The greatest integer function, also known as the floor function, gives the greatest integer less than or equal to its argument. The floor of is usually denoted by or . The action of this function is the same as "rounding down." On a positive argument, this function is the same as "dropping everything after the decimal point," but this is not true for negative values.
- for all real .
- Hermite's Identity:
A useful way to use the floor function is to write , where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems.
Another common definition of the floor function is
where is the fractional part of .
- Let denote the largest integer not exceeding . For example, , and . How many positive integers satisfy the equation .
- Find the integer satisfying . Here denotes the greatest integer less than or equal to .
(1999-2000 Hong Kong IMO Prelim)
- What is the units (i.e., rightmost) digit of
(1986 Putnam Exam, A-2)
- If is a positive real number, and is a positive integer, prove that
where denotes the greatest integer less than or equal to .
(1981 USAMO, #5) (Discussion 1) (Discussion 2)
- Let denote the integer part of , i.e., the greatest integer not exceeding . If is a positive integer, express as a simple function of the sum
(1986 IMO, #6)