or Addition

For practical reasons, we begin the class tree with Additive. Starting with Associative and Unital, or using Semigroup and Monoid from base tends to confuse the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.

\a -> zero + a == a

\a -> a + zero == a

\a b c -> (a + b) + c == a + (b + c)

\a b -> a + b == b + a

By convention, (+) is regarded as commutative, but this is not universal, and the introduction of another symbol which means non-commutative addition seems a bit dogmatic.

>>> zero + 1
1

>>> 1 + 1
2

Compute the sum of a Foldable.

>>> sum [0..10]
55

Compute the accumulating sum of a Traversable.

>>> accsum [0..10]
[0,1,3,6,10,15,21,28,36,45,55]

or Subtraction

\a -> a - a == zero

\a -> negate a == zero - a

\a -> negate a + a == zero

\a -> a + negate a == zero

>>> negate 1
-1

>>> 1 - 2
-1

or Multiplication

For practical reasons, we begin the class tree with Additive and Multiplicative. Starting with Associative and Unital, or using Semigroup and Monoid from base tends to confuse the interface once you start having to disinguish between (say) monoidal addition and monoidal multiplication.

\a -> one * a == a

\a -> a * one == a

\a b c -> (a * b) * c == a * (b * c)

By convention, (*) is regarded as not necessarily commutative, but this is not universal, and the introduction of another symbol which means commutative multiplication seems a bit dogmatic.

>>> one * 2
2

>>> 2 * 3
6

Compute the product of a Foldable.

>>> product [1..5]
120

Compute the accumulating product of a Traversable.

>>> accproduct [1..5]
[1,2,6,24,120]

or Division

Though unusual, the term Divisive usefully fits in with the grammer of other classes and avoids name clashes that occur with some popular libraries.

\(a :: Double) -> a / a ~= one || a == zero

\(a :: Double) -> recip a ~= one / a || a == zero

\(a :: Double) -> recip a * a ~= one || a == zero

\(a :: Double) -> a * recip a ~= one || a == zero

>>> recip 2.0
0.5

>>> 1 / 2
0.5

Distributive

\a b c -> a * (b + c) == a * b + a * c

\a b c -> (a + b) * c == a * c + b * c

\a -> zero * a == zero

\a -> a * zero == zero

The sneaking in of the Absorption laws here glosses over the possibility that the multiplicative zero element does not have to correspond with the additive unital zero.

A Ring is an abelian group under addition (Unital, Associative, Commutative, Invertible) and monoidal under multiplication (Unital, Associative), and where multiplication distributes over addition.

\a -> zero + a == a
\a -> a + zero == a
\a b c -> (a + b) + c == a + (b + c)
\a b -> a + b == b + a
\a -> a - a == zero
\a -> negate a == zero - a
\a -> negate a + a == zero
\a -> a + negate a == zero
\a -> one * a == a
\a -> a * one == a
\a b c -> (a * b) * c == a * (b * c)
\a b c -> a * (b + c) == a * b + a * c
\a b c -> (a + b) * c == a * c + b * c
\a -> zero * a == zero
\a -> a * zero == zero

A StarSemiring is a semiring with an additional unary operator (star) satisfying:

\a -> star a == one + a * star a

A Kleene Algebra is a Star Semiring with idempotent addition.

a * x + x = a ==> star a * x + x = x
x * a + x = a ==> x * star a + x = x

Involutive Ring

adj (a + b) ==> adj a + adj b
adj (a * b) ==> adj a * adj b
adj one ==> one
adj (adj a) ==> a

Note: elements for which adj a == a are called "self-adjoint".

Defining two requires adding the multiplicative unital to itself. In other words, the concept of two is a Ring one.

>>> two
2

A Field is a set on which addition, subtraction, multiplication, and division are defined. It is also assumed that multiplication is distributive over addition.

