator by the denominator. This rule is evidently the reverse of the former; and the reason of it is manifest from the nature of common division. To reduce a whole number to an equivalent fraction, having a given denominator. * MULTIPLY the whole number by the given denominator; then set the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 9 to a fraction whose denominator shall be 7. Here 9 × 7 63: then 3 is the Answer; = For 36379, the Proof. 2. Reduce 12 to a fraction whose denominator shall be 13. 3. Reduce 27 to a fraction whose denominator shall be 11. Ans. 158. Ans. 97. CASE V. To reduce a compound fraction to an equivalent simple one. + MULTIPLY all the numerators together for a numerator, and all the denominators together for a denominator, and they will form the simple fraction sought. When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction by one of the former cases. And, when it can be done, any two terms of the fraction the same number, and the quotients used instead of them. terms that are common, they may be omitted, or cancelled. may be divided by Or, when there are * Multiplication and Division being here equally used, the result must be the same as the quantity first proposed. The truth of this rule may be shown as follows: Let the compound fraction be 3 of 5. Now of is÷÷3, which is ; consequently of will be × 2 or f; that is, the numerators are multiplied together, and also the denominators, as in the Rule. When the compound fraction consists of more than two single ones; having first reduced two of them as above, then the resulting fraction and a third will be the same as a compound fraction of two parts; and so on to the last of all. |