Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.11

11. What is the value of[x] + [-x] where x is a real number?
Solution:  

We consider the following expression
[x]+[-x]
We consider following two cases

Case 1

If x is an integer, then so is -x

Also,
[x]=x
[-x]=-x
Therefore,
[x]+[-x]=x+(-x)
=x-x
=0

Case 2

If x is not an integer, then so is not -x

Let us take x to be a rational number, that is,
x=1/p
Where p\neq0,-1,1

And is an integer

Therefore,
[x]+[-x]=[1/p]+[-1/p]
=0+(-1)
=-1

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