represents the domain of non-positive rational numbers, as in xNonPositiveRationals.


  • xNonPositiveRationals evaluates immediately if x is a numeric quantity.
  • Simplify[exprNonPositiveRationals,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.


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Basic Examples  (3)

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

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

Find non-positive rational solutions of an equation:

Scope  (5)

Test domain membership of a numeric expression:

Make domain membership assumptions:

Specify the default domain over which Reduce should work:

Test whether several numbers are non-positive rationals:

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

TraditionalForm formatting:

Properties & Relations  (4)

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

NonPositiveRationals contains NonPositiveIntegers:

NonPositiveRationals is contained in NonPositiveReals, Algebraics and Complexes:

NonPositiveRationals is disjoint from PositiveRationals:

NonPositiveRationals intersects NonNegativeRationals:

Wolfram Research (2019), NonPositiveRationals, Wolfram Language function,


Wolfram Research (2019), NonPositiveRationals, Wolfram Language function,


Wolfram Language. 2019. "NonPositiveRationals." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2019). NonPositiveRationals. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_nonpositiverationals, author="Wolfram Research", title="{NonPositiveRationals}", year="2019", howpublished="\url{}", note=[Accessed: 17-June-2024 ]}


@online{reference.wolfram_2024_nonpositiverationals, organization={Wolfram Research}, title={NonPositiveRationals}, year={2019}, url={}, note=[Accessed: 17-June-2024 ]}