Definition and Properties
In the following formulas, x and y are real numbers, k, m, and n are integers, and is the set of integers (positive, negative, and zero).
Floor and ceiling may be defined by the set equations
Since there is exactly one integer in a half-open interval of length one, for any real x there are unique integers m and n satisfying
Then and may also be taken as the definition of floor and ceiling.
Read more about this topic: Floor And Ceiling Functions
Famous quotes containing the words definition and/or properties:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)