base-4.14.0.0: 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)

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 Double #

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 #

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 #

Since: base-2.1

Instance details

Defined in GHC.Num

Num Int8 #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int16 #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int32 #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Int64 #

Since: base-2.1

Instance details

Defined in GHC.Int

Num Integer #

Since: base-2.1

Instance details

Defined in GHC.Num

Num Natural #

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 Word #

Since: base-2.1

Instance details

Defined in GHC.Num

Num Word8 #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word16 #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word32 #

Since: base-2.1

Instance details

Defined in GHC.Word

Num Word64 #

Since: base-2.1

Instance details

Defined in GHC.Word

Num IntPtr # 
Instance details

Defined in Foreign.Ptr

Num WordPtr # 
Instance details

Defined in Foreign.Ptr

Num CUIntMax # 
Instance details

Defined in Foreign.C.Types

Num CIntMax # 
Instance details

Defined in Foreign.C.Types

Num CUIntPtr # 
Instance details

Defined in Foreign.C.Types

Num CIntPtr # 
Instance details

Defined in Foreign.C.Types

Num CSUSeconds # 
Instance details

Defined in Foreign.C.Types

Num CUSeconds # 
Instance details

Defined in Foreign.C.Types

Num CTime # 
Instance details

Defined in Foreign.C.Types

Num CClock # 
Instance details

Defined in Foreign.C.Types

Num CSigAtomic # 
Instance details

Defined in Foreign.C.Types

Num CWchar # 
Instance details

Defined in Foreign.C.Types

Num CSize # 
Instance details

Defined in Foreign.C.Types

Num CPtrdiff # 
Instance details

Defined in Foreign.C.Types

Num CDouble # 
Instance details

Defined in Foreign.C.Types

Num CFloat # 
Instance details

Defined in Foreign.C.Types

Num CBool # 
Instance details

Defined in Foreign.C.Types

Num CULLong # 
Instance details

Defined in Foreign.C.Types

Num CLLong # 
Instance details

Defined in Foreign.C.Types

Num CULong # 
Instance details

Defined in Foreign.C.Types

Num CLong # 
Instance details

Defined in Foreign.C.Types

Num CUInt # 
Instance details

Defined in Foreign.C.Types

Num CInt # 
Instance details

Defined in Foreign.C.Types

Num CUShort # 
Instance details

Defined in Foreign.C.Types

Num CShort # 
Instance details

Defined in Foreign.C.Types

Num CUChar # 
Instance details

Defined in Foreign.C.Types

Num CSChar # 
Instance details

Defined in Foreign.C.Types

Num CChar # 
Instance details

Defined in Foreign.C.Types

Num Fd # 
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 CNfds # 
Instance details

Defined in System.Posix.Types

Num CSocklen # 
Instance details

Defined in System.Posix.Types

Num CKey # 
Instance details

Defined in System.Posix.Types

Num CId # 
Instance details

Defined in System.Posix.Types

Num CFsFilCnt # 
Instance details

Defined in System.Posix.Types

Num CFsBlkCnt # 
Instance details

Defined in System.Posix.Types

Num CClockId # 
Instance details

Defined in System.Posix.Types

Num CBlkCnt # 
Instance details

Defined in System.Posix.Types

Num CBlkSize # 
Instance details

Defined in System.Posix.Types

Num CRLim # 
Instance details

Defined in System.Posix.Types

Num CTcflag # 
Instance details

Defined in System.Posix.Types

Num CSpeed # 
Instance details

Defined in System.Posix.Types

Num CCc # 
Instance details

Defined in System.Posix.Types

Num CUid # 
Instance details

Defined in System.Posix.Types

Num CNlink # 
Instance details

Defined in System.Posix.Types

Num CGid # 
Instance details

Defined in System.Posix.Types

Num CSsize # 
Instance details

Defined in System.Posix.Types

Num CPid # 
Instance details

Defined in System.Posix.Types

Num COff # 
Instance details

Defined in System.Posix.Types

Num CMode # 
Instance details

Defined in System.Posix.Types

Num CIno # 
Instance details

Defined in System.Posix.Types

Num CDev # 
Instance details

Defined in System.Posix.Types

Integral a => Num (Ratio a) #

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 #

Num a => Num (Down a) #

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 (Product a) #

Since: base-4.7.0.0

Instance details

Defined in Data.Semigroup.Internal

Num a => Num (Sum a) #

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 #

Num a => Num (Identity a) #

Since: base-4.9.0.0

Instance details

Defined in Data.Functor.Identity

Num a => Num (Max a) #

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) #

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 #

RealFloat a => Num (Complex a) #

Since: base-2.1

Instance details

Defined in Data.Complex

Num a => Num (Op a b) # 
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 #

HasResolution a => Num (Fixed a) #

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

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 #

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

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 a => Num (Const a b) #

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 #

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.