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floor (Definition)

The floor of a real number is the greatest integer less than or equal to the number. The floor of $ x$ is usually denoted by $ \lfloor x\rfloor$.

Some examples:

  • $ \lfloor 6.2\rfloor=6$,
  • $ \lfloor 0.4\rfloor=0$,
  • $ \lfloor 7\rfloor=7$,
  • $ \lfloor -5.1\rfloor=-6$,
  • $ \lfloor \pi\rfloor=3$,
  • $ \lfloor -4\rfloor=-4$.

Note that this function is not the integer part ($ [x]$), since $ \lfloor -3.5\rfloor = -4$ and $ [ -3.5]=-3$. However, both functions agree for non-negative numbers.

The notation for floor and ceiling was introduced by Iverson in 1962[1]. In some texts however, the bracket notation is used to denote the floor function (although they actually work with integer part) so it is sometimes also called the bracket function.

Bibliography

1
N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.



"floor" is owned by yark. [ full author list (2) | owner history (2) ]
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See Also: Beatty's theorem, ceiling

Other names:  floor function, bracket function, greatest integer function, greatest integer less than or equal to
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Cross-references: ceiling, integer part, function, number, real number
There are 25 references to this entry.

This is version 16 of floor, born on 2001-10-18, modified 2008-03-16.
Object id is 343, canonical name is Floor.
Accessed 7878 times total.

Classification:
AMS MSC26A09 (Real functions :: Functions of one variable :: Elementary functions)
 11-00 (Number theory :: General reference works )

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