I was just thinking about the equality (for a bit of fun)
$\lfloor ab \rfloor = \lfloor a\rfloor\lfloor b \rfloor$
for non-integer $a$ and $b$. I was wondering if anyone could point me to some lecture notes or something such where this features, since I can't find anything on google. Alternatively, is there some trivial condition that I'm overlooking? I've been scribbling away for a while just writing some things down and was wondering if anybody could tell me anything about the conditions to place on that equality?
My thought process was as follows: let $a = a_0.a_1a_2a_3...$ and $b=b_0.b_1b_2b_3...$. From this, we can write
$$a = a_0 + \sum_{n=1}^k \frac{a_k}{10^n}\ \ \ \text{and}\ \ \ b = b_0 + \sum_{m=1}^j \frac{b_m}{10^m}$$
and then
\begin{align} ab &= \left(a_0 + \sum_{n=1}^k \frac{a_k}{10^n}\right) \left(b_0 + \sum_{m=1}^j \frac{b_m}{10^m}\right)\\ &= a_0b_0 + b_0\sum_{n=1}^k \frac{a_k}{10^n} + a_0\sum_{m=1}^j \frac{b_m}{10^m} + \left(\sum_{n=1}^k \frac{a_k}{10^n}\right)\left(\sum_{m=1}^j \frac{b_m}{10^m}\right) \end{align}
and then we simply have that
$\lfloor a b \rfloor = \lfloor a \rfloor \lfloor b\rfloor$ iff $$0 < b_0\sum_{n=1}^k \frac{a_k}{10^n} + a_0\sum_{m=1}^j \frac{b_m}{10^m} + \left(\sum_{n=1}^k \frac{a_k}{10^n}\right)\left(\sum_{m=1}^j \frac{b_m}{10^m}\right) <1$$
... more concsely, see @MorganRogers' answer for some better use of notation.