A summary of the rules inherited from super-classes of Field:

zero + a == a
a + zero == a
((a + b) + c) (a + (b + c))
a + b == b + a
a - a == zero
negate a == zero - a
negate a + a == zero
a + negate a == zero
one * a == a
a * one == a
((a * b) * c) == (a * (b * c))
(a * (b + c)) == (a * b + a * c)
((a + b) * c) == (a * c + b * c)
a * zero == zero
zero * a == zero
a / a == one || a == zero
recip a == one / a || a == zero
recip a * a == one || a == zero
a * recip a == one || a == zero

A hyperbolic field class

\a -> a < zero || (sqrt . (**2)) a == a

\a -> a < zero || (log . exp) a ~= a

\a b -> (b < zero) || a <= zero || a == 1 || abs (a ** logBase a b - b) < 10 * epsilon

Conversion from a Field to a Ring

See Field of fractions

\a -> a - one < floor a <= a <= ceiling a < a + one

(\a -> a - one < fromIntegral (floor a :: Int) && fromIntegral (floor a :: Int) <= a && a <= fromIntegral (ceiling a :: Int) && fromIntegral (ceiling a :: Int) <= a + one) :: Double -> Bool

\a -> (round a :: Int) ~= (floor (a + half) :: Int)

Trigonometric Field

The list of laws is quite long: trigonometric identities

infinity is defined for any Field.

>>> one / zero + infinity
Infinity

>>> infinity + 1
Infinity

negative infinity

>>> negInfinity + infinity
NaN

nan is defined as zero/zero

but note the (social) law:

>>> nan == zero / zero
False

A half is a Field because it requires addition, multiplication and division.

>>> half :: Double
0.5

A algebraic structure with element joins: See Semilattice

Associativity: x \/ (y \/ z) == (x \/ y) \/ z
Commutativity: x \/ y == y \/ x
Idempotency:   x \/ x == x

The partial ordering induced by the join-semilattice structure

The partial ordering induced by the join-semilattice structure

A algebraic structure with element meets: See Semilattice

Associativity: x /\ (y /\ z) == (x /\ y) /\ z
Commutativity: x /\ y == y /\ x
Idempotency:   x /\ x == x

The partial ordering induced by the meet-semilattice structure

The partial ordering induced by the meet-semilattice structure

A join-semilattice with an identity element bottom for \/.

Identity: x \/ bottom == x

A meet-semilattice with an identity element top for /\.

Identity: x /\ top == x

Additive Action

flipped additive action

(+.) == flip (.+)

Subtractive Action

right scalar subtraction

(-.) == (+.) . negate

Multiplicative Action

flipped multiplicative action

(*.) == flip (.*)

Divisive Action

right scalar division

(/.) == (*.) . recip

A Module

a .* one == a
(a + b) .* c == (a .* c) + (b .* c)
c *. (a + b) == (c *. a) + (c *. b)
a .* zero == zero
a .* b == b *. a

signum from base is not an operator name in numhask and is replaced by sign. Compare with Norm where there is a change in codomain.

\a -> abs a * sign a ~= a

abs zero == zero, so any value for sign zero is ok. We choose lawful neutral:

>>> sign zero == zero
True

>>> abs (-1)
1

>>> sign (-1)
-1

Norm is a slight generalisation of Signed. The class has the same shape but allows the codomain to be different to the domain.

\a -> norm a >= zero
\a -> norm zero == zero
\a -> a == norm a .* basis a
\a -> norm (basis a) == one

>>> norm (-0.5 :: Double) :: Double
0.5

>>> basis (-0.5 :: Double) :: Double
-1.0

Distance, which combines the Subtractive notion of difference, with Norm.

distance a b >= zero
distance a a == zero
distance a b .* basis (a - b) == a - b

Convert between a "co-ordinated" or "higher-kinded" number and representations of an angle. Typically thought of as polar co-ordinate conversion.

See Polar coordinate system

ray . angle == basis
norm (ray x) == one

Something that has a magnitude and a direction.

Convert from a number to a Polar.

Convert from a Polar to a (coordinated aka higher-kinded) number.

A small number, especially useful for approximate equality.