Kenneth Rosen Edition 7 Exercise 2.3 Question 50 (Page No. 154)

Question

Show that if $x$ is a real number,  and $m$  is an integer, then $\left \lceil x+m \right \rceil = \left \lceil x \right \rceil +m.$

Answer 1

It has been given that $x$ is a real number and $m$ is an integer.

if $x$ is a real number then the number can be represented as $y.z$ where $y$ is the integral part and $z$ is the fractional part.

$\lceil \ x+m \ \rceil = \lceil \ (y+m).z \ \rceil = y+m+1$

and , $\lceil \ x \ \rceil +m  = \lceil \ y.z \ \rceil+m = y+1+m$.


Thus both give the same result.