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Repeating Decimals

by Eugene Chen of the Amador Valley Math Team

Repeating decimals may appear on math competitions. Sometimes, they rely on knowledge that must be learned from sources such as this page.

To start, we introduce some notation:
0.\overline{x} = 0.xxx\dots
In short, if a number has a line on top of it, it is intended to be repeated infinitely.

For example, 0.\overline{7} = 0.777\dots and 0.\overline{2} = 0.222\dots.

A few things to note:

If x is a digit, then 0.\overline{x} = \frac {x}{9}. For example, 0.\overline{7} = \frac {7}{9} and 0.\overline{2} = \frac {2}{9}.

Furthermore, if x is a two-digit string of digits, then 0.\overline{x} = \frac {x}{99}. For example, 0.\overline{02} = \frac {2}{99} and 0.\overline{93} = \frac {93}{99} = \frac {31}{33}.

Some more shortcuts:

If x,y are digits, then 0.x\overline{y} = \frac {9x + y}{90}. Furthermore, if x,y,z are digits, then 0.xy\overline{z} = \frac {90x + 9y + z}{900}. The number of nines and zeroes used depends on how many places repeat.

In general, if x,y are strings of digits, 0.x\overline{y} = \frac {\underline{xy} - x}{999\dots 999 \times 10^k}, where the number of nines is the number of digits in string y and k is the number of digits in string x, and where \underline{xy} represents the numeral formed by putting the string x to the left of string y.

Exercise 1:

Write the following repeated decimals as common fractions.

a) 0.\overline{8}

b) 0.\overline{13}

c) 0.\overline{54}

d) 0.2\overline{88}

e) 0.47\overline{3}


Problem 1:

Prove that if x,y are digits, then 0.x\overline{y} = \frac {9x + y}{90}.

Problem 2:

Prove that if x,y,z are digits, then 0.xy\overline{z} = \frac {90x + 9y + z}{900}.

Challenge 1:

Prove that if x,y are strings of digits, 0.x\overline{y} = \frac {\underline{xy} - x}{999\dots 999 \times 10^k}, where the number of nines is the number of digits in string y and k is the number of digits in string x, and where \underline{xy} represents the numeral formed by putting the string x to the left of string y.
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