MIXED CIRCULATING DECIMALS are those which do not begin to recur, till after a certain number of figures. Thus 128888... ·0113636................, are mixed circulating decimals. The circulating part, or the part which is repeated, is called the PERIOD OF REPETEND. Pure and mixed circulating decimals are generally written down only to the end of the first period, a dot being placed over the first and last figures of that period. Thus 3 represents the pure circulating decimal .333... 96. Pure Circulating Decimals may be converted into their equivalent Vulgar Fractions by the following Rule. RULE. Make the period or repetend the numerator of the fraction, and for the denominator put down as many nines as there are figures in the period or repetend; this fraction, reduced to its lowest terms, will be the fraction required. Note. The fraction is only reduced to its lowest terms for the sake of exhibiting it in its simplest form. It is not of course actually necessary so to reduce it. Exs. Reduce the following pure circulating decimals, 3, 27, 857142, to their respective equivalent vulgar fractions. Proceeding by the Rule given above, The truth of these results will appear from the following considerations. Let the circulating decimal 3333... be represented by a symbol x; therefore and 10 times x= = 10 times 3333... =3.3333... (Art. 86). Now 10 times x, diminished by 1 time x, will leave 9 times x, therefore =3 or 9 times x=3; 1 time x, that is ≈ or ·3333...=3=}. Next, let the circulating decimal 2727... be represented by x. x=·272727... here, since there are two figures in each period, we multiply by 100, and we have 100 times x=100 times 2727... =27·2727... (Art. 86). Therefore 100 times x, diminished by 1 time x, will be equal to Next, let the recurring decimal 857142 be represented by x. here since there are six figures in each period, we multiply 1000000, and we have 1000000 times = 1000000 times 875142... Note 1. The object in each case is to multiply the recurring decimal by such a power of 10, as will bring out the period a whole number. Note 2. The powers of numbers are often expressed by placing a small figure (equivalent to the number of factors and called the INDEX or EXPONENT of the power) at the right hand of the number, a little above the line. Thus 10 × 10, or the second power of 10 is expressed by 10', 10 x 10 x 10, or the third power of 10 10 × 10 x 10 x 10 x 10, or the fifth power of 10 ..... and so on. by 103, by 105, 97. Mixed Circulating Decimals may be converted into their equivalent Vulgar Fractions by the following Rule. RULE. Subtract the figures which do not circulate from the figures taken to the end of the first period, as if both were whole numbers; make the result the numerator; and write down as many nines as there are figures in the circulating part, followed by as many zeros as there are figures in the non-circulating part, for the denominator. Exs. Reduce the following mixed circulating Decimals, 14, ‘0138, 2418, to their respective equivalent vulgar fractions. Proceeding by the Rule given above, The truth of these results will appear from the following considerations. Let the mixed circulating decimal be represented by a in each of the above cases. First, let x=1444... If, by multiplication, we change the decimal in such a manner that the non-circulating part is rendered a whole number, and also change it so that the non-circulating and circulating parts to the end of the first period are rendered a whole number, and then subtract the first result from the second, we shall get rid of the circulating part. Thus, multiplying first by 10 to get the 1 out as a whole number, and then by 100 to get the 14 out as a whole number, we have 10 times x=10 times 1444... = 1'444... 100 times x=14444... ; Here there are three places in the non-recurring part, and one in the recurring part; therefore multiplying first by 1000, and then by 10000, we have Now we have one place in the non-recurring part, and three places in the recurring part; therefore multiplying first by 10, and then by 10000, we have Ex. XXX. 1. Reduce the following vulgar fractions and mixed numbers to 2. Find the vulgar fractions equivalent to the recurring decimals; 98. The value of the circulating decimal '999... is found by Art. (96) to be or 1; but since the difference between 1 and 9=1, between 1 and '99-'01, between 1 and 999=001, &c., it appears that however far we continue the recurring decimal, it can never at any stage be actually = 1. But the recurring decimal is considered = 1, because the difference between 1 and 99... becomes less and less, the more figures we take in the decimal, which thus, in fact, approaches nearer to 1 than by any difference that can be assigned. In like manner, it is in this sense that any vulgar fraction can be said to be the value of a circulating decimal; because there is no assignable difference between their values. 99. In arithmetical operations, where circulating decimals are concerned, and the result is only required to be true to a certain number of decimal places, it will be sufficient to carry on the circulating part to two or three decimal places more than the number required: taking care that the last figure retained be increased by 1, if the succeeding figure be 5, or greater than 5; because, for instance, if we have the mixed decimal '6288, and stop at 628, it is clear that '628 is less, and '629 is greater than the true value of the decimal: but 628 is less than the true value by 000888,... and 629 is greater than the true value by '000111... Now 000111...is less than 000888... Therefore 629 is nearer the true value than '628. Ex. 1. Add together ·33, 0432, 2·345, so as to be correct to 5 places of decimals. |