Question

Question #ae3be

Answer 1

`0.18bar(18)`

where the `bar(18)` means that the 18 repeats for ever

Explanation:

`color(blue)("Introduction to concept")`
It is difficult to explain the traditional approach with just writing so I am going to use a variant on it. It is unlikely that you have seen this before. Really it is the same thing. It just looks different.

It uses an accumulated adjustment approach.

For example

2 is the same as `20xxcolor(green)(1/10)`

20 is the same as `200xx color(green)(1/10)`

200 is the same as `2000 xxcolor(green)(1/10)`

combining this we have `2` is the same as `2000color(green)(xx1/10xx1/10xx1/10)`

This concept is applied every time we wish to divide 11 into a number that is less than 11.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Write 2 as `20xxcolor(green)(1/10)`

`"starting value"->20color(green)(xx1/10)`
`color(magenta)(1)xx11->color(white)("dddd.d")ul(11larr" Subtract")`
`ul(color(white)("dddddddddddddd.")9color(white)("ddddddddddd"))`
Write 9 as `color(white)("dddddd")90color(green)(xx1/10)`
`color(magenta)(8)xx11->color(white)("dddddd")ul(88 larr" Subtract")`
`ul(color(white)("ddddddddddddddd")2color(white)("ddddddddddd"))`
Write 2 as `color(white)("dddddd")20color(green)(xx1/10)`
`color(magenta)(1)xx11->color(white)("dddddd")ul(11 larr" Subtract")`
`ul(color(white)("ddddddddddddddd")9color(white)("dddddddddddd"))`
Write 9 as `color(white)("dddddd")90color(green)(xx1/10) larr" Notice anything ?"`
`color(magenta)(8)xx11->color(white)("dddddd")ul(88 larr" Subtract")`
`ul(color(white)("ddddddddddddddd")2color(white)("ddddddddddd"))`

From this there is a repeating pattern showing.

So we have `color(magenta)(1818) " repeating"->18181818181818.....`

For the 'pink' colours we have `color(magenta)(1818)color(green)(xx1/10xx1/10xx1/10xx1/10) = 0.1818.....`

So for repeating cycle of numbers we have:

`0.181818181..... -> 0.18bar(18)`

where the `bar(18)` means that the 18 repeats for ever

Answer 2

`0.bar18`

Explanation:

Now, when a decimal goes on forever with a pattern, then you can use a bar on top of the pattern. (e.g. `0.985985985...` can be written as `0.bar985`)

Here is an interesting rule...

`1/9=0.bar1`

`1/99=0.bar01`

`1/999=0.bar001`

and so on.

For example, `5/9=0.bar5`

Let's turn `2/11` so that it has a denominator of 9.

`2/11*9/9=>18/99`

Using our first rule, we see that `18/99=0.bar18`.