



 
Hi Jack, It does, but I don't know if it has a name. The basic rule is that the decimal expansion of a fraction that is an ndigit number, with leading zeros allowed, over n nines is consists of the n digits from the numerator Your rule for 11 works because multiplying both the numerator and denominator by 9 converts the your fraction to an equivalent one with denominator 99. A similar rule works for 111 and 999. For example, 55/111 = 495/999 = .495495495... The reason why it works is connected to the meaning of the decimal expansion. The notation 0.abcdef... stands for a/10 + b/100 + c/1000 + d/10^{4} + e/10^{5} +f/10^{6} ... This is the same as ab/100 + cd/(100)^{2} + ef/(100)^{3} + ... Let's suppose that the number in question has a repeating block of length 2 starting at the decimal point, say .ababab... This stands for ab/100 + ab/(100)^{2} + ... = ab/100[1 + 1/(100) + 1/(100)^{2}...] = ab/100[1  (11/100)] (from the formula for the sum of a geometric series) = ab/99. If you have a fraction like ab/99, then the argument in the above paragraph, used from left to right, show that its decimal expansion is .ababab... I hope this helps. Victoria  


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