Prove that $\lfloor a\rfloor + \lfloor a+0.5\rfloor =\lfloor 2a\rfloor $ for every real number $a$.

Question

Show that: $$\lfloor a\rfloor + \lfloor a+0.5\rfloor =\lfloor 2a\rfloor $$ for every real number $a$

I know that you can say that $a=\lfloor a\rfloor+\phi$ with $0\leq \phi<1$. Please help.

Answer 1

Write $a= k+\phi$ where $0\leq \phi<1$ and $k=[a]$. Now consider two cases.

• If $\phi <1/2$ then $$\lfloor a\rfloor + \lfloor a+0.5\rfloor = k+\lfloor k+\phi+0.5\rfloor = 2k+\lfloor\phi+0.5\rfloor=2k =2\lfloor a\rfloor$$
• $\phi \geq 1/2$ then $$\lfloor a\rfloor + \lfloor a+0.5\rfloor = k+\lfloor k+\phi+0.5\rfloor = 2k+\lfloor\phi+0.5\rfloor=2k +1$$ on the other hand we have also $$\lfloor 2a\rfloor=\lfloor 2k+2\phi\rfloor =2k+\lfloor2\phi\rfloor =2k+1$$ where $\lfloor2\phi\rfloor=1$

and we are done.