[Math] Proving that floor(n/2)=n/2 if n is an even integer and floor(n/2)=(n-1)/2 if n is an odd integer.

Question

How would one go about proving the following. Any ideas as to where to start?

For any integer n, the floor of n/2 equals n/2 if n is even and (n-1)/2 if n is odd.

Summarize:

[n/2] = n/2 if n = even

[n/2] = (n-1)/2 if n = odd

Working through it, I try to initially set n = 2n for the even case but am stuck on how to show its a floor…

thanks

Answer 1

You should set $n=2m$ for even numbers, where $m$ is an integer. Then $\frac n2=m$ and the floor of an integer is itself. The odd case is similar.