base-4.14.1.0: 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 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: 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: 2.1

Instance details

Defined in GHC.Float

Num Int Source #

Since: 2.1

Instance details

Defined in GHC.Num

Num Int8 Source #

Since: 2.1

Instance details

Defined in GHC.Int

Num Int16 Source #

Since: 2.1

Instance details

Defined in GHC.Int

Num Int32 Source #

Since: 2.1

Instance details

Defined in GHC.Int

Num Int64 Source #

Since: 2.1

Instance details

Defined in GHC.Int

Num Integer Source #

Since: 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: 4.8.0.0

Instance details

Defined in GHC.Num

Num Word Source #

Since: 2.1

Instance details

Defined in GHC.Num

Num Word8 Source #

Since: 2.1

Instance details

Defined in GHC.Word

Num Word16 Source #

Since: 2.1

Instance details

Defined in GHC.Word

Num Word32 Source #

Since: 2.1

Instance details

Defined in GHC.Word

Num Word64 Source #

Since: 2.1

Instance details

Defined in GHC.Word

Num IntPtr Source # 
Instance details

Defined in Foreign.Ptr

Num WordPtr Source # 
Instance details

Defined in Foreign.Ptr

Num CUIntMax Source # 
Instance details

Defined in Foreign.C.Types

Num CIntMax Source # 
Instance details

Defined in Foreign.C.Types

Num CUIntPtr Source # 
Instance details

Defined in Foreign.C.Types

Num CIntPtr Source # 
Instance details

Defined in Foreign.C.Types

Num CSUSeconds Source # 
Instance details

Defined in Foreign.C.Types

Num CUSeconds Source # 
Instance details

Defined in Foreign.C.Types

Num CTime Source # 
Instance details

Defined in Foreign.C.Types

Num CClock Source # 
Instance details

Defined in Foreign.C.Types

Num CSigAtomic Source # 
Instance details

Defined in Foreign.C.Types

Num CWchar Source # 
Instance details

Defined in Foreign.C.Types

Num CSize Source # 
Instance details

Defined in Foreign.C.Types

Num CPtrdiff 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 CBool Source # 
Instance details

Defined in Foreign.C.Types

Num CULLong Source # 
Instance details

Defined in Foreign.C.Types

Num CLLong Source # 
Instance details

Defined in Foreign.C.Types

Num CULong Source # 
Instance details

Defined in Foreign.C.Types

Num CLong Source # 
Instance details

Defined in Foreign.C.Types

Num CUInt Source # 
Instance details

Defined in Foreign.C.Types

Num CInt Source # 
Instance details

Defined in Foreign.C.Types

Num CUShort Source # 
Instance details

Defined in Foreign.C.Types

Num CShort Source # 
Instance details

Defined in Foreign.C.Types

Num CUChar Source # 
Instance details

Defined in Foreign.C.Types

Num CSChar Source # 
Instance details

Defined in Foreign.C.Types

Num CChar Source # 
Instance details

Defined in Foreign.C.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 CNfds Source # 
Instance details

Defined in System.Posix.Types

Num CSocklen Source # 
Instance details

Defined in System.Posix.Types

Num CKey Source # 
Instance details

Defined in System.Posix.Types

Num CId Source # 
Instance details

Defined in System.Posix.Types

Num CFsFilCnt Source # 
Instance details

Defined in System.Posix.Types

Num CFsBlkCnt Source # 
Instance details

Defined in System.Posix.Types

Num CClockId Source # 
Instance details

Defined in System.Posix.Types

Num CBlkCnt Source # 
Instance details

Defined in System.Posix.Types

Num CBlkSize Source # 
Instance details

Defined in System.Posix.Types

Num CRLim Source # 
Instance details

Defined in System.Posix.Types

Num CTcflag Source # 
Instance details

Defined in System.Posix.Types

Num CSpeed Source # 
Instance details

Defined in System.Posix.Types

Num CCc Source # 
Instance details

Defined in System.Posix.Types

Num CUid Source # 
Instance details

Defined in System.Posix.Types

Num CNlink Source # 
Instance details

Defined in System.Posix.Types

Num CGid Source # 
Instance details

Defined in System.Posix.Types

Num CSsize Source # 
Instance details

Defined in System.Posix.Types

Num CPid Source # 
Instance details

Defined in System.Posix.Types

Num COff Source # 
Instance details

Defined in System.Posix.Types

Num CMode Source # 
Instance details

Defined in System.Posix.Types

Num CIno Source # 
Instance details

Defined in System.Posix.Types

Num CDev Source # 
Instance details

Defined in System.Posix.Types

Integral a => Num (Ratio a) Source #

Since: 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) Source #

Since: 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) Source #

Since: 4.7.0.0

Instance details

Defined in Data.Semigroup.Internal

Num a => Num (Sum a) Source #

Since: 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) Source #

Since: 4.9.0.0

Instance details

Defined in Data.Functor.Identity

Num a => Num (Max a) Source #

Since: 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: 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) Source #

Since: 2.1

Instance details

Defined in Data.Complex

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 #

HasResolution a => Num (Fixed a) Source #

Since: 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) Source #

Since: 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) Source #

Since: 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) Source #

Since: 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.