Question #aee35

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.bar(a_1a_2...a_n)` is the decimal representation of a number (note that the bar denotes a repeating portion).

Let `x = 0.bar(a_1a_2...a_n)`

`=> 10^nx = a_1a_2...a_n.bar(a_1a_2...a_n)`

`=> 10^nx - x = a_1a_2...a_n.bar(a_1a_2...a_n) - 0.bar(a_1a_2...a_n)`

`=> (10^n-1)x = a_1a_2...a_n`

`=> x = (a_1a_2...a_n)/(10^n-1)`

So, in our given example, we have

`0.bar(5) = 5/(10^1-1) = 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*10^7/10^7 = 1234567/10^7`

`color(brown)("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=0.555bar5" " larr" "` the 5's go on for ever

So `10x=5.555bar5`

So `10x-x` is:

`10x=5.555bar5`
`ul(color(white)(10)x=0.555bar5) larr" Subtract"`
`color(white)(1)9x=5`

Divide both sides by 9

`x=5/9`