Floor and ceiling functions
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊ x ⌋ or floor(x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈ x ⌉ or ceil(x).

For example, for floor: ⌊2.4⌋ = 2, ⌊ − 2.4⌋ = − 3, and for ceiling: ⌈2.4⌉ = 3, and ⌈ − 2.4⌉ = − 2.

The floor of x is also called the integral part, integer part, greatest integer, or entier of x, and was historically denoted (among other notations). However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers.

For n an integer, ⌊ n ⌋ = ⌈ n ⌉ = n.

Although floor(x+1) and ceil(x) produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, when x =2.0001; ⌊2.0001+1⌋ = ⌈2.0001⌉ = 3. However, if x =2, then ⌊2+1⌋ = 3, while ⌈2⌉ = 2.
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