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NumeratorDenominator[expr]

gives the list {Numerator[expr],Denominator[expr]} of expr.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Modulus  
Trig  
Applications  
Properties & Relations  
Neat Examples  
See Also
Related Guides
History
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NumeratorDenominator

NumeratorDenominator[expr]

gives the list {Numerator[expr],Denominator[expr]} of expr.

Details and Options

  • Numerator picks out terms that do not have superficially negative exponents. Denominator picks out the remaining terms.
  • An exponent is "superficially negative" if it has a negative number as a factor.
  • The standard representation of rational expressions as products of powers means that you cannot simply use Part to extract numerators.
  • NumeratorDenominator can be used on rational numbers.
  • NumeratorDenominator automatically threads over lists.
  • NumeratorDenominator takes the following options:
  • Modulus 0modulus to assume for integers
    Trig Falsewhether to do trigonometric as well as algebraic transformations

Examples

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Basic Examples  (3)

Extract the numerator and denominator of a rational number:

Wolfram Language code: NumeratorDenominator[2 / 3]

Extract the numerator and denominator of a rational expression:

Wolfram Language code: NumeratorDenominator[(x - 1)(x - 2) / (x - 3) ^ 2]

Extract the numerator and denominator of a symbolic expression:

Wolfram Language code: NumeratorDenominator[Sin[x] ^ a(Sin[x] - 2) ^ -b]

Scope  (7)

Rational numbers:

Wolfram Language code: NumeratorDenominator[3 / 7]

Gaussian rationals:

Wolfram Language code: NumeratorDenominator[3 / 7 + I / 11]

Rational expressions:

Wolfram Language code: NumeratorDenominator[(x - 1) ^ 2 / ((x - 2)(x - 3))]

Select terms with syntactically positive and negative exponents:

Wolfram Language code: expr = a x ^ n y ^ -m Exp[a - b - 2c + 3d]
Wolfram Language code: NumeratorDenominator[expr]

Reconstruct the original expression:

Wolfram Language code: Apply[Divide, %] === expr

NumeratorDenominator automatically threads over lists:

Wolfram Language code: NumeratorDenominator[{1, 2, 3, 4, 5, 6} / 3]

Compute the numerator and denominator over the integers modulo 5:

Wolfram Language code: NumeratorDenominator[24x ^ y (3b) ^ -3, Modulus -> 5]

Compute the numerator and denominator while incorporating common trigonometric identities:

Wolfram Language code: NumeratorDenominator[Tan[x], Trig -> True]

Options  (2)

Modulus  (1)

Find numerators and denominators over integers modulo :

Wolfram Language code: NumeratorDenominator[(1 / 3 x + 3 / 4 y) / (3 / 5x - 1 / 2y), Modulus -> 7]

Trig  (1)

Numerators and denominators of trigonometric functions:

Wolfram Language code: NumeratorDenominator[{Sin[x], Cos[x], Tan[x], Csc[x], Sec[x], Cot[x]}, Trig -> True]

Applications  (2)

Compute the numerator and denominator with respect to a specified modulus:

Wolfram Language code: MatrixPlot@Table[Total@NumeratorDenominator[i / j], {i, 1, 200}, {j, 1, 200}]

Compute the limit at infinity of a rational function:

Wolfram Language code: expr = (x - 1) ^ 2(x - 6) ^ 6 / ((x ^ 3 - 2) ^ 3(x ^ 2 - 3))

Extract the numerator and denominator:

Wolfram Language code: NumeratorDenominator[expr]

Apply Exponent to find the degree of the numerator and denominator:

Wolfram Language code: Exponent[%, x]

We can see that the denominator has a larger degree. Thus, the function approaches zero as grows large:

Wolfram Language code: Plot[expr, {x, 0, 100}]

This can be confirmed using Limit:

Wolfram Language code: Limit[expr, x -> Infinity]

Properties & Relations  (1)

An expression is a quotient of its numerator and denominator:

Wolfram Language code: expr = 5 / 7(x - 1) ^ 2 / (x - 2) ^ 3a ^ b c ^ -d;
Wolfram Language code: {num, den} = NumeratorDenominator[expr]
Wolfram Language code: expr === num / den

Neat Examples  (1)

Plot semitransparent blue squares at positions {i, j} for which i/j is an irreducible fraction:

Wolfram Language code: ListPlot[NumeratorDenominator /@ Flatten@Outer[Divide, Range[20], Range[20]], PlotMarkers -> Blue, AspectRatio -> 1, PlotStyle -> Opacity[0.3]]
Wolfram Research (2019), NumeratorDenominator, Wolfram Language function, https://reference.wolfram.com/language/ref/NumeratorDenominator.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_numeratordenominator, author="Wolfram Research", title="{NumeratorDenominator}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/NumeratorDenominator.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_numeratordenominator, organization={Wolfram Research}, title={NumeratorDenominator}, year={2019}, url={https://reference.wolfram.com/language/ref/NumeratorDenominator.html}, note=[Accessed: 12-June-2026]}