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 17:07, 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. Unknown error_msg)\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