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.

abs x * signum x == x

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

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

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

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

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