@funnyorangutan-1 Thanks for your very polite and curious responses. The first question about why \(\frac{1}{3} = 0.\overline{333}\ldots \) is quite long, so if you don't mind, I'll put that in it's own post, here. https://forum.poshenloh.com/topic/545/why-is-0-3333 Answer to why \(\frac{1}{3} = 0.\overline{333}\ldots \)
For your other question, about why \( 0.\overline{222}\ldots = \frac{2}{9} \text{ and } 0.\overline{444}\ldots = \frac{4}{9}\) and so on, maybe this diagram will help a bit:
You can treat \(0.\overline{111}\ldots\) as a starting point from which you can multiply by any factor in order to get any fraction of the form \(\frac{n}{9},\) where \(n\) is an integer.
"Oh, but didn't you leave out the \(\frac{3}{9} \text{ and } \frac{6}{9}?\)"
Very perceptive question!
I left those out because \(\frac{3}{9}\) and \(\frac{6}{9}\) aren't in simplified form. You wouldn't put "\(\frac{3}{9}\)" on a test as an answer. Instead, you would put \(\frac{1}{3}.\) And you wouldn't write "\(\frac{6}{9}\)" on a test as an answer either -- you would write \(\frac{2}{3}.\) So even though the pattern holds for multiplying \(0.\overline{111}\ldots\) by both \(3\) and \(6,\) I left it out, but it is very nice to know that everyone holds across math, and that everything works as it should, because
$$ 0.\overline{333}\ldots = \frac{3}{9} = \frac{1}{3} $$
Great, math is consistent!
When the world seems like it's failing you and there's no end in sight......... there's always math!