In this example we find 40=23 X 5, where the highest ex. ponent is 3; therefore the number of decimal places is 3. 2. How many places of decimals will be required to express Ans. 3. 3. How many places of decimals will be required to express Ans. 5. 4. How many places of decimals will be required to express 5. How many places of decimals will be required to express it? 6. How many places of decimals will be required to express Tilo? 45. When the decimal figures obtained by converting a vulgar fraction into decimals do not terminate, they must re. cur in periods, whose number of terms can not exceed the number of units in the denominator, less one. For all the different remainders which occur must be less than the de. nominator ; and therefore their number can not exceed the denominator, less one; and whenever we obtain a remainder like one that has previously occurred, then the decimal figures will begin to repeat. Decimals which recur in this way are called repetends. When the period begins with the first decimal figure it is called a simple repetend. But when other decimal figures occur before the period commences, it is called a compound repetend. A repetend is distinguished from ordinary decimals by a period or dot placed over the first and last figure of the circu, lating period. 46. The following vulgar fractions give simple repetends. b=0.3. =0.142857. =0.052631578947368421. =0.0434782608695652173913. 47. The following ones give compound repetends. =0.16. =0.083. =0.05. =0.045. 48. Whenever the prime factors of the denominator of a vulgar fraction contain neither of the factors 2 and 5, the repetend will be simple. But when they contain one or both of the factors 2 and 5, together with other factors, then the repetend will be compound. 49. Those simple repetends,which have as many terms, less one, as there are units in the denominator, we shall call perfect repetends. The following are some of the perfect repetends. q=0.142857. =0.0434782608695652173913. 50. PERFECT REPETENDS possess some very remarkable properties, which we will explain by means of the following figure. In this figure, the inner circle of figures, commencing at the O, directly under the asterisk, and counting towards the right-hand, is the circulating period of ab. The outer circle of figures, commencing at the same point and counting in the same direction, are the successive remainders, which will occur in the operation of decimating ab, (by rule under Art. 42, page 58.) In this circle of remainders, all the numbers, from 1 to 28 inclusive, occur, but not in numerical order. By inspecting the above figure, we discover the following properties, which are common to all perfect repetends. 1. The sum of any two diametrically opposite figures, of the circle of decimals, will be 9. II. The sum of any two diametrically opposite terms, in the circle of remainders, will make the denominator 29. III. If we subtract the right-hand figure of the denominator from 10, and multiply the remainder by any decimal figure of the inner circle, the right-hand figure of the product will be the same as the right-hand figure of the corresponding remainder of the outer circle. IV. Commencing the circle of decimals at any point, and counting completely round, it will be the perfect repelend of the vulgar fraction, whose denominator is the same as in the first case, but whose numerator is the remainder in the outer circle, standing one place to the left. From the IV. property it follows, that this same circle of decimals expresses the decimal value of all proper vulgar frac. tions whose denominators are 29. The following figure, formed from the perfect repetend of |