# Floor Function

• November 14th 2010, 11:38 AM
dwsmith
Floor Function
Prove:
$\displaystyle\left \lfloor \frac{n}{2} \right \rfloor=\frac{n-1}{2} \ \mbox{if n is odd}$

Since n is odd, $\displaystyle n=2p+1 \ \ni \ p\in\mathbb{Z}$

$\displaystyle\left \lfloor \frac{2p+1}{2} \right \rfloor$ but I am not sure how that will help.
• November 14th 2010, 11:41 AM
roninpro
As a hint, consider $\frac{2p+1}{2}=p+\frac{1}{2}$. Then, $\lfloor p+\frac{1}{2}\rfloor=p$.
• November 14th 2010, 11:42 AM
dwsmith
I have that down as well but don't see the connection.
• November 14th 2010, 11:44 AM
roninpro
Then what is $\frac{n-1}{2}$?

It is $\frac{n-1}{2}=\frac{(2p+1)-1}{2}=\frac{2p}{2}=p$.