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 Question from Jack, a student: I've been working on an equation to support my theory of repeating decimals. Specifically the correlation between the 9 and 11 denominators. I wanted to know if there was already an equation to describe this correlation? Mine is x/11=.b repeating and b=9x, so if you use any single digit number for x (say 4) if 9 x 4 = 36 then 4/11 = .36363636... I just want to know if this theory already exists and if so, what is it called?

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 n-digit number, with leading zeros allowed, over n nines is consists of the n digits from the numerator
repeated over and over. For example, 5/99 = 05/99 = .05050505... and 21/999 = .021021021...

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/104 + e/105 +f/106 ... 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 - (1-1/100)] (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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