# 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**

## As A Graph

The Floor Function is this curious "step" function (like an infinite staircase):

The Floor Function

(Note: 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 "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 rule is: neighbouring integer**closest to zero,**or "just throw away the .65")

So be careful with this function!

## The "Frac" Function

When you use the Floor Function, you "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**

HOWEVER: 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)

Just warning you of the difference.

(Note: some software defines the "frac" function as **x - int(x)**, which leads to different results for negative numbers, so be careful.)