Writing Repeating Decimals as FractionsDate: 11/8/95 at 19:12:32 From: Anonymous Subject: Mathematics Grade 8 When you are expressing a repeating decimal as a fraction, why is it you use 9, 99, 999, or 9999 etc. in the denominator? For example, 0.5555...(5 repeating) equals 5/9. Please help and thanks in advance. Jadon Date: 11/12/95 at 14:37:56 From: Doctor Jeremy Subject: Re: Mathematics Grade 8 It comes from a well-known result in series. If you have the series 1 1 1 1 x = - + -- + -- + ... + -- n a 2 3 n a a a as n increases, the sum gets closer to 1/(a-1). So as you keep going through the digits of .111111..., which is 1 1 1 1 --- + --- + --- + --- + ... 10 2 3 4 10 10 10 your sum gets closer and closer to 1/(10-1) = 1/9. Since .5555 is 5 times that, the sum is 5 times (1/9) or 5/9. Similarly, you get a denominator of 99 because it is 100-1. -Doctor Jeremy, The Geometry Forum |
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