What are the different real numbers in Sage ?

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No computer can grasp the genuine real numbers, so I am trying to understand how sage handles them (and going to write a small tutorial about this). Here is a list of approaching objects i found:

Exact subfields of the reals:

• QQ = RationalField() The genuine rationals. Denominators can be arbitrary big, the computations are exact, so the rounding error is only made when approaching a real by a rational, not during furher computations. Notice that the size of the denominators may grow along the computations, which can therefore become very slow.

• NumberField? number fields described by an irreducible polynomial.

• AA = AlgebraicField() The genuine algebraic numbers. Computations are exact.

• SR = sage.symbolic.ring.SymbolicRing() This is the field made of expressions like sqrt(2+pi). It handles a lot of real numbers but seems very slow. Computations are not necessarily exact, since this field contains floating points real numbers. Is there a way to express any algebraic number in SR ?

• CFF = ContinuedFractionField() reals are represented as a finite list of convergents. Claims it is exact but it actually approaches real numbers by rationals since the list of convergents are finite. However, we could imagine a future version using iterators instead of lists.

sage: CFF(sqrt(2))^2

[1, 1, 1491038293021224]

Floating point numbers:

• RDF = RealDoubleField() Those are the of double precision floating numbers the processor. Definitely the fastest, but a rounding is done at each computation.

• RR = RealField() Floating number with arbitrary (still fixed) many bits of precision. http://www.sagemath.org/doc/reference/rings_numerical/sage/rings/real_mpfr.html Slower than RDF (in particular, atlas won't be used with them). Notice that changing precision back and forth is a bad idea. Notice that NaN and Infinity belong to RDF and RR.

• RLF = RealLazyField() like RealField() but can wait that the user fixes the precision to get one. http://www.sagemath.org/doc/reference/rings_numerical/sage/rings/real_lazy.html

sage: a = RLF(pi)

sage: RealField(100)(a+2)

5.1415926535897932384626433833

• RIF = RealIntervalField() A real number is represented as an interval containing it, and whose extremities are RealFields elements http://www.sagemath.org/doc/reference/rings_numerical/sage/rings/real_mpfi.html

• RLF.interval_field() mixture of the two previous ones.

• sage.rings.real_lazy.LazyAlgebraic

Relations between those fields

• Two fields approaching the real numbers K and L can be compared using the function composite_field(K,L), that finds the "best" field containing both K and L.

sage: composite_field(RDF,QQbar)

Complex Double Field

• Some maps are also defined to embed an abstract number field into the real line.

• The method algebraic_dependency() that allows to guess whether a floating number corresponds to some algebraic number of a given degree.

Is this list complete or am i missing other real representations ? Im i wrong in the previous descriptions ? Could you order them by speed, by number of methods available (without coercion) ? Do you have any remark that could help the beginner to understand the subtilities in the use of those fields in Sage ?

Given an algebraic ... (more)