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NonPositiveRationals
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NonPositiveRationals

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

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

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-s1074
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-l7nnc
Out[1]=1

Find non-positive rational solutions of an equation:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-bemtf7
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Test domain membership of a numeric expression:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-crra11
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e5cq0la3njhz8-f8i88p
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e5cq0la3njhz8-i6yxp8
Out[3]=3

Make domain membership assumptions:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-cn7igo
Out[1]=1

Specify the default domain over which Reduce should work:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-g9ic3l
Out[1]=1

Test whether several numbers are non-positive rationals:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-3pjyzh
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0e5cq0la3njhz8-m7ag79
Out[2]=2

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-4xdcq9

Properties & Relations  (4)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-bus6sg
Out[1]=1

NonPositiveRationals contains NonPositiveIntegers:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-bpv7ri
Out[1]=1

NonPositiveRationals is contained in NonPositiveReals, Algebraics and Complexes:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-culh0e
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e5cq0la3njhz8-i9rm0h
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e5cq0la3njhz8-oij108
Out[3]=3

NonPositiveRationals is disjoint from PositiveRationals:

In[1]:=1
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https://wolfram.com/xid/0e5cq0la3njhz8-pe30t4
Out[1]=1

NonPositiveRationals intersects NonNegativeRationals:

In[2]:=2
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https://wolfram.com/xid/0e5cq0la3njhz8-3ius3z
Out[2]=2
Wolfram Research (2019), NonPositiveRationals, Wolfram Language function, https://reference.wolfram.com/language/ref/NonPositiveRationals.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_nonpositiverationals, author="Wolfram Research", title="{NonPositiveRationals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NonPositiveRationals.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_nonpositiverationals, author="Wolfram Research", title="{NonPositiveRationals}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NonPositiveRationals.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_nonpositiverationals, organization={Wolfram Research}, title={NonPositiveRationals}, year={2019}, url={https://reference.wolfram.com/language/ref/NonPositiveRationals.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_nonpositiverationals, organization={Wolfram Research}, title={NonPositiveRationals}, year={2019}, url={https://reference.wolfram.com/language/ref/NonPositiveRationals.html}, note=[Accessed: 29-March-2025 ]}