# Floor

 `⌊`

Floor (`⌊`) is a monadic scalar function that gives the floor of a real number, that is, the greatest integer tolerantly less than or equal to the given value. This operation is also known as integral part, entier, and round down. Floor shares the glyph `⌊` with the dyadic arithmetic function Minimum. Traditional mathematics derives its notation and name for floor from APL.

## Examples

Floor rounds down the given numbers to the nearest integers.

```      ⌊2 2.8 ¯2 ¯2.8
2 2 ¯2 ¯3
```

Rounding to the nearest integer (rounding up on half) can be achieved by adding 0.5 before applying Floor.

```      ⌊0.5+2 2.3 2.5 2.8
2 2 3 3
```

Integral quotient of division can be found with division followed by Floor.

```      ⌊10 20 30÷3
3 6 10
```

## Properties

The floor of any real number is an integer.

Floor is affected by comparison tolerance. If the given number is tolerantly equal to its ceiling, it is rounded to that number instead.

```      ⎕PP←16
⎕←v←1+0.6×⎕CT×¯2 ¯1 0
0.999999999999988 0.999999999999994 1
⌊v
0 1 1
```

### Complex floor

Main article: Complex Floor

Eugene McDonnell designed the domain extension of Floor to complex numbers. Complex floor maps every complex number to a Gaussian integer, a complex number whose real and imaginary parts are integers. It has an important property that the magnitude of difference between any complex number Z and its floor is less than 1. This extension is currently implemented in Dyalog APL, J, and NARS2000, and is internally used to implement complex ceiling, residue, and GCD.

```      v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8
⌊v
2J2 2J2 2J2 2J2
1>|v-⌊v
1 1 1 1
```
Works in: Dyalog APL