Question

How do you find the repeating decimal 0.1 with 1 repeated as a fraction?

Answer 1

`0.1111111... = 0.bar1 = 1/9`

Explanation:

`0.111111.....`is a recurring decimal which can be written as `0.bar1`

It is useful to know that the decimals which have only 1 digit repeating all come from the fractions which are ninths.

`1/9 = 0.1111..." "2/9 = 0.22222...." "3/9 = 0.333333...`

`4/9 = 0.444444...`

and so on... But why is this?

Let `" "x= 0.111111111." "larr` one digit recurs, multiply by 10.
`" ":.10x = 1.11111111..." "larr` subtract them. `10x-x=9x`

`" "9x = 1.0000000..." "larr` all the way to infinity

Now: `" "9x = 1 hArr x = 1/9`

Dividing will confirm this: `1 div 9 = 0.1111111.....= 0.bar1`

What do you make of `0.99999..... ?`