Definition and properties

In the afterward formulas, x and y are absolute numbers, k, m, and n are integers, and is the set of integers (positive, negative, and zero).

Floor and beam may be authentic by the set equations

Since there is absolutely one accumulation in a half-open breach of breadth one, for any absolute x there are different integers m and n satisfying

Then and may aswell be taken as the analogue of attic and ceiling.

editEquivalences

These formulas can be acclimated to abridge expressions involving floors and ceilings.12

In the accent of adjustment theory, the attic action is a residuated mapping, that is, allotment of a Galois connection: it is the high adjoint of the action that embeds the integers into the reals.

These formulas appearance how abacus integers to the arguments affect the functions:

The aloft are not necessarily accurate if n is not an integer; however:

editRelations a allotment of the functions

It is bright from the definitions that

with adequation if and alone if x is an integer, i.e.

In fact, back for integers n:

Negating the altercation switches attic and beam and changes the sign:

i.e.

Negating the altercation complements the apportioned part:

The floor, ceiling, and apportioned allotment functions are idempotent:

The aftereffect of nested attic or beam functions is the centermost function:

For anchored y, x mod y is idempotent:

Also, from the definitions,

editQuotients

If m and n are integers and n ≠ 0,

If n is positive13

If m is positive14

For m = 2 these imply

More generally,15 for absolute m (See Hermite's identity)

The afterward can be acclimated to catechumen floors to ceilings and vice-versa (m positive)16

If m and n are absolute and coprime, then

Since the right-hand ancillary is balanced in m and n, this implies that

More generally, if m and n are positive,

This is sometimes alleged a advantage law.17

editNested divisions

For absolute integers m,n, and approximate absolute amount x:

editContinuity

None of the functions discussed in this commodity are continuous, but all are piecewise linear. and are piecewise connected functions, with discontinuites at the integers. aswell has discontinuites at the integers, and as a action of x for anchored y is alternate at multiples of y.

is high semi-continuous and and are lower semi-continuous. x mod y is lower semicontinuous for absolute y and high semi-continuous for abrogating y.

editSeries expansions

Since none of the functions discussed in this commodity are continuous, none of them accept a ability alternation expansion. Back attic and beam are not periodic, they do not accept Fourier alternation expansions.

x mod y for anchored y has the Fourier alternation expansion18

in accurate {x} = x mod 1 is accustomed by

At credibility of discontinuity, a Fourier alternation converges to a amount that is the boilerplate of its banned on the larboard and the right, clashing the floor, beam and apportioned allotment functions: for y anchored and x a assorted of y the Fourier alternation accustomed converges to y/2, rather than to x mod y = 0. At credibility of chain the alternation converges to the accurate value.

Using the blueprint {x} = x − floor(x), floor(x) = x − {x} gives