Question

Show that if x is a real number and m is an integer, then ceil(x +m)=ceil(x) + m.

Answer 1

To prove that ceil(x + m) = ceil(x) + m, we can consider the definition of the ceiling function and the properties of addition.

Step-by-step explanation:

To show that ceil(x + m) = ceil(x) + m, we need to consider the definition of the ceiling function and the properties of addition.

1. The ceiling function, denoted as ceil(x), rounds a number up to the nearest integer. For example, ceil(3.2) = 4 and ceil(4) = 4.
2. Let's consider the expression ceil(x + m). When we add an integer m to x, the sum will result in a decimal or an integer. ceil(x + m) will round this sum up to the nearest integer.
3. On the other hand, if we evaluate ceil(x) + m, we first calculate ceil(x), which gives us an integer. Then we add m to that integer. We can see that adding m directly to the integer value of ceil(x) will give us the same integer result as ceil(x + m).

Therefore, we have shown that ceil(x + m) = ceil(x) + m for any real number x and an integer m.