How to transform $2,(9)$ to form $\frac{a}{b}$

Question

How to transform $2,(9)$ to form $\frac{a}{b}$. My attemp:

$$x=2,(9)/\cdot 10$$ $$10x=29,(9)$$ $$10x-x=29,(9)-2(9)$$ $$9x=27$$ $$x=3$$

but do not know if I have done the exact. Please help me

Answer 1

I assume $x, (y)$ is some sort of infinite decimal place notation.

If you have already proven that infinite decimals behave as consistently as finite decimals do under all the same transformations, then yes, this result is a simple consequence.

However, infinite decimals may be treated this way only because they are equivalent to a geometric series:

$$x,(y) = x + \sum_{k=1}^{\infty} y \cdot 10^{-k}$$

which is a geometric series with growth ratio ${1 \over 10}$. So I'd say the only thing the "proof" is missing is the formal statement ${1 \over 10} < 1$ which is necessary to infer convergence. Consider attempting this proof if you were using unary (base 1). Then $x,(y)_1$ would be divergent and your steps would lead you astray.

Answer 2

x,(9) always equates x+1. Everything you did is correct. It might be strange but it is correct answer.