represents the domain of non-positive integers, as in xNonPositiveIntegers.


  • xNonPositiveIntegers evaluates immediately if x is a numeric quantity.
  • Simplify[exprNonPositiveIntegers,assum] can be used to try to determine whether an expression is a non-positive integer under the given assumptions.
  • (x1|x2|)NonPositiveIntegers and {x1,x2,}NonPositiveIntegers test whether all xi are non-positive integers.
  • NonPositiveIntegers is output in StandardForm or TraditionalForm as TemplateBox[{}, NonPositiveIntegers]. This typeset form can be input using npints.


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Basic Examples  (3)

Negative seven is a non-positive integer:

If is an integer, is a non-positive integer:

Find non-positive integer solutions of a Pell equation:

Scope  (6)

Test domain membership of a numeric expression:

Make domain membership assumptions:

Specify the default domain over which a function should work:

Solve an optimization problem over the non-positive integers:

Test whether several numbers are non-positive integers:

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

TraditionalForm formatting:

Applications  (1)

Testing membership in the non-positive integers is a fast way to verify non-positivity of a large list:

Properties & Relations  (3)

Membership in NonPositiveIntegers is equivalent to membership in Integers and non-positivity:

NonPositiveIntegers is contained in NonPositiveReals and NonPositiveRationals:

NonPositiveIntegers is disjoint from PositiveIntegers:

It intersects NonNegativeIntegers:

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


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


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


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


@misc{reference.wolfram_2024_nonpositiveintegers, author="Wolfram Research", title="{NonPositiveIntegers}", year="2019", howpublished="\url{}", note=[Accessed: 13-July-2024 ]}


@online{reference.wolfram_2024_nonpositiveintegers, organization={Wolfram Research}, title={NonPositiveIntegers}, year={2019}, url={}, note=[Accessed: 13-July-2024 ]}