base-4.17.0.0: Basic libraries
Copyright(c) The University of Glasgow 1994-2002
Licensesee libraries/base/LICENSE
Maintainercvs-ghc@haskell.org
Stabilityinternal
Portabilitynon-portable (GHC Extensions)
Safe HaskellTrustworthy
LanguageHaskell2010

GHC.Num

Description

The Num class and the Integer type.

Synopsis

Documentation

class Num a where Source #

Basic numeric class.

The Haskell Report defines no laws for Num. However, (+) and (*) are customarily expected to define a ring and have the following properties:

Associativity of (+)
(x + y) + z = x + (y + z)
Commutativity of (+)
x + y = y + x
fromInteger 0 is the additive identity
x + fromInteger 0 = x
negate gives the additive inverse
x + negate x = fromInteger 0
Associativity of (*)
(x * y) * z = x * (y * z)
fromInteger 1 is the multiplicative identity
x * fromInteger 1 = x and fromInteger 1 * x = x
Distributivity of (*) with respect to (+)
a * (b + c) = (a * b) + (a * c) and (b + c) * a = (b * a) + (c * a)
Coherence with toInteger
if the type also implements Integral, then fromInteger is a left inverse for toInteger, i.e. fromInteger (toInteger i) == i

Note that it isn't customarily expected that a type instance of both Num and Ord implement an ordered ring. Indeed, in base only Integer and Rational do.

Minimal complete definition

(+), (*), abs, signum, fromInteger, (negate | (-))

Methods

(+) :: a -> a -> a infixl 6 Source #

(-) :: a -> a -> a infixl 6 Source #

(*) :: a -> a -> a infixl 7 Source #

negate :: a -> a Source #

Unary negation.

abs :: a -> a Source #

Absolute value.

signum :: a -> a Source #

Sign of a number. The functions abs and signum should satisfy the law:

abs x * signum x == x

For real numbers, the signum is either -1 (negative), 0 (zero) or 1 (positive).

fromInteger :: Integer -> a Source #

Conversion from an Integer. An integer literal represents the application of the function fromInteger to the appropriate value of type Integer, so such literals have type (Num a) => a.

Instances

Instances details
Num CBool Source # 
Instance details

Defined in Foreign.C.Types

Num CChar Source # 
Instance details

Defined in Foreign.C.Types

Num CClock Source # 
Instance details

Defined in Foreign.C.Types

Num CDouble Source # 
Instance details

Defined in Foreign.C.Types

Num CFloat Source # 
Instance details

Defined in Foreign.C.Types

Num CInt Source # 
Instance details

Defined in Foreign.C.Types

Num CIntMax Source # 
Instance details

Defined in Foreign.C.Types

Num CIntPtr Source # 
Instance details

Defined in Foreign.C.Types

Num CLLong Source # 
Instance details

Defined in Foreign.C.Types

Num CLong Source # 
Instance details

Defined in Foreign.C.Types

Num CPtrdiff Source # 
Instance details

Defined in Foreign.C.Types

Num CSChar Source # 
Instance details

Defined in Foreign.C.Types

Num CSUSeconds Source # 
Instance details

Defined in Foreign.C.Types

Num CShort Source # 
Instance details

Defined in Foreign.C.Types

Num CSigAtomic Source # 
Instance details

Defined in Foreign.C.Types

Num CSize Source # 
Instance details

Defined in Foreign.C.Types

Num CTime Source # 
Instance details

Defined in Foreign.C.Types

Num CUChar Source # 
Instance details

Defined in Foreign.C.Types

Num CUInt Source # 
Instance details

Defined in Foreign.C.Types

Num CUIntMax Source # 
Instance details

Defined in Foreign.C.Types

Num CUIntPtr Source # 
Instance details

Defined in Foreign.C.Types

Num CULLong Source # 
Instance details

Defined in Foreign.C.Types

Num CULong Source # 
Instance details

Defined in Foreign.C.Types

Num CUSeconds Source # 
Instance details

Defined in Foreign.C.Types

Num CUShort Source # 
Instance details

Defined in Foreign.C.Types

Num CWchar Source # 
Instance details

Defined in Foreign.C.Types

Num IntPtr Source # 
Instance details

Defined in Foreign.Ptr

Num WordPtr Source # 
Instance details

Defined in Foreign.Ptr

Num Int16 Source #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int32 Source #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int64 Source #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int8 Source #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Word16 Source #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word32 Source #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word64 Source #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word8 Source #

