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NegativeRationals

represents the domain of strictly negative rational numbers, as in x∈NegativeRationals.

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NegativeRationals

NegativeRationals

represents the domain of strictly negative rational numbers, as in x∈NegativeRationals.

Details

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

Examples

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Basic Examples  (3)

-2/3 is a negative rational number:

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

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

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

Find negative rational solutions of an equation:

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

Scope  (5)

Test domain membership of a numeric expression:

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

Make domain membership assumptions:

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

Specify the default domain over which Reduce should work:

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

Test whether several numbers are negative rationals:

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

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

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

TraditionalForm formatting:

Wolfram Language code: NegativeRationals//TraditionalForm

Properties & Relations  (4)

Membership in NegativeRationals is equivalent to membership in Rationals and negativity:

Wolfram Language code: x∈NegativeRationals

NegativeRationals contains NegativeIntegers:

Wolfram Language code: Refine[x∈NegativeRationals, x∈NegativeIntegers]

NegativeRationals is contained in NegativeReals, Algebraics and Complexes:

Wolfram Language code: Refine[x∈NegativeReals, x∈NegativeRationals]
Wolfram Language code: Refine[x∈Algebraics, x∈NegativeRationals]
Wolfram Language code: Refine[x∈ℂ, x∈NegativeRationals]

NegativeRationals is disjoint from NonNegativeRationals and PositiveRationals:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_negativerationals, author="Wolfram Research", title="{NegativeRationals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NegativeRationals.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_negativerationals, organization={Wolfram Research}, title={NegativeRationals}, year={2019}, url={https://reference.wolfram.com/language/ref/NegativeRationals.html}, note=[Accessed: 12-June-2026]}