Question

Recurring decimal to fraction?

How do you write `0.9999` recurring as a fraction. I know how to do ones like:

`0.8dot(8)=8/9` and so on. using `1/9` `1/99` `1/90` etc. The problem is you can't do it with 9 repeating.

Answer 1

`0.bar9=1`

`0.bar8 = 8/9`

Explanation:

let:

`m=0.bar9`

`10m=9.bar9`

subtract the first equation from the second one:

`10m=9.bar9`
`underline("- "m=0.bar9)`
`" "9m=9`

`m=1`

The other problem:

let:

`m=0.bar8`

`10m=8.bar8`

subtract the first equation from the second one:

`10m=8.bar8`
`underline("- "m=0.bar8)`
`" "9m=8`

`m=8/9`