Since: base-2.1

Instance details

Defined in GHC.Word

Num CBlkCnt Source # 
Instance details

Defined in System.Posix.Types

Num CBlkSize Source # 
Instance details

Defined in System.Posix.Types

Num CCc Source # 
Instance details

Defined in System.Posix.Types

Num CClockId Source # 
Instance details

Defined in System.Posix.Types

Num CDev Source # 
Instance details

Defined in System.Posix.Types

Num CFsBlkCnt Source # 
Instance details

Defined in System.Posix.Types

Num CFsFilCnt Source # 
Instance details

Defined in System.Posix.Types

Num CGid Source # 
Instance details

Defined in System.Posix.Types

Num CId Source # 
Instance details

Defined in System.Posix.Types

Num CIno Source # 
Instance details

Defined in System.Posix.Types

Num CKey Source # 
Instance details

Defined in System.Posix.Types

Num CMode Source # 
Instance details

Defined in System.Posix.Types

Num CNfds Source # 
Instance details

Defined in System.Posix.Types

Num CNlink Source # 
Instance details

Defined in System.Posix.Types

Num COff Source # 
Instance details

Defined in System.Posix.Types

Num CPid Source # 
Instance details

Defined in System.Posix.Types

Num CRLim Source # 
Instance details

Defined in System.Posix.Types

Num CSocklen Source # 
Instance details

Defined in System.Posix.Types

Num CSpeed Source # 
Instance details

Defined in System.Posix.Types

Num CSsize Source # 
Instance details

Defined in System.Posix.Types

Num CTcflag Source # 
Instance details

Defined in System.Posix.Types

Num CUid Source # 
Instance details

Defined in System.Posix.Types

Num Fd Source # 
Instance details

Defined in System.Posix.Types

Methods

(+) :: Fd -> Fd -> Fd Source #

(-) :: Fd -> Fd -> Fd Source #

(*) :: Fd -> Fd -> Fd Source #

negate :: Fd -> Fd Source #

abs :: Fd -> Fd Source #

signum :: Fd -> Fd Source #

fromInteger :: Integer -> Fd Source #

Num Integer Source #

Since: base-2.1

Instance details

Defined in GHC.Num

Num Natural Source #

Note that Natural's Num instance isn't a ring: no element but 0 has an additive inverse. It is a semiring though.

Since: base-4.8.0.0

Instance details

Defined in GHC.Num

Num Double Source #

Note that due to the presence of NaN, not all elements of Double have an additive inverse.

>>> 0/0 + (negate 0/0 :: Double)
NaN

Also note that due to the presence of -0, Double's Num instance doesn't have an additive identity

>>> 0 + (-0 :: Double)
0.0

Since: base-2.1

Instance details

Defined in GHC.Float

Num Float Source #

Note that due to the presence of NaN, not all elements of Float have an additive inverse.

>>> 0/0 + (negate 0/0 :: Float)
NaN

Also note that due to the presence of -0, Float's Num instance doesn't have an additive identity

