The floor function (also known as the greatest integer function) of a real number denotes the greatest integer less than or equal to . For example,
In general, is the unique integer satisfying .
Let denote the fractional part of with , for example, Then for any real number . For example, with and .
The key fact that is often enough to solve basic problems involving the floor function.
Find all the values of that satisfy
Let Then This is equivalent to or Since is an integer and is the only integer in that interval, this becomes Any value less than and greater than or equal to will satisfy this equation. Thus, the answer is all the real numbers such that
(1) for any integer
The proofs of these are straightforward. To illustrate, here is a proof of (2). If is an integer, then If is not an integer, then Then and the outsides of the inequality are consecutive integers, so the left side of the inequality must equal by the characterization of the greatest integer function given in the introduction.
Problems involving the floor function of are often simplified by writing , where is an integer and satisfies
If and , what is the value of ?
Write with and as suggested above. Then the first equation becomes Expanding and rearranging the second equation,
so Therefore, and
Definite integrals and sums involving the floor function are quite common in problems and applications. The best strategy is to break up the interval of integration (or summation) into pieces on which the floor function is constant.
Break the integral up into pieces of the form So the integral is the sum of these pieces over all : but by differentiating the geometric series, so the answer is
Let be a prime number, and a positive integer. The largest power of dividing is where
Induct on The base case is clear (both sides are 0), and if it is true for then the largest power of dividing is where where is the largest such that Now it is clear that if divides and otherwise. The number of such that divides is just so which is as desired.
- Omegatron, . Floor function. Retrieved March 30, 2006, from https://commons.wikimedia.org/wiki/File:Floor_function.svg