Rationals

Fractions are created in Julia with the double slash operator //. If a fraction is created from Julia integers, a Julia fraction results, and if either the numerator or denominator is an OSCAR integer of type ZZRingElem, an OSCAR fraction of type QQFieldElem results.

Julia has its own parameterised type Rational{T} for its own fractions, where T is the integer type of the numerator and denominator, e.g. Rational{Int} and Rational{BigInt}. Unlike with Int, all of the basic arithmetic operations on Julia's Rational{Int} are checked for overflow in the numerator and denominator.

The field of rationals

The parent of an OSCAR rational number is the field of rationals. It can be constructed from the ring of integers ZZ using the fraction_field constructor.

For convenience, QQ is already defined to be the field of rational numbers.

julia> S = fraction_field(ZZ)
Rational field

julia> QQ
Rational field

Integer constructors

OSCAR rationals can be created using QQ. Two arguments can be passed to specify numerator and denominator. If a single argument is passed, the denominator is set to 1.

For convenience, QQ also accepts Julia integers and rationals, but will always construct an OSCAR rational.

Naturally, Julia's double slash operator can also be used to construct fractions. However, unlike QQ, the double slash operator only constructs an OSCAR rational if either the numerator or denominator is an OSCAR integer.

An exception is raised if a fraction is constructed with denominator zero.

julia> QQ(1, 2)
1//2

julia> QQ(5)
5

julia> ZZ(3)//5
3//5

julia> 1//ZZ(7)
1//7

julia> QQ(2//3)
2//3

julia> ZZ(3)//0
ERROR: DivideError: integer division error

One can also construct the rational number $0$ with the empty constructor:

julia> QQ()
0

The following special constructors are also provided:

• zero(QQ)
• one(QQ)

julia> zero(QQ)
0

julia> one(QQ)
1

Predicates

• iszero(n::QQFieldElem) -> Bool
• isone(n::QQFieldElem) -> Bool
• is_unit(n::QQFieldElem) -> Bool

The is_unit function will return true iff $n \neq 0$.

julia> iszero(QQ())
true

julia> isone(one(QQ))
true

julia> is_unit(QQ(-2, 3))
true

Properties

• numerator(n::QQFieldElem) -> ZZRingElem
• denominator(n::QQFieldElem) -> ZZRingElem

Return the numerator and denominator respectively, of $n$.

• sign(n::QQFieldElem) -> QQFieldElem

Return the sign of n, i.e. $n/|n|$ if $n \neq 0$, or $0$ otherwise.

julia> sign(QQ(2, 3))
1

julia> sign(QQ())
0

julia> sign(QQ(-1))
-1

• abs(n::QQFieldElem) -> QQFieldElem

Return the absolute value of $n$, i.e. $n$ if $n \geq 0$ and $-n$ otherwise.

julia> abs(QQ(-3, 2))
3//2

• height(n::QQFieldElem) -> ZZRingElem

Return the maximum of the absolute values of the numerator and denominator of $n$.

julia> height(QQ(324987329, -8372492324))
8372492324

• floor(n::QQFieldElem) -> QQFieldElem

Return the greatest integer $m$ (as a rational number) such that $m \leq n$.

• ceil(n::QQFieldElem) -> QQFieldElem

Return the least integer $m$ (as a rational number) such that $m \geq n$.

julia> floor(QQ(-2, 3))
-1

julia> ceil(QQ(7, 2))
4

julia> typeof(ans)
QQFieldElem

julia> ceil(QQ(5))
5

• floor(ZZRingElem, n::QQFieldElem) -> ZZRingElem

Return the greatest integer $m$ such that $m \leq n$.

• ceil(ZZRingElem, n::QQFieldElem) -> ZZRingElem

Return the least integer $m$ such that $m \geq n$.

julia> floor(ZZRingElem, QQ(-2, 3))
-1

julia> ceil(ZZRingElem, QQ(7, 2))
4

julia> typeof(ans)
ZZRingElem

julia> ceil(ZZRingElem, QQ(5))
5

Basic arithmetic

OSCAR provides the basic arithmetic operations +, - and * and comparison operators ==, !=, <, <=, >, >=, including mixed operations between Julia and OSCAR rationals and integers.

[Exact Division]

• divexact(a::QQFieldElem, b::QQFieldElem) -> QQFieldElem
• divexact(a::QQFieldElem, b::Union{ZZRingElem,Base.Integer,Base.Rational}) -> QQFieldElem
• divexact(a::Union{ZZRingElem,Base.Integer,Base.Rational}, b::QQFieldElem) -> QQFieldElem

Return the quotient of $a$ by $b$. Exact division raises an exception if division by zero is attempted.

julia> divexact(QQ(2, 3), QQ(3, 5))
10//9

julia> divexact(QQ(1, 3), ZZ(0))
ERROR: DivideError: integer division error

julia> divexact(QQ(3, 4), ZZ(5))
3//20

julia> divexact(ZZ(6), QQ(2, 3))
9

julia> divexact(QQ(1, 3), 5)
1//15

Powering

• ^(a::QQFieldElem, b::Int) -> QQFieldElem

Return the result of powering $a$ by $b$.

julia> QQ(5, 7)^32
23283064365386962890625//1104427674243920646305299201

julia> QQ(1, 2)^(-2)
4

The following is allowed for convenience.

julia> QQ(0)^0
1

julia> QQ(0)^-2
ERROR: DivideError: integer division error

• is_power(a::QQFieldElem, b::Int) -> Bool, QQFieldElem

Test if $a$ is an $n$-th power. If so, return true and the root, false and any rational otherwise.

• is_perfect_power_with_data(a::QQFieldElem) -> Int, QQFieldElem

Find the largest $n$ such that $a$ is an $n$-th power. Return $n$ and the root.

• root(a::QQFieldElem, b::Int) -> QQFieldElem

Compute an $n$-th root of $a$, raises an error if $a$ is not an $n$-th power.

julia> is_power(QQ(8), 3)
(true, 2)

julia> is_power(QQ(8), 2)
(false, 8)

julia> is_perfect_power_with_data(QQ(9//16))
(2, 3//4)

julia> root(QQ(25//9), 2)
5//3