Question

Show that floor(x + 1/2) is the closest integer to the number x, except when x is midway between two integers, when it is the larger of these two integers.

Answer 1

floor(x + 1/2) is the closest integer to the number x, except when x is midway between two integers, when it is the larger of these two integers.

Step-by-step explanation:

The expression floor(x + 1/2) represents the largest integer that is less than or equal to the value of x + 1/2. To show that floor(x + 1/2) is the closest integer to x, we can consider two cases:

1. If x falls between two integers, say n and n+1, then x + 1/2 falls exactly midway between the two integers. In this case, the value of floor(x + 1/2) is equal to n+1, which is the larger of the two integers. Therefore, floor(x + 1/2) is always the larger of the two integers when x is midway between two integers.
2. If x does not fall between two integers, then x + 1/2 falls closer to either n or n+1. In this case, the value of floor(x + 1/2) will be the integer that is closest to x. For example, if x = 2.7, then x + 1/2 = 3.2 which is closer to 3 than to 4. Thus, floor(x + 1/2) is the closest integer to x when x is not midway between two integers.