Addition of Fractional Part Function

Question

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.

You are welcome. All that matters is that you now know what's going on. — André Nicolas, Mar 15 '15 at 05:01

score 2 · Accepted Answer · answered Mar 15 '15 at 05:09

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 $$\{\{x\}+\{y\}\}=\{x+y\}.$$

1 Answers1