Three notations by Iverson

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