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NonPositiveRationals

represents the domain of non-positive rational numbers, as in x∈NonPositiveRationals.

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NonPositiveRationals

NonPositiveRationals

represents the domain of non-positive rational numbers, as in x∈NonPositiveRationals.

Details

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

Examples

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Basic Examples  (3)

-2/3 is a non-positive rational number:

Wolfram Language code: Element[-2 / 3, NonPositiveRationals]

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

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

Find non-positive rational solutions of an equation:

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

Scope  (5)

Test domain membership of a numeric expression:

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

Make domain membership assumptions:

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

Specify the default domain over which Reduce should work:

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

Test whether several numbers are non-positive rationals:

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

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

Wolfram Language code: {x, y, 1 / 2}∈NonPositiveRationals

TraditionalForm formatting:

Wolfram Language code: NonPositiveRationals//TraditionalForm

Properties & Relations  (4)

Membership in NonPositiveRationals is equivalent to membership in Rationals and non-positivity:

Wolfram Language code: x∈NonPositiveRationals

NonPositiveRationals contains NonPositiveIntegers:

Wolfram Language code: Refine[x∈NonPositiveRationals, x∈NonPositiveIntegers]

NonPositiveRationals is contained in NonPositiveReals, Algebraics and Complexes:

Wolfram Language code: Refine[x∈NonPositiveReals, x∈NonPositiveRationals]
Wolfram Language code: Refine[x∈Algebraics, x∈NonPositiveRationals]
Wolfram Language code: Refine[x∈ℂ, x∈NonPositiveRationals]

NonPositiveRationals is disjoint from PositiveRationals:

Wolfram Language code: Reduce[x∈PositiveRationals, x, NonPositiveRationals]

NonPositiveRationals intersects NonNegativeRationals:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_nonpositiverationals, author="Wolfram Research", title="{NonPositiveRationals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NonPositiveRationals.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_nonpositiverationals, organization={Wolfram Research}, title={NonPositiveRationals}, year={2019}, url={https://reference.wolfram.com/language/ref/NonPositiveRationals.html}, note=[Accessed: 12-June-2026]}