@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:

M1-forum-fractions-with-denominator-9.png

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! 🎉 🎉 🎉