# Derivation of $[x + y] = [x] + [y + x - [x]]$ and the range of the $floor$ $function$

One of the properties of the floor function is this : $$[x + y] = [x] + [y + x - [x]]$$

Please let me know what the derivation of this property is...

Another question that I have is : What is the range of the floor function? Is it $$\Bbb Z$$, the set of all integers? Please let me know

And, I also have a little 'less mathematical question' : Is the derivation of properties of functions, like this one important to be known and there are about 10 properties of the floor function given in my textbook, do I have to learn each one of those? I mean will those properties be used in further concepts or are those just a one time thing?

Thanks

We know $$[x+n]=[x]+n$$ for all n $$\in Z$$
So you RHS=$$[x]+[y+x-[x]]=[x]+[y+x]-[x]=[y+x]=LHS$$ as [x] $$\in Z$$
• Thanks, and what about the range of the $floor$ $function$ and the other $non-mathematical$ question? – Rajdeep Sindhu Apr 10 '20 at 14:40
• Well, if domain is R then range is Z. Also, there are properties of [x] which you will study now but will use it in calculus later. Say x-1<[x] $\le x$ is extensively used in limits. So, better to know and n understand all the properties. And in my opinion derivations help to understand the results better. – Mathsmerizing Apr 10 '20 at 14:40