Is there also an other way to show the equality: $\left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$?

I want to show that:

$$\left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$$

That's what I have tried:

• $\left\lfloor \frac{n}{2}\right\rfloor=\max \{ m \in \mathbb{Z}: m \leq \frac{n}{2}\}$
• $\left\lceil \frac{n}{2} \right\rceil= \min \{ m \in \mathbb{Z}: m \geq \frac{n}{2}\}$

If $n=2k,k \in \mathbb{Z}$,then: $\frac{n}{2} \mathbb{Z}$,so $$\left\lfloor \frac{n}{2}\right\rfloor=\max \left\{ \frac{n}{2}, \frac{n-2}{2}, \dots \right\}=\frac{n}{2} \\ \left\lceil \frac{n}{2} \right\rceil=\min \left\{ \frac{n}{2}, \frac{n+2}{2}, \dots \right\}=\frac{n}{2}$$

Therefore, $\left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$.

If $n=2k+1, k \in \mathbb{Z}$,then $\frac{n}{2} \notin \mathbb{Z}$.So:

$$\left\lfloor \frac{n}{2}\right\rfloor=\max \left\{ \frac{n-1}{2}, \frac{n-3}{2}, \dots \right\}=\frac{n-1}{2}\\ \left\lceil \frac{n}{2} \right\rceil=\min \left\{ \frac{n+1}{2}, \frac{n+3}{2}, \dots \right\}=\frac{n+1}{2}$$

Therefore, $\lfloor \frac{n}{2}\rfloor + \lceil \frac{n}{2} \rceil=\frac{n-1}{2}+\frac{n+1}{2}=n$

Is there also an other way to show the equality or is it the only one?

• Presumably, for $n$ an integer... Jul 26, 2014 at 17:08
• Yes,I have to show that the inequality stands for each integer $n$. Jul 26, 2014 at 17:11
• @ThomasAndrews Is the way I showed the equality the only possible or is there also an other one? Jul 26, 2014 at 23:07
• What do you expect as "an other way" ?
– user65203
Jul 27, 2014 at 8:36

Since the function $f(x)=\{x\}=x-\lfloor x\rfloor$ is periodic with period one, the function: $$g(n) = \left\lfloor\frac{n}{2}\right\rfloor+\left\lceil\frac{n}{2}\right\rceil -n$$ is periodic with period $2$. Since $g(0)=g(1)=0$, $g(n)=0$ for every $n\in\mathbb{N}$.

• Did you conclude this,because of the fact that $f$ is periodic? Jul 26, 2014 at 23:27
• Yes, it is exactly so. Jul 26, 2014 at 23:28
• I understand..thank you!!! Jul 27, 2014 at 12:51

Well, here's another one. In fact, it is possible to prove that for any integer $n$ and any real number $\alpha$, we have $$n=\lfloor (1-\alpha) n\rfloor +\lceil \alpha n\rceil.$$

First, we have $$\alpha n\leq\lceil\alpha n\rceil,$$ therefore, $$n-\lceil\alpha n\rceil\leq (1-\alpha)n,$$ but then, since $n-\lceil\alpha n\rceil$ is an integer, this means $$n-\lceil\alpha n\rceil\leq \lfloor (1-\alpha)n\rfloor.$$

Similarly, we have $$\lfloor (1-\alpha)n\rfloor\leq (1-\alpha)n,$$ therefore, $$\alpha n\leq n-\lfloor (1-\alpha)n\rfloor,$$ but then again, $$\lceil\alpha n\rceil\leq n-\lfloor (1-\alpha)n\rfloor,$$ from the above, we get $$\lfloor (1-\alpha) n\rfloor +\lceil \alpha n\rceil\leq n\leq \lfloor (1-\alpha) n\rfloor +\lceil \alpha n\rceil.$$

Let $$x,y$$ be any two real numbers. Then $$\lfloor x\rfloor+\lceil y\rceil>x+y-1$$, since $$\lfloor x\rfloor>x-1$$ and $$\lceil y\rceil\geq y$$. Similarly $$\lfloor x\rfloor+\lceil y\rceil. So $$\lfloor x\rfloor+\lceil y\rceil$$ differs from $$x+y$$ by strictly less than $$1$$. Since $$\lfloor x\rfloor+\lceil y\rceil$$ is always an integer, if $$x+y$$ is also an integer the two must be equal. This holds in particular for $$x=y=n/2$$.

For even $$n$$, $$\left\lfloor\frac n2\right\rfloor=\left\lceil\frac n2\right\rceil=\frac n2$$.

For odd $$n$$, $$\left\lfloor\frac n2\right\rfloor=\frac{n-1}2$$, and $$\left\lceil\frac n2\right\rceil=\frac{n+1}2$$.

• So,is this the only way to show that the relation is true? Jul 26, 2014 at 17:15
• Er, your way is different, its uses bounds on number sets. What else did you expect, this is rather elementary ?
– user65203
Jul 26, 2014 at 17:17
• Could you explain me further which the difference is? :/ Jul 26, 2014 at 17:18
• You use bounds on number sets, this is unnecessary.
– user65203
Jul 26, 2014 at 17:20
• @YvesDaoust how did you prove the equality in your post? Jul 26, 2014 at 17:39