These properties may be proved by changing the repetends into their equivalent common fractions. 4. Having made two or more repetends similar by the last article, they may be rendered conterminous by the previous one; thus, two or more repetends may always be made similar nd conterminous. 5 If two or more repeating decimals, having several repetends of equal places, be added together, their sum will have a repetend of the same number of places; for, every two sets of repetends will give the same sum. 6. If any repeating decimal be multiplied by any number, the product will be a repeating decimal having the same number of places in the repetend; for, each repetend will be taken the same number of times, and consequently must produce the same product. Examples. 1. Reduce .13'8, 7.5'43' .04'354', to repetends having the same number of places. Since the number of places are now 1, 2, and 3, the least common multiple is 6, and hence each new repetend will contain 6 places; that is, .138.13'888888'; 7.5'43'7.5'434343'; and .04'354'.04'354354'. 2. Reduce 2.4'18', .5'925', .008'497133', to repetends having the same number of places. 3. Reduce the repeating decimals 165.164', '04', .037 to such as are similar and conterminous. 4. Reduce the repeating decimals .5'3, .475', and 1.757', o such as are similar and conterminous. 223. 4. To what brm may two or more repetends be reduced? ADDITION. 224. To add repeating decimals. I. Make the repetends, in each number to be added, similar and conterminous : II. Write the places of the same unit value in the same olumn, and so many figures of the second repetend in each as shall indicate with certainty, how many are to be carried from one repetend to the other: then add as in whole numbers. NOTE.-If all the figures of a repetend are 9's, omit them and add 1 to the figure next at the left. Examples. 1. Add .125, 4.163', 1.7143', and 2.54', together. DISSIMILAR. SIMILAR. SIMILAR AND CONTERMINOUS. = .12 555555555555' .125 = .12'5 4. 163' 4.16'316' 4.16 316316316316' 1.7143' 1.71'4371′ = 1.71437143714371' The true sum = 8.54'854470131697' 1 to carry. 2. Add 67.3'45', 9.'651', .25', 17.47, .5, together. SUBTRACTION. 225. To subtract one repeating decimal from another. I. Make the repetends similar and conterminous : II. Subtract as in finite decimals, observing that when the repetend of the lower line is the larger, 1 must be carried to the first right-hand figure. 224. How do you add repeating decimals?-225. How do you sub tract repeating decimals. Examples 1. From 11.475' take 3.45'735'. DISSIMILAR. 11.475' 3 45'735' SIMILAR. SIMILAR AND CONTERMINOUS. 11.47'57' = 11.47'575757' 3.45'735' = The true difference = 2. From 47.5'3 take 1.757'. 3. From 17.573' take 14.5'7. 4. From 17.4'3 take 12.34'3. 3.45 735735' 575 735 6. From 4.75 take .375. 5. From 1.12754' take.4'7384'. 9. From 1.4'937' take .1475. MULTIPLICATION. 226. To multiply one repeating decimal by another. Change the repeating decimals into their equivalent com mon fractions, then multiply them together, and reduce the product to its equivalent repeating decimal. and since 225000 = 5 × 5 × 5 × 5 × 5 × 2 × 2 × 2 × 9, there will be five places of finite decimals, and one figure in the repetend. NOTE.-Much labor will be saved in this and the next rule by keeping every fraction in its lowest terms; and when two fractions are to be multiplied together, cancel all the factors common to both term before making the multiplication. 2. Multiply .375'4 by 14.75. 3. Multiply .4'253' by 2.57. 4. Multiply .437 by 3.75. 5. Multiply 4.573 by .3'75'. 6. Multiply 3.45'6 by .425. 7. Multiply 1.456′ by 4.2'3. 8. Multiply 45.1'3 by 245'. 9. Multiply .4705`3 by 1.7`35'. DIVISION. 227. To divide one repeating decimal by another. Change the decimals into their equivalent common fractions, and find the quotient of these fractions. quotient into its equivalent decimal. 1. Divide 56.6 by 137. Examples. OPERATION. 56.656 = 510 = 170. Then change the Then, 170137 = 130 X 137=119.41362530'. 2. Divide 24.3'18' by 1.792. | 6. Divide 13.5 169533' by 4.297' 3. Divide 8.5968 by .2'45'. 4. Divide 2.295 by .297'. 5. Divide 47.345 by 1.76'. 7. Divide .'45' by .'118881'. CONTINUED FRACTIONS. 228. A CONTINUED FRACTION has 1 for its numerator, and for its denominator a whole number plus a fraction, which also has a numerator of 1, and for a denominator, a whole number plus a similar fraction, and so on. 1. If we take any irreducible fraction, as 15, and divide both terms by the numerator, it will take the form, 1 15 1 = by making the division. 29 ? 1 + 13, If, now, we divide both terms of 1 by 14, we have, 226. How do you multiply repeating decimals?-227. How do you divide repeating decimals?-228. What are continued fractions? What is the rule for finding the approximate value? ANALYSIS.-Let us analyze this example. If we neglect what comes after 1, the first term of the first denominator, we shall have, = 1, which is called the first approximating fraction. If we neglect what comes after 3, the first term of the second denominator, we shall have, If we neglect what comes after 1, the first term of the third denominator, we shall have, the third approximating fraction; and so on, for fractions which follow. If we stop at the first approximating fraction, the denominator 1 will be less than the true denominator; for, the true denominator is 1 plus a fraction; hence, the value of the first approximating fraction will be too great; that is, it will exceed the value of the given fraction. If we stop at the second, the denominator 3 will be less than the true denominator; hence, will be greater than the number to be added to 1; therefore, 1 + is too large, and 1÷1†, which |