Taking two terms wanting 33 +1807, there would still be to make the sum = }. 468. Another example. Let there be given the infinite progression, 9 Τ 9+ 10+180 + 1000 + 10000 + &c. The first term is 9, the exponent is. So that 1, minus the exponent, =; and 10 the sum required. 9 10 This series is expressed by a decimal fraction, thus 9,9999999, &c. 1 CHAPTER XI. Of Infinite Decimal Fractions. 469. It will be very necessary to shew how a vulgar fraction may be transformed into a decimal fraction; and, conversely, how we may express the value of a decimal fraction by a vulgar fraction. 470. Let it be required, in general, to change the fraction—, into a decimal fraction; as this fraction expresses the quotient of the division of the numerator a by the denominator b, let us write, instead of a, the quantity a,0000000, whose value does not at all differ from that of a, since it contains neither tenth parts, nor hundredth parts, &c. Let us now divide this quantity by the number b, according to the common rules of division, observing to put the point in the proper place, which separates the decimal and the integers. This is the whole operation, which we shall illustrate by some examples. Let there be given first the fraction, the division in decimals will assume this form, Hence it appears, that is equal to 0,5000000 or to 0,5; which is sufficiently evident, since this decimal fraction represents, which is equivalent to 471. Let be the given fraction, and we have, This shews, that the decimal fraction, whose value is 4, cannot, strictly, ever be discontinued, and that it goes on ad infinitum, repeating always the number 3. And, for this reason, it has been already shewn, that the fractions O + TOO + TOOO &c. ad infinitum, added together, make 4. 3 The decimal fraction, which expresses the value of, is also continued ad infinitum; for we have, And besides, this is evident from what we have just said, because is the double of . 472. If be the fraction proposed, we have So that dent, since is equal to 0,2500000, or to 0,25; and this is evi In like manner, we should have for the fraction, 5 75 So that = 0,75; and in fact + 180 = 100 = 1. 4) 5,0000000 Now 1+3= 25 473. In the same manner, 0,4; 0,6; 0,8; 1; } = 1,2, &c. = = = When the denominator is 6, we find = 0,1666666, &c. which is equal to 0,666666 0,5. Now 0,666666, and 0,5 = wherefore 0,1666666 = We find, also, = 0,333333, &c. =; but becomes 0,5000000=. Further, = 0,833333 0,333333 +0,5, that is to say, + 1 = /• = 474. When the denominator is 7, the decimal fractions become more complicated. For example, we find = 0,142857, however it must be observed, that these six figures are repeated continually. To be convinced, therefore, that this decimal fraction precisely expresses the value of 4, we may transform it into a geometrical progression, whose first term is = and the exponent = 100000; and, consequently, the sum 999999 142857 1000000 1-1000000 142857 1000000 (multiplying both terms by 1000000) 475. We may prove, in a manner still more easy, that the decimal fraction which we have found is exactly = ; for substituting for its value the letters, we have s = 0,142857142857142857, 10 s 100 s 1000 s = 10000 s = 100000 s And, dividing by 999999, we have s = 143857 = 4. Wherefore the decimal fraction, which was made = s, is = 476. In the same manner may be transformed into a decimal fraction, which will be 0,28571428, &c. and this enables us to find more easily the value of the decimal fraction which we have supposed = s; because 0,28571428 &c. must be the double of it, and, consequently, = 28. For we have seen that We also find = 0,42857142857 &c. which, according to our supposition, must be 3s; now we have found that 1 So that subtracting 3 s = 0,42857142857 &c. 477. When a proposed fraction, therefore, has the donominator 7, the decimal fraction is infinite, and 6 figures are continually repeated. The reason is, as it is easy to perceive, that when we continue the division we must return, sooner or later, to a remainder which we have had already. Now, in this division, 6 different numbers only can form the remainder, namely, 1, 2, 3, 4, 5, 6; so that, after the sixth division, at furthest, the same figures must return; but when the denominator is such as to lead to a division without remainder, these cases do not happen. 478. Suppose, now, that 8 is the denominator of the fraction proposed; we shall find the following decimal fractions; = 0,125; 0,25; = 0,375; =0,5; § = 3 =0,75; = 0,875; &c. 1 & = 0,625; If the denominator be 9, we have = 0,111 &c. & = 0,222 &c. = 0,333 &c. 3/ If the denominator be 10, we = 0,1; =0,2; To 3 = 0,3. This is evident from the nature of the thing, as also that 이 0,01; that = 0,0024 &c. 0,37; that 25 1000 = 0,256; that 24 479. If 11 be the denominator of the given fraction, we shall have = 0,0909090 &c. Now, suppose it were required to find the value of this decimal fraction; let us call it s, we shall have s = 0,090909, and 10 s = 00,909090; further, 100 s = 9,09090. If, therefore, we subtract from the last the value of s, we shall have 99 s = 9, and consequently s== IT. 11. We shall have, also, 0,181818 &c.; 0,272727 &c.; 0,545454 &c. = = †1 = 480. There is a great number of decimal fractions, therefore, in which one, two, or more figures constantly recur, and which continue thus to infinity. Such fractions are curious, and we shall shew how their values may be easily found. Let us first suppose, that a single figure is constantly repeated, and let us represent it by a, so that s = 0,aaaaaaa. We have 10 s = a,aaaaaaа and subtracting s=0,aaaaaaa we have 9 s = a; wherefore s = a When two figures are repeated, as ab, we have s = 0,abababa. Therefore 100 sab,ababab; and if we subtract s from it, there remains 99 sab; consequently s = ab When three figures, as abc, are found repeated, we have s = 0,abcabcabc; consequently, 1000 s=abc,abcabc; and subtract s from it, there remains 999 s = abc; wherefore $ = and so on. abc 999 8 Whenever, therefore, a decimal fraction of this kind occurs, it is easy to find its value. Let there be given, for example, 0,296296, its value will be 296 9298998 =, dividing both terms by 27. This fraction ought to give again the decimal fraction proposed; and we may easily be convinced that this is the real result, by dividing 8 by 9, and then that quotient by 3, because 27 3 X 9. We have 9) 8,0000000 3) 0,8888888 0,2962962, &c. which is the decimal fraction that was proposed. 481. We shall give a curious example by changing the frac tion 1 1 X2 X3 X4 X5 X6 × 7 x 8 x 9 x 10' tion. The operation is as follows. into a decimal frac 2) 1,00000000000000 3) 0,50000000000000 4) 0,16666666666666 5) 0,04166666666666 |