# 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**

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" 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)

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 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.