Is there a General formula for $\lfloor a+b\rfloor$

ceiling-and-floor-functionssoft-question

This question of mine is a soft question. In this question I would like to ask

Is there a General formula for $\lfloor a+b\rfloor$?

In this question both $a,b \in \Bbb Z$ and $\lfloor \quad \rfloor$ is the floor function. I tried searching online for this but found none. Any help/hint is appreciated

Thanks in advance 🙂

Edit : Can $\lfloor a+b \rfloor = \lfloor a\rfloor + \lfloor a + b -\lfloor a\rfloor \rfloor$?

I tried to prove it in this way.

We know that $$\lfloor x+n\rfloor = \lfloor x\rfloor + n$$
if $n\in \Bbb Z$

So, $$\lfloor a \rfloor + \lfloor a+b-\lfloor a\rfloor \rfloor = \lfloor a \rfloor + \lfloor a+b\rfloor – \lfloor a \rfloor = \lfloor a+b\rfloor$$

Is this correct?

Best Answer

The floor function restricted to $\mathbb Z$ is the identity function. So you can simply skip it! It ain't don't doing nothing!

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