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PositiveRationals

represents the domain of strictly positive rational numbers, as in x∈PositiveRationals.

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PositiveRationals

PositiveRationals

represents the domain of strictly positive rational numbers, as in x∈PositiveRationals.

Details

  • x∈PositiveRationals evaluates immediately if x is a numeric quantity.
  • Simplify[expr∈PositiveRationals,assum] can be used to try to determine whether an expression corresponds to a positive rational number under the given assumptions.
  • (x1|x2|…)∈PositiveRationals and {x1,x2,…}∈PositiveRationals test whether all xi are positive rational numbers.
  • The domain of positive integers is taken to be a subset of the domain of positive rationals.
  • PositiveRationals is output in StandardForm or TraditionalForm as TemplateBox[{}, PositiveRationals]. This typeset form can be input using prats.

Examples

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Basic Examples  (3)

2/3 is a positive rational number:

Wolfram Language code: Element[2 / 3, PositiveRationals]

A sum of positive rational numbers is a positive rational number:

Wolfram Language code: Simplify[x + y∈PositiveRationals, (x | y)∈PositiveRationals]

Find positive rational solutions of an equation:

Wolfram Language code: Reduce[(2x ^ 2 - 1)(4x ^ 2 - 1) == 0, x, PositiveRationals]

Scope  (5)

Test domain membership of a numeric expression:

Wolfram Language code: Element[21 / 32, PositiveRationals]
Wolfram Language code: Element[0, PositiveRationals]
Wolfram Language code: Element[Pi, PositiveRationals]

Make domain membership assumptions:

Wolfram Language code: Refine[x^2 + 2 x y + (y^3/3)∈PositiveRationals, (x | y)∈PositiveRationals]

Specify the default domain over which Reduce should work:

Wolfram Language code: Reduce[(x ^ 2 - 4 / 9)(x ^ 3 - 2) == 0, x, PositiveRationals]

Test whether several numbers are positive rationals:

Wolfram Language code: (x | y | 1 / 2)∈PositiveRationals

If any number is explicitly not a positive rational, the result is False:

Wolfram Language code: {x, y, 0}∈PositiveRationals

TraditionalForm formatting:

Wolfram Language code: PositiveRationals//TraditionalForm

Properties & Relations  (4)

Membership in PositiveRationals is equivalent to membership in Rationals along with positivity:

Wolfram Language code: x∈PositiveRationals

PositiveRationals contains PositiveIntegers:

Wolfram Language code: Refine[x∈PositiveRationals, x∈PositiveIntegers]

PositiveRationals is contained in PositiveReals, Algebraics and Complexes:

Wolfram Language code: Refine[x∈PositiveReals, x∈PositiveRationals]
Wolfram Language code: Refine[x∈Algebraics, x∈PositiveRationals]
Wolfram Language code: Refine[x∈ℂ, x∈PositiveRationals]

PositiveRationals is disjoint from NonPositiveRationals and NegativeRationals:

Wolfram Language code: Refine[x∈NonPositiveRationals, x∈PositiveRationals]
Wolfram Language code: Refine[x∈NegativeRationals, x∈PositiveRationals]
Wolfram Research (2019), PositiveRationals, Wolfram Language function, https://reference.wolfram.com/language/ref/PositiveRationals.html.

Text

Wolfram Research (2019), PositiveRationals, Wolfram Language function, https://reference.wolfram.com/language/ref/PositiveRationals.html.

CMS

Wolfram Language. 2019. "PositiveRationals." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PositiveRationals.html.

APA

Wolfram Language. (2019). PositiveRationals. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PositiveRationals.html

BibTeX

@misc{reference.wolfram_2026_positiverationals, author="Wolfram Research", title="{PositiveRationals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PositiveRationals.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_positiverationals, organization={Wolfram Research}, title={PositiveRationals}, year={2019}, url={https://reference.wolfram.com/language/ref/PositiveRationals.html}, note=[Accessed: 12-June-2026]}