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?
Are there any other numbers like $0.999\ldots$?
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0Looks like this person wants to invent a .999 tag and is adding it to these old questions. I'm not sure I agree with it but there it is. – 2013-11-21
4 Answers
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.
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 $0.\overline{25} = \frac{25}{99}$.
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0@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. – 2011-09-17
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 $0. \overline{23} = 23/99$
$0. \overline{123} = 123/999 = 61/333$
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1The 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 $N=10^{0}a_{0}+10^{1}a_{1}+\cdots +10^{p-1}a_{p-1}.$ – 2011-09-17
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 $2$).
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0http://en.wikipedia.org/wiki/Balanced_ternary – 2011-09-18