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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?

Eric Wofsey
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  • Yes, it is a finite number. It is less than 1 and greater than 0, for example. –  Aug 31 '19 at 03:22

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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.

CyclotomicField
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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 ^^

Fractal Admirer
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  • This is really helpful. Now I think I have a clear idea of infinite. In 0.2222... numbers of 2 is infinite but 0.222... it self is a finite number. Am I right? –  Aug 31 '19 at 03:39
  • @Shekhar Yes, that's right. – saulspatz Aug 31 '19 at 05:59