WOLFRAM

Wolfram Language & System Documentation Center

Rationals

represents the domain of rational numbers, as in x∈Rationals.

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

Rationals

Rationals

represents the domain of rational numbers, as in x∈Rationals.

Details

  • x∈Rationals evaluates immediately if x is a numeric quantity.
  • Simplify[expr∈Rationals,assum] can be used to try to determine whether an expression corresponds to a rational number under the given assumptions.
  • (x1|x2|…)∈Rationals and {x1,x2,…}∈Rationals test whether all xi are rational numbers.
  • The domain of integers is taken to be a subset of the domain of rationals.
  • Rationals is output in StandardForm or TraditionalForm as TemplateBox[{}, Rationals]. This typeset form can be input using rats.

Examples

open all close all

Basic Examples  (3)

2/3 is a rational number:

Wolfram Language code: Element[2 / 3, Rationals]

A sum of rational numbers is a rational number:

Wolfram Language code: Simplify[Element[x + y, Rationals], Element[x | y, Rationals]]

Find rational solutions of an equation:

Wolfram Language code: Reduce[(2x ^ 2 - 1)(4x ^ 2 - 1) == 0, x, Rationals]

Scope  (6)

Test domain membership of a numeric expression:

Wolfram Language code: Element[21 / 32, Rationals]
Wolfram Language code: Element[Sqrt[2], Rationals]
Wolfram Language code: Element[Pi, Rationals]

For some numeric expressions, domain membership may not be resolved automatically:

Wolfram Language code: {Element[7 Sqrt[2] + 2 Sqrt[7], Rationals], Element[Sqrt[5 + 2Sqrt[6]] - Sqrt[2] - Sqrt[3], Rationals]}

Use RootReduce to resolve domain membership of algebraic number expressions:

Wolfram Language code: RootReduce[%]

Make domain membership assumptions:

Wolfram Language code: Refine[Element[x ^ 2 - 2x y + y ^ 3 / 3, Rationals], Element[x | y, Rationals]]

Specify the default domain over which Reduce should work:

Wolfram Language code: Reduce[(x ^ 3 - 8 / 27)(x ^ 3 - 2) == 0, x, Rationals]

Test whether several numbers are rational:

Wolfram Language code: (x | y | 1 / 2)∈Rationals

If any number is explicitly irrational, the result is False:

Wolfram Language code: {x, y, Pi}∈Rationals

TraditionalForm formatting:

Wolfram Language code: Rationals//TraditionalForm

Properties & Relations  (2)

Rationals contains Integers and Primes:

Wolfram Language code: Refine[Element[x, Rationals], Element[x, #]]& /@ {Integers, Primes}

Rationals is contained in Complexes, Reals, and Algebraics:

Wolfram Language code: Refine[Element[x, #], Element[x, Rationals]]& /@ {Complexes, Reals, Algebraics}
Wolfram Research (1999), Rationals, Wolfram Language function, https://reference.wolfram.com/language/ref/Rationals.html (updated 2017).

Text

Wolfram Research (1999), Rationals, Wolfram Language function, https://reference.wolfram.com/language/ref/Rationals.html (updated 2017).

CMS

Wolfram Language. 1999. "Rationals." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/Rationals.html.

APA

Wolfram Language. (1999). Rationals. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Rationals.html

BibTeX

@misc{reference.wolfram_2026_rationals, author="Wolfram Research", title="{Rationals}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Rationals.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_rationals, organization={Wolfram Research}, title={Rationals}, year={2017}, url={https://reference.wolfram.com/language/ref/Rationals.html}, note=[Accessed: 12-June-2026]}