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Here are the decimal representations of a few fractions:
1/2=.5
1/3=.33333...
1/4=.25
1/5=.2
1/6=.16666...
1/7=.142857 142857 ...
1/8=.125
1/9=.11111...
1/10=.1
1/11=.09090909...
1/12=.0833333...
Some of these are repeating decimals (1/3=.33333... or 1/7=.142857 142857 ...), and the rest come out exact (1/5=.2), to a certain number of decimal places. Some of these fractions (1/12=.0833333...) go a ways before they start repeating. All factions are of one of these two kinds. Numbers like pi and e do not repeat, ever. Whether a fraction is repeating or comes out even after some decimal places depends upon the fact that we are using base 10. 1/5 is of course 2/10, and does not repeat.
1/37 is .027 027 027... That is a fairly short cycle, in which to repeat. 1/47 is 0.0212765957446808510638297872340425531914893617 02127659574468... 1/47 takes a long time (46 digits) before it repeats. What is the difference between these two numbers? 37 and 47 are both primes. Is it a mystery? It is easy to figure out, actually. Let's say you want a certain number to repeat every n digits, in a repeating decimal:
.000000xyz000000xyz000000xyz...
In this example our repeating number (xyz) is three digits long, and n is 9. Well this can be represented as a Geometric Series. And the fraction (perhaps unreduced) is xyz/999999999, with 9 9's in the denominator, or n 9's in general. Well, 37 divides 999 (999=3x3x3x37), so we would expect it to have a cycle of 3, as it does not divide 9 or 99. And it does (1/37=.027 027...). 11 divides 99, so it has a cycle of 2 (1/11=.01010101...).
Well, the amazing thing is not so much that 1/37 has such a short cycle, but that 47 divides no string of 9's (or 1's) until it gets to 46 9's (or 1's). 1/47=0212765957446808510638297872340425531914893617/9999999999999999999999999999999999999999999999 My calculator doesn't verify this, as it gives only 15 digits of accuracy. It turns out that 1/p, where p is prime, repeats in p-1 or fewer digits. In our original list of fractions, we see that 1/7=.142857 142857 ... is the only one that does this, repeating in six digits. Actually 1/2=.5 repeats with a cycle of one, because it is .500000..., with repeated zeros.
A string of 9's is 9 times a Repunit. See Repunit Primes.