# Question #aee35

Oct 19, 2016

$0. \overline{5} = \frac{5}{9}$ is a rational number

#### Explanation:

As it happens, any real number whose decimal representation has a portion which repeats indefinitely or terminates* can be expressed as a ratio of integers, and thus is rational.

We can find the fraction using a bit of algebra.

Suppose $0. \overline{{a}_{1} {a}_{2.} . . {a}_{n}}$ is the decimal representation of a number (note that the bar denotes a repeating portion).

Let $x = 0. \overline{{a}_{1} {a}_{2.} . . {a}_{n}}$

$\implies {10}^{n} x = {a}_{1} {a}_{2.} . . {a}_{n} . \overline{{a}_{1} {a}_{2.} . . {a}_{n}}$

$\implies {10}^{n} x - x = {a}_{1} {a}_{2.} . . {a}_{n} . \overline{{a}_{1} {a}_{2.} . . {a}_{n}} - 0. \overline{{a}_{1} {a}_{2.} . . {a}_{n}}$

$\implies \left({10}^{n} - 1\right) x = {a}_{1} {a}_{2.} . . {a}_{n}$

$\implies x = \frac{{a}_{1} {a}_{2.} . . {a}_{n}}{{10}^{n} - 1}$

So, in our given example, we have

$0. \overline{5} = \frac{5}{{10}^{1} - 1} = \frac{5}{9}$

Note that this process can be generalized to handle cases in which the repeating portion starts after nonrepeating digits as well.

*A terminating decimal can be converted to a fraction by multiplying and dividing by an appropriate power of $10$. E.g.

$0.1234567 = 0.1234567 \cdot {10}^{7} / {10}^{7} = \frac{1234567}{10} ^ 7$

Oct 20, 2016

very difficult to demonstrate this without algebra

This is a rational number as it may be written as the fraction $\frac{5}{9}$

#### Explanation:

$\textcolor{b r o w n}{\text{Note}}$
If the use of the $x$ confuses you do the following: every time you see $x$ think of it as saying "the unknown fraction value"

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let $x = 0.555 \overline{5} \text{ " larr" }$ the 5's go on for ever

So $10 x = 5.555 \overline{5}$

So $10 x - x$ is:

$10 x = 5.555 \overline{5}$
$\underline{\textcolor{w h i t e}{10} x = 0.555 \overline{5}} \leftarrow \text{ Subtract}$
$\textcolor{w h i t e}{1} 9 x = 5$

Divide both sides by 9

$x = \frac{5}{9}$