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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

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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] $\in Z$

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  • $\begingroup$ Thanks, and what about the range of the $floor$ $function$ and the other $non-mathematical$ question? $\endgroup$ – Rajdeep Sindhu Apr 10 '20 at 14:40
  • $\begingroup$ 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. $\endgroup$ – Mathsmerizing Apr 10 '20 at 14:40
  • $\begingroup$ Thank You, it was really helpful ) $\endgroup$ – Rajdeep Sindhu Apr 10 '20 at 14:44
  • $\begingroup$ You are welcome. Actually i have done a series on greatest integer function. It might be of some help to you youtube.com/… $\endgroup$ – Mathsmerizing Apr 10 '20 at 14:45
  • $\begingroup$ Thanks again...! $\endgroup$ – Rajdeep Sindhu Apr 10 '20 at 14:47

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