Multiplication of repeating decimal $0.3333\overline{3}$ by $3$

Question

Let's start considering a simple fractions like $\dfrac {1}{2}$ and $\dfrac {1}{3}$.

If I choose to represent those fraction using decimal representation, I get, respectively, $0.5$ and $0.3333\overline{3}$ (a repeating decimal).

That is where my question begins.

If I multiply either $\dfrac {1}{2}$ or $0.5$ by $2$, I end up with $1$, as well as, if I multiply $\dfrac {1}{3}$ by $3$.

Nonetheless, if I decide to multiply $0.3333\overline{3}$ by $3$, I will not get $1$, but instead, $0.9999\overline{9}$

What am I missing here?

*Note that my question is different than the question Adding repeating decimals

Answer 1

Hint: compute the difference between $1$ and $0.9\bar9$. How much is that ? What do you conclude ?

Answer 2

$0.9999$ repeating is equal to $1$. There is some crucial thing you should understand: Every finite decimal of the form $0.999$ is $not$ equal to $0.999$ repeating. The latter represents the number one and no other number. Watch this video and the others related to the construction of reals by the professor Francis Su: https://www.youtube.com/watch?v=sqEyWLGvvdw