is_real method should exist for elements of AA, ZZ, QQ,..., and be fixed where it is already defined

[seblabbe]

The function is_real is useful but it is not practical because it it not implemented for every types of numbers in Sage. And I do not want to do type checking before being able to use it in my code.

sage: CC(3).is_real()
True
sage: RR(3).is_real()
True
sage: SR(3).is_real()
True
sage: ZZ(3).is_real()
Traceback (most recent call last):
...
AttributeError: 'sage.rings.integer.Integer' object has no attribute 'is_real'
sage: QQ(3).is_real()
Traceback (most recent call last):
...
AttributeError: 'sage.rings.rational.Rational' object has no attribute 'is_real'
sage: AA(3).is_real()
Traceback (most recent call last)
...
AttributeError: 'AlgebraicReal' object has no attribute 'is_real'
sage: AlgebraicNumber(3).is_real()
Traceback (most recent call last)
...
AttributeError: 'AlgebraicNumber' object has no attribute 'is_real'

sage: RDF(3).is_real()
AttributeError: 'sage.rings.real_double.RealDoubleElement' object has no attribute 'is_real'

sage: QQbar(3).is_real()
AttributeError: 'AlgebraicNumber' object has no attribute 'is_real'

I am sure I forget some other rings...

We need more of this kind of consistency in Sage. is_imaginary is another example.

The .is_real() method does not handle NaN and +/-Infinity correctly, for example:

sage: CC(NaN).is_real()
True

Component: algebra

Issue created by migration from https://trac.sagemath.org/ticket/16436

[sagetrac-tmonteil]

comment:1

Hi Sebastien. Note that you can also use coercion with the poorest real representation:

sage: AA(3) in RealField(2)
True
sage: CC(3) in RealField(2)
True
sage: RR(3) in RealField(2)
True
sage: SR(3) in RealField(2)
True
sage: ZZ(3) in RealField(2)
True
sage: QQ(3) in RealField(2)
True
sage: AA(3) in RealField(2)
True
sage: QQbar(3) in RealField(2)
True

though you need to be careful with NaN and +/-Infinity that also belong to RealField(2). By the way:

sage: SR(Infinity).is_real()
True

[seblabbe]

comment:3

Replying to @sagetrac-tmonteil:

Hi Sebastien. Note that you can also use coercion with the poorest real representation:

I have been using monkey tricks like that for too long now like .n().imag() == 0. I want a clean solution. It should be normal to ask if some eigenvalue of some matrix is a real number.

sage: M = identity_matrix(RR,4)
sage: [a.is_real() for a in M.eigenvalues()]
[True, True, True, True]

sage: M = identity_matrix(ZZ,4)
sage: [a.is_real() for a in M.eigenvalues()]
Traceback (most recent call last):
...
AttributeError: 'sage.rings.rational.Rational' object has no attribute 'is_real'

sage: M = random_matrix(ZZ,4)
sage: [a.is_real() for a in M.eigenvalues()]
Traceback (most recent call last):
...
AttributeError: 'AlgebraicNumber' object has no attribute 'is_real'

[sagetrac-tmonteil]

comment:5

If, as i proposed on sage-devel we could have an overlay (say, RR) for real fields, the syntax CC(3) in RR would be very readable. The only problem with CC(3) in RealField(2) is about NaN and +/-Infinity, but you should notice that the same problem exists with the current .is_real() methods, so the solution is not clean either.

[rwst]

comment:8

Replying to @sagetrac-tmonteil:

Hi Sebastien. Note that you can also use coercion with the poorest real representation:

sage: AA(3) in RealField(2)
True

See #12967 and #17984 for why x in RR will determine elementship in RR not in the mathematical R. For this is_real is meant.

[kedlaya]

comment:9

One clean way to implement is_real might be to compare x with its conjugate, which is implemented quite often:

sage: def is_real(x):
....:     return(x == x.conjugate())
....:
sage: is_real(ZZ(3))
True
sage: is_real(QQ(3))
True
sage: is_real(AA(3))
True
sage: is_real(RR(3))
True
sage: is_real(CC(3))
True

One could handle is_imaginary similarly.