# Difference between revisions of "Floor function"

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*<math>\lfloor 3.14 \rfloor = 3</math> | *<math>\lfloor 3.14 \rfloor = 3</math> | ||

− | *<math>\lfloor 5 \rfloor = | + | *<math>\lfloor 5 \rfloor = ? |

− | *<math>\lfloor -3.2 \rfloor = -4< | + | *</math>\lfloor -3.2 \rfloor = -4<math> |

− | A useful way to use the floor function is to write <math>\lfloor x \rfloor=\lfloor y+k \rfloor | + | A useful way to use the floor function is to write </math>\lfloor x \rfloor=\lfloor y+k \rfloor$, where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems. |

==Alternate Definition== | ==Alternate Definition== |

## Revision as of 18:07, 19 September 2020

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.

## Properties

- for all real .
- Hermite's Identity:

## Examples

- $\lfloor 5 \rfloor = ?

- $ (Error compiling LaTeX. ! Missing $ inserted.)\lfloor -3.2 \rfloor = -4\lfloor x \rfloor=\lfloor y+k \rfloor$, where y is an integer and k is the leftover stuff after the decimal point. This can greatly simplify many problems.

## Alternate Definition

Another common definition of the floor function is

where is the fractional part of .

## Olympiad Problems

- [1981 USAMO #5] If is a positive real number, and is a positive integer, prove that

where denotes the greatest integer less than or equal to .

- [1968 IMO #6] 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