>>> 0 + (-0 :: Float)
0.0

Since: base-2.1

Instance details

Defined in GHC.Float

Num Int Source #

Since: base-2.1

Instance details

Defined in GHC.Num

Num Word Source #

Since: base-2.1

Instance details

Defined in GHC.Num

RealFloat a => Num (Complex a) Source #

Since: base-2.1

Instance details

Defined in Data.Complex

Num a => Num (Identity a) Source #

Since: base-4.9.0.0

Instance details

Defined in Data.Functor.Identity

Num a => Num (Down a) Source #

Since: base-4.11.0.0

Instance details

Defined in Data.Ord

Methods

(+) :: Down a -> Down a -> Down a Source #

(-) :: Down a -> Down a -> Down a Source #

(*) :: Down a -> Down a -> Down a Source #

negate :: Down a -> Down a Source #

abs :: Down a -> Down a Source #

signum :: Down a -> Down a Source #

fromInteger :: Integer -> Down a Source #

Num a => Num (Max a) Source #

Since: base-4.9.0.0

Instance details

Defined in Data.Semigroup

Methods

(+) :: Max a -> Max a -> Max a Source #

(-) :: Max a -> Max a -> Max a Source #

(*) :: Max a -> Max a -> Max a Source #

negate :: Max a -> Max a Source #

abs :: Max a -> Max a Source #

signum :: Max a -> Max a Source #

fromInteger :: Integer -> Max a Source #

Num a => Num (Min a) Source #

Since: base-4.9.0.0

Instance details

Defined in Data.Semigroup

Methods

(+) :: Min a -> Min a -> Min a Source #

(-) :: Min a -> Min a -> Min a Source #

(*) :: Min a -> Min a -> Min a Source #

negate :: Min a -> Min a Source #

abs :: Min a -> Min a Source #

signum :: Min a -> Min a Source #

fromInteger :: Integer -> Min a Source #

Num a => Num (Product a) Source #

Since: base-4.7.0.0

Instance details

Defined in Data.Semigroup.Internal

Num a => Num (Sum a) Source #

Since: base-4.7.0.0

Instance details

Defined in Data.Semigroup.Internal

Methods

(+) :: Sum a -> Sum a -> Sum a Source #

(-) :: Sum a -> Sum a -> Sum a Source #

(*) :: Sum a -> Sum a -> Sum a Source #

negate :: Sum a -> Sum a Source #

abs :: Sum a -> Sum a Source #

signum :: Sum a -> Sum a Source #

fromInteger :: Integer -> Sum a Source #

Integral a => Num (Ratio a) Source #

Since: base-2.0.1

Instance details

Defined in GHC.Real

Methods

(+) :: Ratio a -> Ratio a -> Ratio a Source #

(-) :: Ratio a -> Ratio a -> Ratio a Source #

(*) :: Ratio a -> Ratio a -> Ratio a Source #

negate :: Ratio a -> Ratio a Source #

abs :: Ratio a -> Ratio a Source #

signum :: Ratio a -> Ratio a Source #

fromInteger :: Integer -> Ratio a Source #

HasResolution a => Num (Fixed a) Source #

Since: base-2.1

Instance details

Defined in Data.Fixed

Methods

(+) :: Fixed a -> Fixed a -> Fixed a Source #

(-) :: Fixed a -> Fixed a -> Fixed a Source #

(*) :: Fixed a -> Fixed a -> Fixed a Source #

negate :: Fixed a -> Fixed a Source #

abs :: Fixed a -> Fixed a Source #

signum :: Fixed a -> Fixed a Source #

fromInteger :: Integer -> Fixed a Source #

Num a => Num (Op a b) Source # 
Instance details

Defined in Data.Functor.Contravariant

Methods

(+) :: Op a b -> Op a b -> Op a b Source #

(-) :: Op a b -> Op a b -> Op a b Source #

(*) :: Op a b -> Op a b -> Op a b Source #

negate :: Op a b -> Op a b Source #

abs :: Op a b -> Op a b Source #

signum :: Op a b -> Op a b Source #

fromInteger :: Integer -> Op a b Source #

Num a => Num (Const a b) Source #

Since: base-4.9.0.0

Instance details

Defined in Data.Functor.Const

Methods

(+) :: Const a b -> Const a b -> Const a b Source #

(-) :: Const a b -> Const a b -> Const a b Source #

(*) :: Const a b -> Const a b -> Const a b Source #

negate :: Const a b -> Const a b Source #

abs :: Const a b -> Const a b Source #

signum :: Const a b -> Const a b Source #

fromInteger :: Integer -> Const a b Source #

(Applicative f, Num a) => Num (Ap f a) Source #

Note that even if the underlying Num and Applicative instances are lawful, for most Applicatives, this instance will not be lawful. If you use this instance with the list Applicative, the following customary laws will not hold:

Commutativity:

>>> Ap [10,20] + Ap [1,2]
Ap {getAp = [11,12,21,22]}
>>> Ap [1,2] + Ap [10,20]
Ap {getAp = [11,21,12,22]}

Additive inverse:

>>> Ap [] + negate (Ap [])
Ap {getAp = []}
>>> fromInteger 0 :: Ap [] Int
Ap {getAp = [0]}

Distributivity:

>>> Ap [1,2] * (3 + 4)
Ap {getAp = [7,14]}
>>> (Ap [1,2] * 3) + (Ap [1,2] * 4)
Ap {getAp = [7,11,10,14]}

Since: base-4.12.0.0

Instance details

Defined in Data.Monoid

Methods

(+) :: Ap f a -> Ap f a -> Ap f a Source #

(-) :: Ap f a -> Ap f a -> Ap f a Source #

(*) :: Ap f a -> Ap f a -> Ap f a Source #

negate :: Ap f a -> Ap f a Source #

abs :: Ap f a -> Ap f a Source #

signum :: Ap f a -> Ap f a Source #

fromInteger :: Integer -> Ap f a Source #

Num (f a) => Num (Alt f a) Source #

Since: base-4.8.0.0

Instance details

Defined in Data.Semigroup.Internal

Methods

(+) :: Alt f a -> Alt f a -> Alt f a Source #

(-) :: Alt f a -> Alt f a -> Alt f a Source #

(*) :: Alt f a -> Alt f a -> Alt f a Source #

negate :: Alt f a -> Alt f a Source #

abs :: Alt f a -> Alt f a Source #

signum :: Alt f a -> Alt f a Source #

fromInteger :: Integer -> Alt f a Source #

quotRemInteger :: Integer -> Integer -> (# Integer, Integer #) Source #

Deprecated: Use integerQuotRem# instead

subtract :: Num a => a -> a -> a Source #

the same as flip (-).

Because - is treated specially in the Haskell grammar, (- e) is not a section, but an application of prefix negation. However, (subtract exp) is equivalent to the disallowed section.