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Wolfram Language & System Documentation Center
Rational
  • See Also
    • Rationals
    • Integer
    • Real
    • Numerator
    • Denominator
    • Rationalize
    • RealValuedNumberQ
  • Related Guides
    • Representation of Numbers
    • Atomic Elements of Expressions
    • WDF (Wolfram Data Framework)
  • Tech Notes
    • Types of Numbers
    • See Also
      • Rationals
      • Integer
      • Real
      • Numerator
      • Denominator
      • Rationalize
      • RealValuedNumberQ
    • Related Guides
      • Representation of Numbers
      • Atomic Elements of Expressions
      • WDF (Wolfram Data Framework)
    • Tech Notes
      • Types of Numbers

Rational

is the head used for rational numbers.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
History
Cite this Page
BUILT-IN SYMBOL
  • See Also
    • Rationals
    • Integer
    • Real
    • Numerator
    • Denominator
    • Rationalize
    • RealValuedNumberQ
  • Related Guides
    • Representation of Numbers
    • Atomic Elements of Expressions
    • WDF (Wolfram Data Framework)
  • Tech Notes
    • Types of Numbers
    • See Also
      • Rationals
      • Integer
      • Real
      • Numerator
      • Denominator
      • Rationalize
      • RealValuedNumberQ
    • Related Guides
      • Representation of Numbers
      • Atomic Elements of Expressions
      • WDF (Wolfram Data Framework)
    • Tech Notes
      • Types of Numbers

Rational

Rational

is the head used for rational numbers.

Details

  • You can enter a rational number in the form n/m.
  • The pattern object _Rational can be used to stand for a rational number. It cannot stand for a single integer.
  • You have to use Numerator and Denominator to extract parts of Rational numbers.

Examples

open all close all

Basic Examples  (1)

Enter a rational number:

Wolfram Language code: 22 / 7

Rational is the Head for rational numbers:

Wolfram Language code: Head[%]

Scope  (7)

Enter a rational number with very big integers in the numerator and denominator:

Wolfram Language code: 1237918739182739817238917127398123 / 12809812308120812038038101

Rational numbers are represented with the smallest possible positive denominator:

Wolfram Language code: 7 / 49

The FullForm of a rational number is Rational[numerator,denominator]:

Wolfram Language code: FullForm[22 / 7]

Enter a rational using the FullForm:

Wolfram Language code: Rational[22, 7]

You have to use Numerator and Denominator to extract parts of Rational numbers:

Wolfram Language code: r = 22 / 7;
Wolfram Language code: {Numerator[r], Denominator[r]}

Part does not work:

Wolfram Language code: r[[1]]

The pattern object _Rational can be used to stand for a rational number:

Wolfram Language code: MatchQ[22 / 7, _Rational]

It cannot stand for a single integer:

Wolfram Language code: MatchQ[6 / 3, _Rational]

A rule that replaces all rationals with their reciprocals:

Wolfram Language code: rule = x_Rational :> Denominator[x] / Numerator[x];
Wolfram Language code: f[22 / 7, 201 / 64, x / y] /. rule

An alternate way to write the rule:

Wolfram Language code: f[22 / 7, 201 / 64, x / y] /. Rational[n_, d_] :> d / n

Applications  (1)

Define a function that only applies to rational numbers:

Wolfram Language code: f[r_Rational] := Module[{x = Numerator[r], y = Denominator[r]}, (x ^ 2 + 2y ^ 2) / (2x y)]
Wolfram Language code: Nest[f, 3 / 2, 6]

This is a close approximation to :

Wolfram Language code: Block[{$MaxExtraPrecision = ∞}, N[% - Sqrt[2], 20]]

An alternative definition of the function:

Wolfram Language code: g[Rational[x_, y_]] := (x ^ 2 + 2y ^ 2) / (2x y)
Wolfram Language code: Nest[g, 3 / 2, 6]

Properties & Relations  (5)

Rationals are numbers:

Wolfram Language code: NumberQ[22 / 7]

Rationals are atomic objects with no subexpressions:

Wolfram Language code: AtomQ[22 / 7]

Rationals are exact numbers:

Wolfram Language code: ExactNumberQ[22 / 7]

Denominator of a rational is positive:

Wolfram Language code: r = -14 / 21
Wolfram Language code: Denominator[r]

Numerator and Denominator of a rational are relatively prime:

Wolfram Language code: GCD[Numerator[r], Denominator[r]]

Use Rationals to indicate assumptions and domain conditions:

Wolfram Language code: Reduce[1 / 2 - 6x + 10x ^ 2 - x ^ 99 / 2 + x ^ 100 == 0, x, Rationals]

Possible Issues  (1)

Numbers entered in the form n/m only become Rational numbers on evaluation:

Wolfram Language code: SetAttributes[f, HoldAll]; f[x_Rational] := Numerator[x] - Denominator[x]
Wolfram Language code: f[22 / 7]
Wolfram Language code: f[Evaluate[22 / 7]]

The unevaluated form is expressed in terms of Times and Power:

Wolfram Language code: FullForm[HoldForm[22 / 7]]

See Also

Rationals  Integer  Real  Numerator  Denominator  Rationalize  RealValuedNumberQ

Tech Notes

    ▪
  • Types of Numbers

Related Guides

    ▪
  • Representation of Numbers
  • ▪
  • Atomic Elements of Expressions
  • ▪
  • WDF (Wolfram Data Framework)

History

Introduced in 1988 (1.0)

Wolfram Research (1988), Rational, Wolfram Language function, https://reference.wolfram.com/language/ref/Rational.html.

Text

Wolfram Research (1988), Rational, Wolfram Language function, https://reference.wolfram.com/language/ref/Rational.html.

CMS

Wolfram Language. 1988. "Rational." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Rational.html.

APA

Wolfram Language. (1988). Rational. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Rational.html

BibTeX

@misc{reference.wolfram_2026_rational, author="Wolfram Research", title="{Rational}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Rational.html}", note=[Accessed: 02-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_rational, organization={Wolfram Research}, title={Rational}, year={1988}, url={https://reference.wolfram.com/language/ref/Rational.html}, note=[Accessed: 02-October-2026]}

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