Fractions and long division.

Question

$\frac{1}{9}=0.111\dots$

$9\times \frac{1}{9} = 0.999\dots$

$1=0.999\dots$

What is the problem here? Thanks for any help.

Answer 1

there is no problem, you just expressed the same number in two different ways or repesantations, it's just like you say $\frac{1}{2}=\frac{2}{4}$, the more rigorous way to write $0.9999999...$ is $9 \lim_{n\to +\infty}\sum_{i=1}^{n}(\frac{1}{10})^i$ and we have: $$9 \lim_{n\to +\infty}\sum_{i=1}^{n}(\frac{1}{10})^i=1 $$