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In a manner similar to how the value $1$ can be represented as $0.(9)$ too, are there any other values that exhibit this property when represented in base 10?

Mikhail Katz
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Paul Manta
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4 Answers4

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Any number that ends in an infinite series of $9$'s is equal to the number changing all the $9$'s to $0$'s and incrementing the previous place by $1$. So $0.5=0.4999\ldots , 0.1328=0.132799999\ldots$ etc.

Ross Millikan
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Every repeating decimal can be represented as a fraction.

Example: Represent $0.\overline{25}$ as a fraction.

First, let $x = 0.\overline{25}$.

Next, multiply both sides of the equation by a power of ten to move the decimal place after the first repeat. In this case, I should choose 100 and get $$100x = 25.\overline{25}.$$

Notice that since there are an infinite number of 25's after the decimal place in $0.\overline{25}$, moving two of them in front of the decimal still leaves an infinite number of them after the decimal. That means we can write that as $$100x = 25 + x.$$

Now we can solve this for $x$.

$$ \begin{align*} 99x &= 25\\ x &= \frac{25}{99} \end{align*} $$

So, $x$ is both equal to $0.\overline{25}$ and $\frac{25}{99}$. That must mean $0.\overline{25} = \frac{25}{99}$.

Austin Mohr
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    This isn't really what my question was about, but now I know where the rules to convert repeating decimal fractions to ordinary ones come from. :) – Paul Manta Sep 17 '11 at 23:01
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    @Paul, then I guess my answer isn't really about your question either! Hope it helps, in any case. – The Chaz 2.0 Sep 17 '11 at 23:03
  • @Paul I interpreted your question as "Are there any other repeating decimals that can be represented as fractions?" Was that not what you intended? – Austin Mohr Sep 17 '11 at 23:06
  • @Austin I know all rational fractions can be represented as both decimal and ordinary fractions. (I also know that if an ordinary cannot be precisely represented as a decimal, that is a limitation of the number base, not a property of the value.) My question was about numbers that, from an infinitezimal point of view, should not represent the same value, but which in actuality they do. – Paul Manta Sep 17 '11 at 23:13
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To generalize Austin's answer (and since I don't know what "this property" is, exactly):

The number $0.\overline{a_1a_2...a_n}$ is equal to $ \frac{ a_1a_2 ... a_n}{99\dots9}$, where there are $n$ nines in the denominator.

So $0. \overline{23} = 23/99$

$0. \overline{123} = 123/999 = 61/333$

Brian M. Scott
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The Chaz 2.0
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  • And I can't, for the life of me, figure out how to fix that LaTex. I intended for there to be 99....9 in the denominator (which could be more precisely represented as nine times a sum of powers of ten...) – The Chaz 2.0 Sep 17 '11 at 23:00
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    Or a power of ten minus one. – anon Sep 17 '11 at 23:05
  • Thanks, Brian! And @anon, let's not get into another discussion a lá http://math.stackexchange.com/q/11/7850 :) – The Chaz 2.0 Sep 17 '11 at 23:12
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    The number $0.\overline{a_{p-1}a_{p-2}\ldots a_{1}a_{0}}$ is the sum of a geometric series

    $$0.\overline{a_{p-1}a_{p-2}\ldots a_{1}a_{0}}=\dfrac{N}{10^{p}}+\dfrac{N}{10^{2p}}+\cdots =\dfrac{N/10^{p}}{1-10^{-p}}=\dfrac{N}{10^{p}-1},$$

    where

    $$N=10^{0}a_{0}+10^{1}a_{1}+\cdots +10^{p-1}a_{p-1}.$$

     – Américo Tavares Sep 17 '11 at 23:19
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I seem to recall reading somewhere (therefore it's true!! (?)) that Johannes Kepler proposed a base-3 numeral system with three digits: $0$, $1$, and $-1$. In that system, the number $1/2$ can be represented in two different ways: $$ 1.,\ -1,\ -1,\ -1,\ \ldots, $$ and $$ 0.,\ 1,\ 1,\ 1, \ \ldots\ . $$ And similarly for every binary rational number (i.e. rational number whose denominator is a power of $2$).

Michael Hardy
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