# Arithmetic

##### Arithmetic Operators

These unary and binary operators perform arithmetic on numeric or complex vectors (or objects which can be coerced to them).

- Keywords
- arith

##### Usage

```
+ x
- x
x + y
x - y
x * y
x / y
x ^ y
x %% y
x %/% y
```

##### Arguments

- x, y
numeric or complex vectors or objects which can be coerced to such, or other objects for which methods have been written.

##### Details

The unary and binary arithmetic operators are generic functions:
methods can be written for them individually or via the
`Ops`

group generic function. (See
`Ops`

for how dispatch is computed.)

If applied to arrays the result will be an array if this is sensible (for example it will not if the recycling rule has been invoked).

Logical vectors will be coerced to integer or numeric vectors,
`FALSE`

having value zero and `TRUE`

having value one.

`1 ^ y`

and `y ^ 0`

are `1`

, *always*.
`x ^ y`

should also give the proper limit result when
either (numeric) argument is infinite (one of `Inf`

or
`-Inf`

).

Objects such as arrays or time-series can be operated on this way provided they are conformable.

For double arguments, `%%`

can be subject to catastrophic loss of
accuracy if `x`

is much larger than `y`

, and a warning is
given if this is detected.

`%%`

and `x %/% y`

can be used for non-integer `y`

,
e.g.`1 %/% 0.2`

, but the results are subject to representation
error and so may be platform-dependent. Because the IEC 60559
representation of `0.2`

is a binary fraction slightly larger than
`0.2`

, the answer to `1 %/% 0.2`

should be `4`

but
most platforms give `5`

.

Users are sometimes surprised by the value returned, for example why
`(-8)^(1/3)`

is `NaN`

. For double inputs, R makes
use of IEC 60559 arithmetic on all platforms, together with the C
system function `pow` for the `^`

operator. The relevant
standards define the result in many corner cases. In particular, the
result in the example above is mandated by the C99 standard. On many
Unix-alike systems the command `man pow`

gives details of the
values in a large number of corner cases.

Arithmetic on type double in R is supposed to be done in ‘round to nearest, ties to even’ mode, but this does depend on the compiler and FPU being set up correctly.

##### Value

Unary `+`

and unary `-`

return a numeric or complex vector.
All attributes (including class) are preserved if there is no
coercion: logical `x`

is coerced to integer and names, dims and
dimnames are preserved.

The binary operators return vectors containing the result of the element
by element operations. If involving a zero-length vector the result
has length zero. Otherwise, the elements of shorter vectors are recycled
as necessary (with a `warning`

when they are recycled only
*fractionally*). The operators are `+`

for addition,
`-`

for subtraction, `*`

for multiplication, `/`

for
division and `^`

for exponentiation.

`%%`

indicates `x mod y`

(“x modulo y”) and
`%/%`

indicates integer division. It is guaranteed that

`x == (x %% y) + y * (x %/% y)`

(up to rounding error)

unless `y == 0`

where the result of `%%`

is
`NA_integer_`

or `NaN`

(depending on the
`typeof`

of the arguments) or for some non-finite
arguments, e.g., when the RHS of the identity above
amounts to `Inf - Inf`

.

If either argument is complex the result will be complex, otherwise if
one or both arguments are numeric, the result will be numeric. If
both arguments are of type integer, the type of the result of
`/`

and `^`

is numeric and for the other operators it
is integer (with overflow, which occurs at
\(\pm(2^{31} - 1)\),
returned as `NA_integer_`

with a warning).

The rules for determining the attributes of the result are rather
complicated. Most attributes are taken from the longer argument.
Names will be copied from the first if it is the same length as the
answer, otherwise from the second if that is. If the arguments are
the same length, attributes will be copied from both, with those of
the first argument taking precedence when the same attribute is
present in both arguments. For time series, these operations are
allowed only if the series are compatible, when the class and
`tsp`

attribute of whichever is a time series (the same,
if both are) are used. For arrays (and an array result) the
dimensions and dimnames are taken from first argument if it is an
array, otherwise the second.

##### Note

All logical operations involving a zero-length vector have a zero-length result.

The binary operators are sometimes called as functions as
e.g.``&`(x, y)`

: see the description of how
argument-matching is done in `Ops`

.

`**`

is translated in the parser to `^`

, but this was
undocumented for many years. It appears as an index entry in Becker
*et al* (1988), pointing to the help for `Deprecated`

but
is not actually mentioned on that page. Even though it had been
deprecated in S for 20 years, it was still accepted in R in 2008.

##### S4 methods

These operators are members of the S4 `Arith`

group generic,
and so methods can be written for them individually as well as for the
group generic (or the `Ops`

group generic), with arguments
`c(e1, e2)`

(with `e2`

missing for a unary operator).

##### Implementation limits

R is dependent on OS services (and they on FPUs) for floating-point
arithmetic. On all current R platforms IEC 60559 (also known as IEEE
754) arithmetic is used, but some things in those standards are
optional. In particular, the support for *denormal* aka
*subnormal* numbers
(those outside the range given by `.Machine`

) may differ
between platforms and even between calculations on a single platform.

Another potential issue is signed zeroes: on IEC 60559 platforms there
are two zeroes with internal representations differing by sign. Where
possible R treats them as the same, but for example direct output
from C code often does not do so and may output `-0.0` (and on
Windows whether it does so or not depends on the version of Windows).
One place in R where the difference might be seen is in division by
zero: `1/x`

is `Inf`

or `-Inf`

depending on the sign of
zero `x`

. Another place is
`identical(0, -0, num.eq = FALSE)`

.

##### References

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

D. Goldberg (1991).
What Every Computer Scientist Should Know about Floating-Point
Arithmetic.
*ACM Computing Surveys*, **23**(1), 5--48.
10.1145/103162.103163.

Postscript version available at http://www.validlab.com/goldberg/paper.ps. Extended PDF version at http://www.validlab.com/goldberg/paper.pdf.

For the IEC 60559 (aka IEEE 754) standard: https://www.iso.org/standard/57469.html and https://en.wikipedia.org/wiki/IEEE_754.

##### See Also

`sqrt`

for miscellaneous and `Special`

for special
mathematical functions.

`Syntax`

for operator precedence.

`%*%`

for matrix multiplication.

##### Examples

`library(base)`

```
# NOT RUN {
x <- -1:12
x + 1
2 * x + 3
x %% 2 #-- is periodic
x %/% 5
x %% Inf # now is defined by limit (gave NaN in earlier versions of R)
# }
```

*Documentation reproduced from package base, version 3.6.2, License: Part of R 3.6.2*