Final answer:
To show the relationship between the ceiling and floor functions for a real number, one needs to understand that for an integer x, both ceil(x) and floor(x) equal x, leading to a difference of 0. For a non-integer x, ceil(x) is the next integer greater, and floor(x) is the highest integer less than x, resulting in a difference of 1.
Step-by-step explanation:
To show that if x is a real number, then ceil(x) − floor(x) equals 1 if x is not an integer and equals 0 if x is an integer, we need to understand the functions ceil (ceiling) and floor.
The ceil or ceiling of a number is the smallest integer that is greater than or equal to x. The floor of a number is the largest integer that is less than or equal to x. When x is an integer, ceil(x) and floor(x) are both equal to x itself, so their difference is 0.
For a non-integer real number x, the ceiling of x will be the next integer greater than x, and the floor of x will be the integer just less than x. As a result, the difference between them will always be:
ceil(x) − floor(x) = (floor(x) + 1) − floor(x)
ceil(x) − floor(x) = floor(x) + 1 − floor(x)
ceil(x) − floor(x) = 1
Thus, when x is not an integer real number, ceil(x) - floor(x) = 1; and when x is an integer, ceil(x) - floor(x) = 0.