# Floor and Ceiling Functions

The floor and ceiling functions give us 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 we 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

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" and an open dot means "not including".

### 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)

And this is the Ceiling Function: The Ceiling Function

## The "Int" Function

The "Int" function (short for "integer") is like the "Floor" function, BUT some calculators and computer programs show different results when given negative numbers:

• Some say int(−3.65) = −4 (the same as the Floor function)
• Others say int(−3.65) = −3 (the neighbouring integer closest to zero, or "just throw away the .65")

So be careful with this function!

## The "Frac" Function

With the Floor Function, we "throw away" the fractional part. That part is called the "frac" or "fractional part" function:

frac(x) = x − floor(x)

It looks like a sawtooth: The Frac Function

### Example: what is frac(3.65)?

frac(x) = x − floor(x)

So: frac(3.65) = 3.65 − floor(3.65) = 3.65 − 3 = 0.65

### Example: what is frac(−3.65)?

frac(x) = x − floor(x)

So: frac(−3.65) = (−3.65) − floor(−3.65) = (−3.65) − (−4) = −3.65 + 4 = 0.35

BUT many calculators and computer programs use frac(x) = x − int(x), and so their result depends on how they calculate int(x):

• Some say frac(−3.65) = 0.35 i.e −3.65 − (−4)
• Others say frac(−3.65) = −0.65 i.e. −3.65 − (−3)

So be careful using this function with negative values.