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 $\lfloor x\rfloor$ .

The real function $x \mapsto \lfloor{x}\rfloor$ is monotonically nondecreasing and satisfies $$x-1 < \lfloor{x}\rfloor \leqq x$$ for all $x$ . The function is continuous everywhere except in the integer points $0,\,\pm1,\,\pm2,\,\ldots$ where it is only continuous from the right. One has $$\lfloor\lfloor{x}\rfloor\rfloor \;=\; \lfloor{x}\rfloor,$$ i.e. the function is idempotent.

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.