# Ceiling Function Calculator

Welcome to Omni's **ceiling function calculator** — the perfect place to fall in love with this popular math operation. In the short article below, we not only give the **formal definition** of the ceiling function, but also

- Explain
**intuitively**what the ceiling function does to a number; - Show how to
**graph the ceiling function**; - Discuss what is the most popular
**symbol**for the ceiling function; and - Go together through some
**examples of ceiling function computation**.

## Ceiling function in math

The ceiling function maps a real number `x`

to the **smallest** integer number that is **greater than or equal to** `x`

:

🙋 In the formula above you can see the most widespread ceiling function symbol. It looks like square brackets `[ ]`

with their bottom part missing (so what remains is the... ceiling! Clever, right?). In programming languages, you most often find this function under the command `ceil(x)`

.

To get a better understanding of the ceiling function definition, let's go through a few examples together.

## Examples of ceiling function computation

#### Example 1

Let's compute the ceiling of $11.2$.

We pose the question dictated by the definition of the ceiling function: *what are the integers that are greater than (or equal to) $11.2$?*

There are lots of such integers: $12, 13, 14, 15, \ldots$. But we need the smallest one. Clearly, it is $12$. So $\lceil 11.2 \rceil = 12$. Don't forget that you can verify this result with the ceiling function calculator!

#### Example 2

Let's compute the ceiling of $-5$.

The integers that are greater than or equal to $-5$ are: $-5, -4, -3, \ldots$. The smallest one is $-5$, so $\lceil -5 \rceil = -5$. Note how crucial it is here to remember the "or equal to" part of the definition!

#### Example 3

The last challenge — the ceiling of a non-integer negative number! Let's compute the ceil of $-2.3$.

What are the integers that are greater than (or equal to) $-2.3$?

If you think for a bit, you can easily see that the desired integers are $-2, -1, 0, 1\ldots$. The smallest one is $-2$, and so $\lceil -2.3 \rceil = -2$.

As you can see from what we calculated above, the **ceiling function** **rounds** **a number up to the nearest integer**. If we're already at an integer, there's no need for rounding, and so the ceil function does not affect integers. Logical, right?

🙋 Not enough examples? To generate more, you can put random numbers into our ceiling function calculator and see what comes out!

## Graph of the ceiling function

Once you're done playing with our ceiling function calculator, it's high time we discussed **how to graph the ceiling function**. Here it is, in all its glory:

Looking at it, you can easily guess why we say that the ceiling function (along with its cousin, the floor function) belongs to the family of so-called **step-functions**.

## FAQ

### What does the ceil function do?

The ceil function transforms a real number into the **smallest integer that is greater than or equal to this number**. It's like rounding a number up to the nearest integer.

### What is the domain of the floor and ceiling function?

The **domain** of the floor and ceiling function is the set of all **real numbers**. The image, in turn, is the set of integers.

### How do I type the ceiling function in LaTeX?

The LaTeX code for `⌈`

is `\lceil`

, and for `⌉`

it's `\rceil`

. Hence, to get `⌈x⌉`

you can type `\lceil x \rceil`

.

### What is the ceiling of pi?

The ceiling of the number pi is 4. This is because pi is approximately equal to `3.14`

, and so the smallest integer that is greater than pi is `4`

.

### How do I calculate the ceiling of a number?

To determine the ceiling of a number:

- If your number is an integer, then the ceiling is equal to this number. You're done!
- Otherwise, write down the integers that are greater than your number.
- Pick the smallest of the integers you've written down.
- That's it! You've found the ceiling of your number.