Hence 10 n 1 10 n 2 10 n 2 10 n 1 n 2 1 is divisible

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Hence 10 n 1 - 10 n 2 = 10 n 2 (10 n 1 - n 2 - 1) is divisible by q , and so 10 n - 1, where n = n 1 - n 2 , is divisible by q . Hence r may be expressed in the form P/ (10 n - 1), or in the form P 10 n + P 10 2 n + . . . , i.e. as a pure recurring decimal with n figures. If on the other hand q = 2 α 5 β Q , where Q is prime to 10, and m is the greater of α and β , then 10 m r has a denominator prime to 10, and is therefore expressible as the sum of an integer and a pure recurring decimal. But this is not true of 10 μ r , for any value of μ less than m ; hence the decimal for r has exactly m non-recurring figures. 6. To the results of Exs. 2–5 we must add that of Ex. i . 3. Finally, if we observe that . 9 = 9 10 + 9 10 2 + 9 10 3 + · · · = 1 , we see that every terminating decimal can also be expressed as a mixed recurring decimal whose recurring part is composed entirely of 9’s. For example, . 217 = . 216 9. Thus every proper fraction can be expressed as a recurring decimal, and conversely. 7. Decimals in general. The expression of irrational numbers as non-recurring decimals. Any decimal, whether recurring or not, corresponds
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[IV : 78] LIMITS OF FUNCTIONS OF A 174 to a definite number between 0 and 1. For the decimal .a 1 a 2 a 3 a 4 . . . stands for the series a 1 10 + a 2 10 2 + a 3 10 3 + . . . . Since all the digits a r are positive, the sum s n of the first n terms of this series increases with n , and it is certainly not greater than . 9 or 1. Hence s n tends to a limit between 0 and 1. Moreover no two decimals can correspond to the same number (except in the special case noticed in Ex. 6). For suppose that .a 1 a 2 a 3 . . . , .b 1 b 2 b 3 . . . are two decimals which agree as far as the figures a r - 1 , b r - 1 , while a r > b r . Then a r = b r + 1 > b r .b r +1 b r +2 . . . (unless b r +1 , b r +2 , . . . are all 9’s), and so .a 1 a 2 . . . a r a r +1 · · · > .b 1 b 2 . . . b r b r +1 . . . . It follows that the expression of a rational fraction as a recurring decimal (Exs. 2–6) is unique. It also follows that every decimal which does not recur represents some irrational number between 0 and 1. Conversely, any such number can be expressed as such a decimal. For it must lie in one of the intervals 0 , 1 / 10; 1 / 10 , 2 / 10; . . . ; 9 / 10 , 1 . If it lies between r/ 10 and ( r + 1) / 10, then the first figure is r . By subdividing this interval into 10 parts we can determine the second figure; and so on. But (Exs. 3, 4) the decimal cannot recur. Thus, for example, the decimal 1 . 414 . . . , obtained by the ordinary process for the extraction of 2, cannot recur. 8. The decimals . 101 001 000 100 001 0 . . . and . 202 002 000 200 002 0 . . . , in which the number of zeros between two 1’s or 2’s increases by one at each stage, represent irrational numbers. 9. The decimal . 111 010 100 010 10 . . . , in which the n th figure is 1 if n is prime, and zero otherwise, represents an irrational number. [Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine m and p so that m , m + p , m + 2 p , m + 3 p , . . . are all prime numbers; and this is absurd, since the series includes m + mp .] * * All the results of Exs. xxix may be extended, with suitable modifications, to decimals in any scale of notation. For a fuller discussion see Bromwich, Infinite Series , Appendix I.
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