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Reals

represents the domain of real numbers, as in x∈Reals.

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Basic Examples  
Scope  
Properties & Relations  
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Reals

Reals

represents the domain of real numbers, as in x∈Reals.

Details

  • x∈Reals evaluates immediately if x is a numeric quantity.
  • Simplify[expr∈Reals,assum] can be used to try to determine whether an expression corresponds to a real number under the given assumptions.
  • (x1|x2|…)∈Reals and {x1,x2,…}∈Reals test whether all xi are real numbers.
  • Within Simplify and similar functions, objects that satisfy inequalities are always assumed to be real.
  • Reals is output in StandardForm or TraditionalForm as TemplateBox[{}, Reals]. This typeset form can be input using reals.

Examples

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Basic Examples  (3)

is a real number:

Wolfram Language code: Element[Pi ^ E, Reals]

If is a real number, so is :

Wolfram Language code: Simplify[Element[E ^ Sin[x], Reals], Element[x, Reals]]

Find real solutions of an equation:

Wolfram Language code: Reduce[x ^ 5 - 2x + 1 == 0, x, Reals]

Scope  (5)

Test domain membership of a numeric expression:

Wolfram Language code: Element[Sin[7], Reals]

Make domain membership assumptions:

Wolfram Language code: Refine[Im[x + I y], Element[x | y, Reals]]
Wolfram Language code: Integrate[Abs[1 - Abs[x + 2]], x, Assumptions -> Element[x, Reals]]

Specify the default domain over which a function should work:

Wolfram Language code: Reduce[E ^ x - 2x == 3, x, Reals]
Wolfram Language code: Resolve[Exists[x, y == Sqrt[x]], Reals]
Wolfram Language code: PiecewiseExpand[Abs[x + Sign[y]], Reals]

Test whether several numbers are real numbers:

Wolfram Language code: (x | y | 1)∈Reals

If any number is explicitly non-real, the result is False:

Wolfram Language code: {x, y, I}∈Reals

TraditionalForm formatting:

Wolfram Language code: Reals//TraditionalForm

Properties & Relations  (3)

Reals contains Rationals, Integers, and Primes:

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

Reals is contained in Complexes:

Wolfram Language code: Refine[Element[x, Complexes], Element[x, Reals]]

Reals neither contains nor is contained in Algebraics:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_reals, author="Wolfram Research", title="{Reals}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/Reals.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_reals, organization={Wolfram Research}, title={Reals}, year={2017}, url={https://reference.wolfram.com/language/ref/Reals.html}, note=[Accessed: 13-June-2026]}