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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]
However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the function rounds towards negative infinity. For a given number x ∈ R − {\displaystyle x\in \mathbb {R} _{-}} , the function ceil {\displaystyle \operatorname {ceil} } is used instead
Rectangular function; Floor function: Largest integer less than or equal to a given number. Ceiling function: Smallest integer larger than or equal to a given number. Sign function: Returns only the sign of a number, as +1, −1 or 0. Absolute value: distance to the origin (zero point)
(The notation ⌈ ⌉ denotes the ceiling function). The essential property of a quantizer is having a countable set of possible output values smaller than the set of possible input values. The members of the set of output values may have integer, rational, or real values.
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.
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.
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Floor and ceiling functions; Formula; Formula calculator; Free variables and bound variables; G. Galactic algorithm; ... Software calculator; Steinhaus–Moser notation;