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As the responses show, there are a number of quick "probably real" tests. In general, the problem is undecidable, however. This is an easy corollary of Richardson's theorem, which says that it is impossible to decide if two real expressions $x$ and $y$ are equal. Assuming Richardson's theorem, note that $(x-y)i$ is real if and only if $x=y$.

As a more mundane example, that arises in common practice with Mathematica, consider the polynomial $p(x)=13x^3-13x-1$. It's easy to see that all three roots are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

Given the undecidability, perhaps you might as well use some numerical evaluation, as in

realQ[x_] := Internal`RealValuedNumericQ[Chop[N[x]]]

This seems to work for all the examples on this page, although, we could certainly find examples where it fails.

As the responses show, there are a number of quick "probably real" tests. In general, the problem is undecidable, however. This is an easy corollary of Richardson's theorem, which says that it is impossible to decide if two real expressions $x$ and $y$ are equal. Assuming Richardson's theorem, note that $(x-y)i$ is real if and only if $x=y$.

As a more mundane example, that arises in common practice with Mathematica, consider the polynomial $p(x)=13x^3-13x-1$. It's easy to see that all three roots are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

As the responses show, there are a number of quick "probably real" tests. I'm sure that the problem is undecidable, in general, though. For example, it's easy to see that all three of the roots of the polynomial $p(x)=13x^3-13x-1$ are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

Not surprisingly,

realQ[x_] := Internal`RealValuedNumericQ[Chop[N[x]]]

works for these examples but must certainly be fraught with its own problems.

As the responses show, there are a number of quick "probably real" tests. In general, the problem is undecidable, however. This is an easy corollary of Richardson's theorem, which says that it is impossible to decide if two real expressions $x$ and $y$ are equal. Assuming Richardson's theorem, note that $(x-y)i$ is real if and only if $x=y$.

As a more mundane example, that arises in common practice with Mathematica, consider the polynomial $p(x)=13x^3-13x-1$. It's easy to see that all three roots are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

Given the undecidability, perhaps you might as well use some numerical evaluation, as in

realQ[x_] := Internal`RealValuedNumericQ[Chop[N[x]]]

This seems to work for all the examples on this page, although, we could certainly find examples where it fails.

As the responses show, there are a number of quick "probably real" tests. I'm sure that the problem is undecidable, in general, though. For example, it's easy to see that all three of the roots of the polynomial $p(x)=13x^3-13x-1$ are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

As the responses show, there are a number of quick "probably real" tests. I'm sure that the problem is undecidable, in general, though. For example, it's easy to see that all three of the roots of the polynomial $p(x)=13x^3-13x-1$ are real (even if they don't look it), yet they don't pass any of the test here.

roots = x /. Solve[13 x^3 - 13 x - 1 == 0, x]
Internal`RealValuedNumericQ /@ roots

Not surprisingly,

realQ[x_] := Internal`RealValuedNumericQ[Chop[N[x]]]

works for these examples but must certainly be fraught with its own problems.