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.

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One thought on “Three notations by Iverson

  1. 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.

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