71] LIMITS OF FUNCTIONS OF AN INTEGRAL VARIABLE 147 
6. To the results of Exs. 2—5 we must add that of Ex. I. 4. Finally, if 
we observe that 
.9 = 1+A + A + —i 
KK10 2 10 3 
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 = '2169. Thus every proper fraction can be expressed as a re 
curring decimal, and conversely. 
7. Decimals in general. The expression of irrational numbers as 
non-recurring decimals. Any decimal, whether recurring or not, corresponds 
to a definite number between 0 and 1. For the decimal •a 1 a 2 a 3 a i ... stands 
for the series 
a l , ^2 , ( H_ , 
10 + 10 2 + 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: also it is certainly less 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 ..., 'l>ib 2 b 3 ... are 
two decimals which agree as far as the figures a r _,, Z>,._ 1} while a r >h r . 
Then cf,.>6,.+1>Z) ( .. 6,. +1 6,. +2 ... (unless 6 r + 1 , b r + 2 , ... are all 9’s), and so 
ct\a 2 ,..cicely+j...b\ b 2 . "bp bp + j.... 
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 in r/10, (/• +1)/10 the first figure is r: by subdividing this interval 
into 10 parts we can determine the second figure; and so on. 
Thus we see that the decimal T414..., obtained by the ordinary process 
for the extraction of x /2, cannot recur. 
8. The decimals T010010001000010... and -2020020002000020..., in 
which the number of zeros between two l’s or 2’s increases by one at each 
stage, represent irrational numbers. 
9. The decimal -11101010001010..., in which the nth 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 in and p so that m, m+p, m + 2p, 
m + Zp,... are all prime numbers; and this is absurd, since the series includes 
rn + mp.]* 
* All the results of Exs. XXXI. 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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