The floor of *y* is the greatest integer less than or equal to *y* and is denoted ⌊*y*⌋.

Similarly, the ceiling of *y* is the smallest integer greater than or equal to *y* and is denoted ⌈*y*⌉.

Both of these notations were introduced by Kenneth Iverson. Before Iverson’s notation caught on, you might see [*x*] for the floor of *x*, and I don’t know whether there was a notation for ceiling.

There was also a lack of standardization over whether [*x*] meant to round *x* down or round it to the nearest integer. Iverson’s notation caught on because it’s both mnemonic and symmetrical.

Iverson also invented the notation of using a Boolean expression inside square brackets to indicate the function that is 1 when the argument is true and 0 when it is false. I find this notation very convenient. I’ve used it on projects for two different clients recently.

Here’s an equation from Concrete Mathematics using all three Iverson notations discussed here:

⌈*x*⌉ – ⌊*x*⌋ = [*x* is not an integer].

In words, the ceiling of *x* minus the floor of *x* is 1 when *x* is not an integer and 0 when *x* is an integer.

## Related links

- Four handy notations
- Manipulating sums
- Notation as Tool of Thought by Kenneth Iverson

See also Donald Knuth’s article “Two notes on notation” (American Mathematical Monthly, Vol. 99, No. 5 from May 1992, pp. 403-422) which is worth the read.