What is 0.22222... equal to? Here '...' Represent infinite time.

Question

I have seen that 0.99999... equal to 1. But what about 0.2222...? Do it also equal to some finite number? If yes then what is it? And how do you know?

Answer 1

I think you're misreading what a finite number and what a periodic tithe are.

Let $a$ be a number, $a$ is finite if and only if $|a| < +\infty$, so $0.\overline{2}$ is finite, because $- \infty < 0 < 0.\overline{2} < 1 < + \infty$.

What is not finite is the number of terms of the sum $$0 + 0.2 + 0.02 + 0.002 + ... = \displaystyle\sum_{k=1}^{+\infty} \frac{2}{10^k} = 0.\overline{2}$$

I hope that I've helped ^^

Answer 2

Yes it's $\frac{2}{9}$ and we can prove this as follows.

First set $x = 0.222...$ so that $10x=2.222..$. Then, subtracting the second equation from the first give us $9x=2$ and so $x=\frac{2}{9}$ and we're done.