Question

Can a repeating decimal be equal to an integer?

Answer 1

No, it will allways turn out to be a fraction.

Explanation:

I will not delve into how you turn a repeating decimal into a fraction, but just one example:

`0.333....=1/3`

There is one exeption though (see example above):

`0.999....=3*0.333....=3*1/3=1`

Answer 2

Yes

Explanation:

The general term of a geometric series can be written:

`a_n = a*r^(n-1)`

where `a` is the initial term and `r` the common ratio.

When `abs(r) < 1` then its sum to infinity converges and is given by the formula:

`sum_(n=1)^oo ar^(n-1) = a/(1-r)`

So for example:

`0.999... = 9/10+9/100+9/1000+...`

is given by `a=9` and `r=1/10`

which has sum:

`sum_(n=1)^oo 9/10*(1/10)^(n-1) = (9/10)/(1-1/10) = (9/10)/(9/10) = 1`

So `0.bar(9) = 0.999... = 1`

In fact, any integer can be expressed as a repeating decimal using `9`'s.

For example:

`12345 = 12344.999... = 12344.bar(9)`

`-5 = -4.999... = -4.bar(9)`