# Round: Rounding of Numbers

 Round R Documentation

## Rounding of Numbers

### Description

`ceiling` takes a single numeric argument `x` and returns a numeric vector containing the smallest integers not less than the corresponding elements of `x`.

`floor` takes a single numeric argument `x` and returns a numeric vector containing the largest integers not greater than the corresponding elements of `x`.

`trunc` takes a single numeric argument `x` and returns a numeric vector containing the integers formed by truncating the values in `x` toward `0`.

`round` rounds the values in its first argument to the specified number of decimal places (default 0). See ‘Details’ about “round to even” when rounding off a 5.

`signif` rounds the values in its first argument to the specified number of significant digits. Hence, for `numeric` `x`, `signif(x, dig)` is the same as `round(x, dig - ceiling(log10(abs(x))))`. For `complex` `x`, this is not the case, see the ‘Details’.

### Usage

```ceiling(x)
floor(x)
trunc(x, ...)

round(x, digits = 0)
signif(x, digits = 6)
```

### Arguments

 `x` a numeric vector. Or, for `round` and `signif`, a complex vector. `digits` integer indicating the number of decimal places (`round`) or significant digits (`signif`) to be used. Negative values are allowed (see ‘Details’). `...` arguments to be passed to methods.

### Details

These are generic functions: methods can be defined for them individually or via the `Math` group generic.

Note that for rounding off a 5, the IEC 60559 standard (see also ‘IEEE 754’) is expected to be used, ‘go to the even digit’. Therefore `round(0.5)` is `0` and `round(-1.5)` is `-2`. However, this is dependent on OS services and on representation error (since e.g. `0.15` is not represented exactly, the rounding rule applies to the represented number and not to the printed number, and so `round(0.15, 1)` could be either `0.1` or `0.2`).

Rounding to a negative number of digits means rounding to a power of ten, so for example `round(x, digits = -2)` rounds to the nearest hundred.

For `signif` the recognized values of `digits` are `1...22`, and non-missing values are rounded to the nearest integer in that range. Complex numbers are rounded to retain the specified number of digits in the larger of the components. Each element of the vector is rounded individually, unlike printing.

These are all primitive functions.

### S4 methods

These are all (internally) S4 generic.

`ceiling`, `floor` and `trunc` are members of the `Math` group generic. As an S4 generic, `trunc` has only one argument.

`round` and `signif` are members of the `Math2` group generic.

### Warning

The realities of computer arithmetic can cause unexpected results, especially with `floor` and `ceiling`. For example, we ‘know’ that `floor(log(x, base = 8))` for `x = 8` is `1`, but `0` has been seen on an R platform. It is normally necessary to use a tolerance.

Rounding to decimal digits in binary arithmetic is non-trivial (when `digits != 0`) and may be surprising. Be aware that most decimal fractions are not exactly representable in binary double precision. In R 4.0.0, the algorithm for `round(x, d)`, for d > 0, has been improved to measure and round “to nearest even”, contrary to earlier versions of R (or also to `sprintf()` or `format()` based rounding).

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

The ISO/IEC/IEEE 60559:2011 standard is available for money from https://www.iso.org.

The IEEE 754:2008 standard is more openly documented, e.g, at https://en.wikipedia.org/wiki/IEEE_754.

`as.integer`. Package round's `roundX()` for several versions or implementations of rounding, including some previous and the current R version (as `version = "3d.C"`).

### Examples

```round(.5 + -2:4) # IEEE / IEC rounding: -2  0  0  2  2  4  4
## (this is *good* behaviour -- do *NOT* report it as bug !)

( x1 <- seq(-2, 4, by = .5) )
round(x1) #-- IEEE / IEC rounding !
x1[trunc(x1) != floor(x1)]
x1[round(x1) != floor(x1 + .5)]
(non.int <- ceiling(x1) != floor(x1))

x2 <- pi * 100^(-1:3)
round(x2, 3)
signif(x2, 3)
```