# Fractional part function of $\frac{y}{n} + k$

I am wondering why is $$\{\frac{y}{n} + k\} = \{\frac{y}{n}\}$$ where $$k$$ is an integer. $$\{x\}$$ is a fractional part function where $$\{x\} = x - \lfloor x\rfloor$$. I know it makes sense logically, but when I try to prove it I can not seem to get rid of $$k$$.

$$\{\frac{y}{n} + k\} = \frac{y}{n} + k - \lfloor\frac{y}{n} + k\rfloor$$. How does $$k$$ disappear from this equation?

• It would help if you stated any particular restrictions on the variables $n,k,y$ since some of the properties of the floor, ceiling, etc., functions are tied to such restrictions. I imagine the result doesn't old true, for example, for all $n,k,y$ in the reals. – Eevee Trainer Feb 14 '19 at 9:22
• Edited, $k$ is an integer. – Michael Munta Feb 14 '19 at 9:29

First of all, in general, $$\{\frac yn+k\}\neq \{\frac yn\}$$. For example, $$y=0,n=1,k=\frac12$$ are a counterexample. But yes, the equality holds if $$k$$ is an integer. Then, it can even be written more simply as $$\{x+k\}=\{x\}$$, where $$x$$ can be any real number.
The function $$\lfloor x\rfloor$$ has the property that $$\lfloor x+n\rfloor = \lfloor x \rfloor + n$$ for all $$n\in\mathbb N$$.
• I know of this property, but $n$ is still positive so it can not cancel out. – Michael Munta Feb 14 '19 at 10:01
• @MichaelMunta $\{x + k\} = x+k - \lfloor x+k\rfloor = x+k-(\lfloor x\rfloor + k) = \dots$. – 5xum Feb 14 '19 at 10:10