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:
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.
There are lots of integers less than 2.31.
So which one do we choose? Choose the largest one, which is 2
The largest 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
As A Graph
The Floor Function is this curious "step" function (like an infinite staircase):
The Floor Function
(a solid dot means "including"
an open dot means "not including")
If it looks confusing, just imagine you are at some x-value (say x=1.5), and see what y-value you get ... does it make sense now?
Example: at x=2 we meet an open dot at y=1 (so it does not include x=2), and a solid dot at y=2 (which does include x=2) so the answer is y=2
And this is the Ceiling Function:
The Ceiling Function
The "Frac" Function
When you use the Floor Function, you "throw away" the fractional part. That part is called the "frac" function:
frac(x) = x - floor(x)
It looks like a sawtooth:
Example: what is frac(3.65)?
Answer: frac(x) = x - floor(x) = 3.65 - 3 = 0.65
Example: what is frac(-3.65)?
Answer: frac(x) = x - floor(x) = (-3.65) - (-4) = -3.65+4 = 0.35
(Note that it isn't 0.65)
|
Home
