WOLFRAM

Wolfram Language & System Documentation Center

NonPositiveReals

represents the domain of non-positive real numbers.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
Cite this Page

NonPositiveReals

NonPositiveReals

represents the domain of non-positive real numbers.

Details

  • x∈NonPositiveReals evaluates immediately if x is a numeric quantity.
  • Simplify[expr∈NonPositiveReals,assum] can be used to try to determine whether an expression corresponds to a non-positive real number under the given assumptions.
  • (x1|x2|…)∈NonPositiveReals and {x1,x2,…}∈NonPositiveReals test whether all xi are non-positive real numbers.
  • NonPositiveReals is output in StandardForm and TraditionalForm as . This typeset form can be input using npreals.

Examples

open all close all

Basic Examples  (3)

is a non-positive real number:

Wolfram Language code: Element[-Pi ^ E, NonPositiveReals]

If is a real number, is a non-positive real number:

Wolfram Language code: Simplify[Sin[x] - 1∈NonPositiveReals, x∈ℝ]

Find non-positive real solutions of an equation:

Wolfram Language code: Reduce[x ^ 5 - 2x + 1 == 0, x, NonPositiveReals]

Scope  (4)

Test if a numeric quantity is non-positive:

Wolfram Language code: Sin[1]∈NonPositiveReals

Make domain membership assumptions:

Wolfram Language code: Refine[Sign[Im[x + I y]], (x | y)∈NonPositiveReals]
Wolfram Language code: Integrate[Abs[1 - Abs[x - 2]], x, Assumptions -> x∈NonPositiveReals]

Specify the default domain over which a function should work:

Wolfram Language code: Reduce[E ^ x - 2x == 3, x, NonPositiveReals]
Wolfram Language code: Resolve[Exists[x, y == Sqrt[x]], NonPositiveReals]

Test whether several numbers are non-positive reals:

Wolfram Language code: (x | y | -1)∈NonPositiveReals

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

Wolfram Language code: {x, y, 1}∈NonPositiveReals

Applications  (1)

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

Wolfram Language code: list = RandomReal[{-1, 0}, 1000000];
Wolfram Language code: AbsoluteTiming[list∈NonPositiveReals]
Wolfram Language code: AbsoluteTiming[AllTrue[list, NonPositive]]

Properties & Relations  (4)

Membership in NonPositiveReals is equivalent to membership in Reals along with non-positivity:

Wolfram Language code: x∈NonPositiveReals

NonPositiveReals contains NonPositiveRationals and NonPositiveIntegers:

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

NonPositiveReals is contained in Complexes:

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

NonPositiveReals is disjoint from PositiveReals:

Wolfram Language code: Reduce[x∈PositiveReals, x, NonPositiveReals]

It intersects NonNegativeReals:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_nonpositivereals, author="Wolfram Research", title="{NonPositiveReals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NonPositiveReals.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_nonpositivereals, organization={Wolfram Research}, title={NonPositiveReals}, year={2019}, url={https://reference.wolfram.com/language/ref/NonPositiveReals.html}, note=[Accessed: 12-June-2026]}