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floor∗
yark†
2013-03-21 12:29:55
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 bxc.
y
3
Graph y = bxc
2
1
x
−4
−3
−2
−1
1
2
3
4
−1
−2
−3
The real function x 7→ bxc is monotonically nondecreasing and satisfies
x − 1 < bxc 5 x
for all x. The function is continuous everywhere except in the integer points
0, ±1, ±2, . . . where it is only continuous from the right. One has
bbxcc = bxc,
∗ hFloori
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1
i.e. the function is idempotent.
Some examples:
• b6.2c = 6,
• b0.4c = 0,
• b7c = 7,
• b−5.1c = −6,
• bπc = 3,
• b−4c = −4.
Note that this function is not the integer part ([x]), since b−3.5c = −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[?]. 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.
References
[1] N. Higham, Handbook of writing for the mathematical sciences, Society for
Industrial and Applied Mathematics, 1998.
2