---
## Definition
> [!tldr] Definition
> 1. The **floor function**, denoted $f(x) = \lfloor x \rfloor$, is a [[Function|function]] from the set of real numbers to the set of [[Integers|integers]] defined by taking the input and rounding down to the next lower integer.
> 2. The **ceiling function**, denoted $g(x) = \lceil x \rceil$, is a [[Function|function]] from the set of real numbers to the set of [[Integers|integers]] defined by taking the input and rounding up to the next higher integer.
Notes:
- In Python, the floor and ceiling functions are accessed by first loading the `math` library and then accessing `floor` or `ceil` as methods:
```python
import math
math.floor(3.4)
# Result: 3
import math # Not necessary to load this a second time
math.ceil(3.4)
# Result: 4
```
* In *Mathematica*, the floor and ceiling functions are `Floor` and `Ceiling`. ([Floor documentation](https://reference.wolfram.com/language/ref/Floor.html); [Ceiling documentation](https://reference.wolfram.com/language/ref/Ceiling.html))
```mathematica
Floor[-4.5]
> -5
Ceiling[-4.5]
> -4
```
## Examples
| Expression to evaluate | Result |
| :-----------------------: | ---------------------------------------- |
| $\lfloor 3.4 \rfloor$ | $3$ |
| $\lceil 3.4 \rceil$ | $4$ |
| $\lfloor 3.99999\rfloor$ | $3$ |
| $\lceil 3.0000001 \rceil$ | $4$ |
| $\lfloor 5 \rfloor$ | $5$ |
| $\lfloor -4.3 \rfloor$ | $-5$ (go down to the next lower integer) |
| $\lceil -5.6 \rceil$ | $-5$ (go up to the next higher integer) |
## Resources
