GHC.Num

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

GHC.Real.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 Data.Ratio.Rational do.

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.