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RationalExpressionQ
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RationalExpressionQ

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gives True if expr is structurally a rational expression in x, and False otherwise.

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RationalExpressionQ[expr,{x,y,…}]

gives True if expr is structurally a rational expression in x,y,…, and False otherwise.

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RationalExpressionQ[expr,{x,y,…},test]

gives True if expr is structurally a rational expression in x,y,… with coefficients satisfying test, and False otherwise.

Details

  • A rational expression in x,y,… is an expression constructed with x,y,… and coefficients not containing x,y,…, using Plus, Times and integer Power.
  • RationalExpressionQ[expr,vars,NumericQ] tests whether expr is a rational expression in vars with numeric coefficients.

Examples

open allclose all

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

Test whether an expression is rational in the specified variable:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-g6tr6q
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-knbt4b
Out[2]=2

Test whether an expression is rational in the specified set of variables:

In[1]:=1
✖
https://wolfram.com/xid/0dc113yeqh73-kbfaad
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-xck30
Out[2]=2

Test whether an expression is rational with numeric coefficients:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-bzryi5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-fas7o
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

Multilevel fractions are rational expressions:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-2tq6r
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-dnfhp0
Out[2]=2

Coefficients of rational expressions may involve arbitrary functions:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-giw0ti
Out[1]=1

Variables need not be symbols:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-dz4vqv
Out[1]=1

Variables need not be independent of each other:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-hgepl8
Out[1]=1

Properties & Relations  (2)Properties of the function, and connections to other functions

Together represents rational expressions as ratios of polynomials:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-fljtqf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-elfmlw
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc113yeqh73-cgsf38
Out[3]=3

Use NumeratorDenominator to extract the numerator and the denominator:

In[4]:=4
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https://wolfram.com/xid/0dc113yeqh73-h1c0ra
Out[4]=4

Use PolynomialExpressionQ to verify that the resulting expressions are polynomials:

In[5]:=5
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https://wolfram.com/xid/0dc113yeqh73-bzymig
Out[5]=5

Rational expressions represent functions that are singular at zeros of the denominators:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-h8gr0o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-myodjv
Out[2]=2

Use FunctionSingularities to find the singularities:

In[3]:=3
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https://wolfram.com/xid/0dc113yeqh73-ehkb7l
Out[3]=3

Outside zeros of the denominators, rational expressions represent analytic functions:

In[4]:=4
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https://wolfram.com/xid/0dc113yeqh73-nye4gv
Out[4]=4

Possible Issues  (3)Common pitfalls and unexpected behavior

A rational expression may not represent a rational function due to hidden division by zero:

In[1]:=1
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https://wolfram.com/xid/0dc113yeqh73-ks8v4o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc113yeqh73-eg8tdt
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc113yeqh73-e836en
Out[3]=3

A nonrational expression may represent a rational function:

In[1]:=1
✖
https://wolfram.com/xid/0dc113yeqh73-csoh4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0dc113yeqh73-cbur58
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc113yeqh73-gdtgeh
Out[3]=3

RationalExpressionQ is purely syntactic:

In[1]:=1
✖
https://wolfram.com/xid/0dc113yeqh73-fwp82z
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc113yeqh73-cxyfu3
Out[2]=2

Syntactically, Tan[x] is a coefficient, free of Sin[x] and Cos[x]:

In[3]:=3
✖
https://wolfram.com/xid/0dc113yeqh73-j93zev
Out[3]=3
Wolfram Research (2020), RationalExpressionQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RationalExpressionQ.html.
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Wolfram Research (2020), RationalExpressionQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RationalExpressionQ.html.

Text

Wolfram Research (2020), RationalExpressionQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RationalExpressionQ.html.

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Wolfram Research (2020), RationalExpressionQ, Wolfram Language function, https://reference.wolfram.com/language/ref/RationalExpressionQ.html.

CMS

Wolfram Language. 2020. "RationalExpressionQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RationalExpressionQ.html.

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Wolfram Language. 2020. "RationalExpressionQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RationalExpressionQ.html.

APA

Wolfram Language. (2020). RationalExpressionQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RationalExpressionQ.html

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Wolfram Language. (2020). RationalExpressionQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RationalExpressionQ.html

BibTeX

@misc{reference.wolfram_2025_rationalexpressionq, author="Wolfram Research", title="{RationalExpressionQ}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/RationalExpressionQ.html}", note=[Accessed: 23-May-2025 ]}

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@misc{reference.wolfram_2025_rationalexpressionq, author="Wolfram Research", title="{RationalExpressionQ}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/RationalExpressionQ.html}", note=[Accessed: 23-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_rationalexpressionq, organization={Wolfram Research}, title={RationalExpressionQ}, year={2020}, url={https://reference.wolfram.com/language/ref/RationalExpressionQ.html}, note=[Accessed: 23-May-2025 ]}

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@online{reference.wolfram_2025_rationalexpressionq, organization={Wolfram Research}, title={RationalExpressionQ}, year={2020}, url={https://reference.wolfram.com/language/ref/RationalExpressionQ.html}, note=[Accessed: 23-May-2025 ]}