What is ⌊0.9 recurring ⌋? For a ceiling and floor function, the number is taken to 0 decimal places. Does this process mean that 0.9 recurring inside a floor function would go to 0? Or would the mathematician take 0.9 recurring to be equal to 1, thus making the answer 1?
And if 0.9 recurring does equal 1, does that mean (by definition) that ⌊1⌋ = 0?

You have to look at the $.9$ recurring as a sum... then you'll know the answer.
$$\bar{.9} = \sum_{i=1}^{\infty} \frac{9}{10^{i}}$$
So, $$\lfloor \bar{.9}\rfloor = \left\lfloor \sum_{i=1}^{\infty}\frac{9}{10^{i}}\right\rfloor=\lfloor 1 \rfloor = 1.$$  You cannot split up the floor function over a sum, i.e. $\lfloor a+b\rfloor \neq \lfloor a\rfloor + \lfloor b\rfloor$.
