Question

How do you write `2/11` as a decimal?

Answer 1

`0.18dot1dot8`
The two dots above the last 1 and 8 indicate that they repeat indefinitely. You may also use a dash above them.

Answer 2

Suppose you do not have a calculator.

`0.18bar(18)`

Explanation:

For another IMPORTANT example have a look at
https://socratic.org/s/aKi5x5nx

It is important as it shows how to deal with zeros. In the above we have the answer `0.5909090909...` which has a lot of zeros.
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`color(blue)("Introduction to method")`

We avoid the decimal point until write at the end

11 is more that 2 but we can and may write 2 as `20xx1/10` where the `1/10` is an adjustment. The 20 is more that 11 so the division is a bit more strait forward.

We reintroduce the decimal at the very end by multiplying the answer by EXERY adjuster of `1/10`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

`color(white)("dddddddd")20color(green)(xx1/10)larr color(brown)(" changed the 2")`
`color(magenta)(1)xx11->ul(11 larr" Subtract")`
`color(white)("ddddddddd")9 larr" Remainder"`

/////////////////////////////////////////////////////////////////////////
`color(white)("ddddddddd")90 larrcolor(green)(xx1/10) larrcolor(brown)(" changed the remainder")`
`color(magenta)(8)xx11->color(white)("d")ul(88 larr" Subtract")`
`color(white)("dddddddddd")2 larr" Remainder"`

////////////////////////////////////////////////////////////////////////////
`color(white)("ddddddddd")20color(green)(xx1/10)larrcolor(brown)(" changed the remainder")`
`color(magenta)(1)xx11-> color(white)("d")ul(11 larr" Subtract")`
`color(white)("dddddddddd")9`

/////////////////////////////////////////////////////////////////////////
`color(white)("dddddddddd")90color(green)(xx1/10) larrcolor(brown)(" changed the remainder")`
`color(magenta)(8)xx11->color(white)("dd")ul(88 larr" Subtract")`
`color(white)("dddddddddd")2 larr" Remainder"`

We are getting a pattern of repeats so we may stop at this point as we can see what that pattern is.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting what we have got so far together")`

`color(magenta)(1818)color(green)(xx1/10xx1/10xx1/10xx1/10)color(white)("d")=color(white)("d")0.181818`

As this is a repeating pattern we have `0.18181818181818....` going on for ever.

If we put a bar over a repeating par it mathematically indicates that they repeat for ever. So we can write:

`0.18bar(18)`