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.

  • 1
    $\begingroup$ Let $x=y=3/4$. Or $x=y=1/2$. Or $x=3.7$ and $y=19.3$. $\endgroup$ Mar 15, 2015 at 4:55
  • $\begingroup$ You are welcome. All that matters is that you now know what's going on. $\endgroup$ Mar 15, 2015 at 5:01
  • $\begingroup$ @AndréNicolas: it makes a better reference in an answer. $\endgroup$
    – robjohn
    Mar 15, 2015 at 5:03

1 Answer 1


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


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