What’s 0.222222222... as a simplified fraction?

2/9

I would have said 2/10 but math is not my forte.

@Xuph Don’t try to be irrational with us on Fluther!

Exactly two times 1/9 .

@janbb 2/10 is 0.2.

@Zaku Yeah, I know that much but I thought you’d just round down.

x = .222222…
10x = 2.22222… = x + 2
10x = x+2
9x = 2
x = 2/9

This same idea can be used to find the fraction representation for any repeating decimal, which means it can be used for any rational number.

@LostInParadise Got it. Thanks.

From a more formal point of view, .22222… can be considered a geometric series. I just so happen to have a Web page on that subject.

The easiest way of seeing that the answer is 2/9 is to reason as follows.

Consider .111111… If we multiply it by 9, we get .99999…., which is 1, so .11111… = 1/9, and .222222…. must be twice that value, or 2/9.

Suppose we had .123123123123123… . We can use similar reasoning to express the value as a fraction. Consider .001001001001001…. If we multiply this by 999, we get .99999…, which again is 1, so .0001001001001… is 1/999, and .123123123… is 123 times greater, and so equals 123/999.