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 Nonnegative and nines (Posted on 2010-02-16)
Show that that every positive fraction is expressible in the form:

A                B
------- + -----------------
10C       (10D)*(nines)

where each of A, B, C and D is a nonnegative integer and "nines" represents a string of one or more 9s.

 No Solution Yet Submitted by K Sengupta Rating: 1.0000 (1 votes)

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 No problem! Comment 1 of 1
Well, this seems pretty straightforward.  The first term is the non-repeating portion of the fraction and the 2nd term is the repeating portion, where the number of 9's is the period of repetition.

For instance, 1/7 = .142857142857142857 ... = 142857/999999

53/35 = 1.5142857142857142857 ...
= 15/10 + 142857/10*999999

The representation is not unique.
1/3 = .333333.. = 3/9 = 3/10+ 3/90 = 33/100 + 3/900, etc.

1/7 = 142857/999999
= 1/10 + 428571/9999990
= 14/100 + 285714285714/99999999999900

I think it possible to require C = D, and still have every positive fraction expressible in that form.

 Posted by Steve Herman on 2010-02-16 20:26:04

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