Are there any other numbers like $0.999\ldots$?

Question

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?

Answer 1

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.

Answer 2

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 $0.\overline{25} = \frac{25}{99}$.

Answer 3

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

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

Answer 4

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 $2$).