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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 least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
In mathematics, an integer-valued function is a function whose values are integers.In other words, it is a function that assigns an integer to each member of its domain.. The floor and ceiling functions are examples of integer-valued functions of a real variable, but on real numbers and, generally, on (non-disconnected) topological spaces integer-valued functions are not especially useful.
The ceiling function ⌈ ⌉ from R to Z (with the usual order in each case) is residuated, with residual mapping the natural embedding of Z into R. The embedding of Z into R is also residuated. Its residual is the floor function x ↦ ⌊ x ⌋ {\displaystyle x\mapsto \lfloor x\rfloor } .
For arbitrary n and m, this generalizes to + = ⌊ / ⌋ + = ⌈ / ⌉, where ⌊ ⌋ and ⌈ ⌉ denote the floor and ceiling functions, respectively. Though the principle's most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence .
the floor, ceiling and fractional part functions are idempotent; the real part function () of a complex number, is idempotent. the subgroup generated function from the power set of a group to itself is idempotent; the convex hull function from the power set of an affine space over the reals to itself is idempotent;
Here is an example on tips given in a restaurant. Tips using a $1 bin width, skewed right, unimodal. ... The braces indicate the ceiling function. Square-root choice
However, Square brackets, as in = 3, are sometimes used to denote the floor function, which rounds a real number down to the next integer. Conversely, some authors use outwards pointing square brackets to denote the ceiling function, as in ]π[ = 4. Braces, as in {π} < 1 / 7, may denote the fractional part of a real number.
For example, 1.4 rounded is 1, the floor of 1.4 is 1, the ceiling of 1.4 is 2. 1.6 rounded is 2, the floor of 1.6 is 1, the ceiling of 1.6 is 2. So the floor of a fraction is always down; the ceiling of a fraction is always up; rounding can be up or down depending upon