Repeating Decimals

Words to Remember

"Tell me, I'll forget.
Show me, I'll remember.
Involve me, I'll understand."
– Chinese proverb

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The Amador Valley Math Team seeks to foster a deeper understanding and enjoyment into the field of competitive and recreational mathematics through the school and the community.

Lessons‎ > ‎

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

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

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

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

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

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

and

is the number of digits in string

, and where $\underline{xy}$ represents the numeral formed by putting the string

to the left of string

.

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

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

Problem 2:

Prove that if

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

Challenge 1:

Prove that if