my question is simple. For some reason I can't seem to deduce whether the statement:
{x} + {y} = {x+y}
Is true, where $x,y \in \mathbb{Q} $ and {x} denotes the fractional part of x.
This really is a somewhat stupid question and I give thanks to anyone willing to waste their time answering it.
Take for example $x=y=\frac{3}{4}$. Then $\{x\}+\{y\}=1.5$, while $\{x+y\}=0.5$.
One can find many other examples where the equality fails. Take for instance $x=7.3$ and $y=15.7$.
Remark: A result very close in spirit to the false $\{x\}+\{y\}=\{x+y\}$ does hold. It is $$\{\{x\}+\{y\}\}=\{x+y\}.$$