Editing Floor and ceiling functions (section)

===Rounding===
For an arbitrary real number <math>x</math>, [[rounding]] <math>x</math> to the nearest integer with [[Rounding#Tie-breaking|tie breaking]] towards positive infinity is given by
:<math>\text{rpi}(x)=\left\lfloor x+\tfrac{1}{2}\right\rfloor = \left\lceil \tfrac12\lfloor 2x \rfloor \right\rceil;</math>
rounding towards negative infinity is given as
:<math>\text{rni}(x)=\left\lceil x-\tfrac{1}{2}\right\rceil = \left\lfloor \tfrac12 \lceil 2x \rceil \right\rfloor.</math>

If tie-breaking is away from 0, then the rounding function is
:<math>\text{ri}(x) = \sgn(x)\left\lfloor|x|+\tfrac{1}{2}\right\rfloor</math>
(where <math>\sgn</math> is the [[sign function]]), and [[Rounding#Rounding half to even|rounding towards even]] can be expressed with the more cumbersome
:<math>\lfloor x\rceil=\left\lfloor x+\tfrac{1}{2}\right\rfloor+\left\lceil\tfrac14(2x-1)\right\rceil-\left\lfloor\tfrac14(2x-1)\right\rfloor-1,</math>
which is the above expression for rounding towards positive infinity <math>\text{rpi}(x)</math> minus an [[integer|integrality]] [[indicator function|indicator]] for <math>\tfrac14(2x-1)</math>.

Rounding a [[real number]] <math>x</math> to the nearest integer value forms a very basic type of [[Quantization (signal processing)|quantizer]] – a ''uniform'' one.  A typical (''mid-tread'') uniform quantizer with a quantization ''step size'' equal to some value <math>\Delta</math> can be expressed as

:<math>Q(x) = \Delta \cdot  \left\lfloor \frac{x}{\Delta} + \frac{1}{2} \right\rfloor</math>,