Floor and Ceiling Functions
Definitions
Let x be a real number. The floor function of x, denoted by ⌊x⌋ or floor (x), is defined to be the greatest integer that is less than or equal to x.
The ceiling function of x, denoted by ⌈x⌉ or ceil (x), is defined to be the least integer that is greater than or equal to x.
For example,
It follows from the definitions that the floor and ceiling functions have type
Graphs of the Floor and Ceiling Functions
The floor and ceiling functions look like a staircase and have a jump discontinuity at each integer point.
Properties of the Floor and Ceiling Functions
There are many interesting and useful properties involving the floor and ceiling functions, some of which are listed below. The number
Fractional Part Function
The fractional part of a number
For example,
The graph of the fractional part function looks like a sawtooth wave, with a period of
The range of fractional part function is the half-open interval
Some other properties of the fractional part are