FriCAS now modified equality so the problem is not present.
I try :
fricas
a := 3 + sqrt 5
fricas
(-a = (a^2)^(1/2))::Boolean
Type: Boolean
But
fricas
(sqrt 2 = - sqrt 2)::Boolean
Type: Boolean
I expect the same result, and I prefer false.
Do you consider
to be a particular (single) solution
of
, e.g. the positive one, or
the general solution of
this polynomial? I prefer the latter, so I think both
(2) and
(3)
should be
true
.
Axiom's sqrt does not seem to be multiple valued, so therefore I
disagree with you Bill, it cannot be the general solution of the
polynomial. As with most other systems, sqrt is taken to be the
positive root only, and if it comes from the solution of a quadratic,
the +/- multiple valued-ness must be encoded elsewhere.
e.g.
fricas
sqrt(2)::Float
Type: Float
gives only one value.
Also note that axiom is not even self-consistent here:
fricas
-a::Float
Type: Float
fricas
(a^2)^(1/2)::Float
Type: Expression(Float)
which clearly are not equal.
If you write:
fricas
-a=(a^2)^(1/2)
Type: Equation(AlgebraicNumber
?)
fricas
coerce(%%(7))$Equation(AlgebraicNumber)
Type: Boolean
you are asking about equality in the domain of
AlgebraicNumber? (whatever that happens to be). But
if you write:
fricas
coerce(%%(7)::Equation(Float))$Equation(Float)
Type: Boolean
then you are referring to equality in the domain
Float. These need not be the same thing. But I agree
that Axiom does not have an entirely consistent
treatment of sqrt
even within a given domain. :(
From:
http://wiki.axiom-developer.org/axiom--test--1/src/algebra/ConstantSpad
In AlgebraicNumber? we find:
a=b == trueEqual((a-b)::Rep,0::Rep)
In InnerAlgebraicNumber? it says:
trueEqual(a,b) ==
-- if two algebraic numbers have the same norm (after deleting repeated
-- roots, then they are certainly conjugates. Note that we start with a
-- monic polynomial, so don't have to check for constant factors.
-- this will be fooled by sqrt(2) and -sqrt(2), but the = in
-- AlgebraicNumber knows what to do about this.
but unfortunately is seems that the special treatment of sqrt(2)
(and similar algebraic numbers) was never implemented. :(
- InnerAlgebraicNumber? (IAN) is described as
- Algebraic closure of the rational numbers.
- while AlgebraicNumber? (AN) is described as
- Algebraic closure of the rational numbers, with mathematical =
Apparently IAN implements trueEqual
as the equivalence relation
defined by conjugation:
trueEqual : (%,%) -> Boolean
++ trueEqual(x,y) tries to determine if the two numbers are equal
fricas
trueEqual(sqrt(2),-sqrt(2))$IAN
Type: Boolean
This is quite strange. As I understand it,
AN
is almost
IAN
(only three functions have over-riding implementations:
zero?
,
one?
, and
=
) and yet,
=
in
AN
is not transitive, but apparently,
trueEqual
in
IAN
is:
fricas
a:AN:=sqrt(5)+3
fricas
b:=(a^2)^(1/2)
fricas
(a=b)::Boolean
Type: Boolean
fricas
(a=-b)::Boolean
Type: Boolean
fricas
(b=-b)::Boolean
Type: Boolean
fricas
f:IAN:=sqrt(5)+3
Type: InnerAlgebraicNumber
?
fricas
g:=(f^2)^(1/2)
Type: InnerAlgebraicNumber
?
fricas
trueEqual(f,-g)
Type: Boolean
fricas
trueEqual(f,g)
Type: Boolean
fricas
trueEqual(g,-g)
Type: Boolean
But this is deceptive since trueEqual(g, -g)
is not trueEqual(g+g,0)
. Note we cannot use trueEqual(a,b)
or trueEqual(a-b,0)
since there is no coercion from AN
to IAN
. Moreover, trueEqual
is not transitive either.
fricas
trueEqual(g+g,0)
- not the same as trueEqual(g, -g)
Type: Boolean
fricas
trueEqual(-f,g)
Type: Boolean
fricas
trueEqual(f,-f)
Type: Boolean
There is also =
in IAN
, which is implemented using =
in EXPR INT
.
Perhaps trueEqual
should be called conjugate?
instead and exported from AN
, and =
in AN
is really not possible. Also, trueEqual
is only used in constant.spad
and changing its name will not affect any other constructors. But
http://mathworld.wolfram.com/ConjugateElements.html
Two elements alpha, beta of a field K, which is an extension
field of a field F, are called conjugate (over F) if they are
both algebraic over F and have the same minimal polynomial.
...
This conjugacy relation is an equivalence relation on the set
of algebraic elements in a given extension K of the field F.
So if trueEqual
is supposed to implement conjugate?
then
the bug must be in trueEqual
, I think.
Some details of the algorithm are here:
http://www.csd.uwo.ca/~watt/pub/reprints/1995-issac-dynev.pdf
See especially reference [DDD].
Anonymous wrote:
I expect the same result, and I prefer false.
This result is possible using the RealClosure package:
fricas
Ran := RealClosure( Fraction(Integer) )
Type: Type
fricas
a1:Ran := 3 + sqrt 5
fricas
(-a1 = (a1^2)^(1/2))::Boolean
Type: Boolean
fricas
a2:=sqrt(2)$Ran
fricas
a3:=-sqrt(2)$Ran
fricas
(a2=a3)::Boolean
Type: Boolean