Floor and Ceiling Functions Usage

Floor and Ceiling Functions

The floor and ceiling functions give you the nearest integer up or down.

Example: What is the floor and ceiling of 2.31?

The Floor of 2.31 is 2
The Ceiling of 2.31 is 3

Floor and Ceiling of Integers

What if you want the floor or ceiling of a number that is already an integer?

That's easy: no change!

Example: What is the floor and ceiling of 5?

The Floor of 5 is 5
The Ceiling of 5 is 5

Here are some example values for you:

x	Floor	Ceiling
-1.1	-2	-1
0	0	0
1.01	1	2
2.9	2	3
3	3	3

Symbols

The symbols for floor and ceiling are like the square brackets [ ] with the top or bottom part missing:

But I prefer to use the word form: floor(x) and ceil(x)

Definitions

How do we give this a formal definition?

Example: How do we define the floor of 2.31?

Well, it has to be an integer ...

... and it has to be less than (or maybe equal to) 2.31, right?

• 2 is less than 2.31 ...
• but 1 is also less than 2.31,
• and so is 0, and -1, -2, -3, etc.

Oh no! There are lots of integers less than 2.31.

So which one do we choose?

Choose the greatest one (which is 2 in this case)

So we get:

The greatest integer that is less than (or equal to) 2.31 is 2

Which leads to our definition:

Floor Function: the greatest integer that is less than or equal to x

Likewise for Ceiling:

Ceiling Function: the least integer that is greater than or equal to x