orbital.math
Interface Fraction

All Superinterfaces:

Arithmetic, Normed

public interface Fraction
extends Arithmetic

Representation of a fraction a⁄s ∈ S^-1M = M_S.

S^-1M := M_S := M×S/~ = {a⁄s ¦ a∈M ∧ s∈S} with congruence relation ~ defined by

a⁄s ~ b⁄t :⇔ ∃sʹ∈S sʹ⋅(a⋅t-b⋅s)=0 (⇔ a⋅t-b⋅s = 0 if M is an integrity domain)

is the R_S-module (or ring or monoid) of fractions of the R-module (or commutative ring with 1 or monoid) M with denominators in S≤(R,⋅). This means that S≤(R,⋅) is a submonoid of the multiplicative group of R. S^-1M is also called localisation of M to S. If p⊲R is prime, then R_p := R_R\p is called localisation of R in p and is a local ring.

Especially, for an integrity domain R, Quot(R) := R₍₀₎ = (R∖{0})^-1R is the field of fractions of R, and for R-modules M, M_R∖{0} is a Quot(R)-vector space. If the underlying integrity domain R has an order, its field of fractions supports a unique order that restricts to the order on R (an extends Comparable).

A fraction a⁄s ∈ S^-1M with numerator a and denominator s is usually written as

  a
  s
 

There is the canonical embedding homomorphism ι_S:a↦a⁄1 which is injective if and only if S does not contain zero divisors. The ring of fractions R_Sis the presenting object of the presentable functor

Rng1	→	Ens
R'	↦	{φ∈HomRng1(R,R') ¦ φ(S)⊆(R')×}

Therefore it enjoys the following universal mapping property

∀φ:R→R' homomorphism of rings with 1 with φ(S)⊆(R')^×
∃!φ̃:R_S→R' homomorphism of rings with 1 where φ=φ̃∘ι_S

For R-modules M and S≤(R,⋅) it is

∀I⊴R M_S ≅ M ⊗_R R_S

Author:

André Platzer

See Also:

ValueFactory.fraction(Arithmetic,Arithmetic), "N. Bourbaki, Algebra I.2.4: Monoid of fractions of a commutative monoid.", "N. Bourbaki, Algebra VI.2.2: Ordered fields."

Field Summary

Fields inherited from interface orbital.math.Arithmetic

numerical

Method Summary

Fraction	add(Fraction bt) Adds two fractions returning a third as a result.
Arithmetic	denominator() Returns the denominator component.
Fraction	divide(Fraction bt) Divides two fractions returning a third as a result.
Fraction	multiply(Fraction bt) Multiplies two fractions returning a third as a result.
Arithmetic	numerator() Returns the numerator component.
Arithmetic	scale(Arithmetic alpha) Multiplies a scalar with this arithmetic object returning the result.
Fraction	subtract(Fraction bt) Subtracts two fractions returning a third as a result.

Methods inherited from interface orbital.math.Arithmetic

add, divide, equals, inverse, isOne, isZero, minus, multiply, one, power, subtract, toString, valueFactory, zero

Methods inherited from interface orbital.math.Normed

norm

Method Detail

numerator

Arithmetic numerator()

Returns the numerator component.

Returns:

a of this fraction a⁄s.

denominator

Arithmetic denominator()

Returns the denominator component.

Returns:

s of this fraction a⁄s.

add

Fraction add(Fraction bt)

Adds two fractions returning a third as a result.

Returns:

a⁄s + b⁄t := (a⋅t+b⋅s)⁄(s⋅t).

subtract

Fraction subtract(Fraction bt)

Subtracts two fractions returning a third as a result.

Returns:

a⁄s - b⁄t := (a⋅t-b⋅s)⁄(s⋅t).

multiply

Fraction multiply(Fraction bt)

Multiplies two fractions returning a third as a result.

Returns:

a⁄s ⋅ b⁄t := (a⋅b)⁄(s⋅t).

divide

Fraction divide(Fraction bt)

Divides two fractions returning a third as a result.

Returns:

a⁄s ∕ b⁄t := (a⋅t)⁄(s⋅b).

Throws:

java.lang.ArithmeticException - if b∉S and b is not invertible.

Preconditions:

b∈S ∨ b invertible anyway

scale

Arithmetic scale(Arithmetic alpha)

Multiplies a scalar with this arithmetic object returning the result.

Specified by:

scale in interface Arithmetic

Parameters:

alpha - the factor α to scale this arithmetic object with (per law of action of scalar multiplication).

Returns:

α·this

See Also:

Arithmetic.multiply(Arithmetic)

Postconditions:

RES instanceof Fraction