The fractions ${2\over 9}$ and ${22\over 99}$ both have the same decimal value $0.22222\ldots$ But obviously they are not equal. What causes this situation?
And also, what is the correct rational form of $0.2$ recurring? ${2\over 9}$ or ${22\over 99}$ or anything else?
On
In terms of repeating decimals $$\frac{1}{9} = .1111111111\cdots$$. Any multiple of this value leads to a repeating similar pattern. As examples defined by this problem: \begin{align} .2222\cdots &= \frac{2}{9} \\ &= \frac{22}{99} \\ &= \frac{222}{999}\\ &= \cdots \end{align}
On
To answer in a more formal way than you are probably looking for, rational numbers are defined as an equivalence class. The form of the class looks like $[(a,b)]$ where $a,b$ are both integers and $b\ne 0$. In this sense, $[(a,b)]=\frac a b$. We call it an equivalence class because there's a whole set of members that are equivalent to each other. For instance $\frac 1 2=\frac 2 4$, so in this parlance we'd say $(1,2)\in [(2,4)]$, or visa versa....they are all part of the same set. Each fraction has an infinite number of representations, that's what we call the particular pair of integers used to form them. The rule of when two fractions are equal is the rule of cross multiplication: $(a,b)\sim (c,d)$ if $ad=bc$
So in short: Rational numbers have an infinite number of representations, all for the same number. No particular representation is more "proper" than another, although it's common to use "reduced" form, where you require the gcd of $a$ and $b$ to be 1, and $b>0$,
$$\frac29 = \frac29\cdot\big(1\big) = \frac29\cdot\left(\frac{11}{11}\right) =\frac{2\cdot 11}{9\cdot 11} =\frac{22}{99}$$
Placing rational numbers into decimal form you will find that they either terminate ($1/4$ for example), or repeat for ever, like the $2/9$ you found. I recommend trying some long division on integers divided by $11$, or $7$. You ultimately will see why the patterns repeat based on the remainder from the previous digit.