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/Haskell 8

GHC.Num

Copyright (c) The University of Glasgow 1994-2002
License see libraries/base/LICENSE
Maintainer [email protected]
Stability internal
Portability non-portable (GHC Extensions)
Safe Haskell Trustworthy
Language Haskell2010

Description

The Num class and the Integer type.

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

Methods

(+) :: Int -> Int -> Int Source

(-) :: Int -> Int -> Int Source

(*) :: Int -> Int -> Int Source

negate :: Int -> Int Source

abs :: Int -> Int Source

signum :: Int -> Int Source

fromInteger :: Integer -> Int Source

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 CKey
Instance details

Defined in System.Posix.Types

Num CId
Instance details

Defined in System.Posix.Types

Methods

(+) :: CId -> CId -> CId Source

(-) :: CId -> CId -> CId Source

(*) :: CId -> CId -> CId Source

negate :: CId -> CId Source

abs :: CId -> CId Source

signum :: CId -> CId Source

fromInteger :: Integer -> CId Source

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

Methods

(+) :: CCc -> CCc -> CCc Source

(-) :: CCc -> CCc -> CCc Source

(*) :: CCc -> CCc -> CCc Source

negate :: CCc -> CCc Source

abs :: CCc -> CCc Source

signum :: CCc -> CCc Source

fromInteger :: Integer -> CCc Source

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

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

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

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)

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.

module GHC.Integer

module GHC.Natural

© The University of Glasgow and others
Licensed under a BSD-style license (see top of the page).
https://downloads.haskell.org/~ghc/8.8.3/docs/html/libraries/base-4.13.0.0/GHC-Num.html