Edit detail for Sqrt3Demo revision 3 of 4
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Editor: kratt6
Time: 2008/08/16 06:37:55 GMT-7 |
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changed: - - Alternatively, we could also use Renaud Rioboo's 'RECLOS package, which has both mathematical equality and ordering. Unfortunately, it is not as easy to use - most importantly, you have to "name" your real roots, if you want simple answers: \begin{axiom} RAN ==> RECLOS FRAC INT x1 := (sqrt(3)$RAN-3)*(sqrt(3)$RAN+1)/6 xx1 := -1/sqrt(3)$RAN (x1=xx1)@Boolean \end{axiom} It's preferable to give names to the roots: \begin{axiom} s3 := sqrt(3)$RAN (s3-3)*(s3+1)/6 \end{axiom} 'AlgebraicNumber' doesn't like the following: \begin{axiom} f3 := sqrt(3,5)$RAN f25 := sqrt(1/25,5)$RAN; f32 := sqrt(32/5,5)$RAN; f27 := sqrt(27/5,5)$RAN; sqrt(f32-f27,3)-f25*(1+f3-f3^2) \end{axiom} Although the main point of 'RECLOS' is supposed do be mathematical ordering and approximation, I could not find a convincing exmaple. From the 'examples' section of 'RECLOS': \begin{axiom} s := sqrt(190)$RAN+sqrt(1751)$RAN-sqrt(208)$RAN-sqrt(1698)$RAN approximate(s, 10^-15)::Float \end{axiom} But we get the same without 'RECLOS': \begin{axiom} t := sqrt(190)+sqrt(1751)-sqrt(208)-sqrt(1698) digits(30); numeric t - approximate(s, 10^-30)::Float \end{axiom}
Some demo involving the algebraic number 
.
axiom
t1 := (sqrt(3)-3)*(sqrt(3)+1)/6
 | (1) |
Type: AlgebraicNumber?
axiom
tt1 := -1/sqrt(3)
 | (2) |
Type: AlgebraicNumber?
axiom
t2 := sqrt(3)/6
 | (3) |
Type: AlgebraicNumber?
axiom
t1+t2
 | (4) |
Type: AlgebraicNumber?
axiom
tt1+t2
 | (5) |
Type: AlgebraicNumber?
Note that in PanAxiom? the above are not generic expressions but of type AlgebraicNumber?.
Alternatively, we could also use Renaud Rioboo's 'RECLOS package, which has both mathematical equality and ordering. Unfortunately, it is not as easy to use - most importantly, you have to "name" your real roots, if you want simple answers:
axiom
RAN ==> RECLOS FRAC INT
Type: Void
axiom
x1 := (sqrt(3)$RAN-3)*(sqrt(3)$RAN+1)/6
 | (6) |
Type: RealClosure? Fraction Integer
axiom
xx1 := -1/sqrt(3)$RAN
 | (7) |
Type: RealClosure? Fraction Integer
axiom
(x1=xx1)@Boolean
 | (8) |
Type: Boolean
It's preferable to give names to the roots:
axiom
s3 := sqrt(3)$RAN
 | (9) |
Type: RealClosure? Fraction Integer
axiom
(s3-3)*(s3+1)/6
 | (10) |
Type: RealClosure? Fraction Integer
AlgebraicNumber doesn't like the following:
axiom
f3 := sqrt(3,5)$RAN
 | (11) |
Type: RealClosure? Fraction Integer
axiom
f25 := sqrt(1/25,5)$RAN;
Type: RealClosure? Fraction Integer
axiom
f32 := sqrt(32/5,5)$RAN;
Type: RealClosure? Fraction Integer
axiom
f27 := sqrt(27/5,5)$RAN;
Type: RealClosure? Fraction Integer
axiom
sqrt(f32-f27,3)-f25*(1+f3-f3^2)
 | (12) |
Type: RealClosure? Fraction Integer
Although the main point of RECLOS is supposed do be mathematical ordering and approximation, I could not find a convincing exmaple. From the examples section of 'RECLOS':
axiom
s := sqrt(190)$RAN+sqrt(1751)$RAN-sqrt(208)$RAN-sqrt(1698)$RAN
 | (13) |
Type: RealClosure? Fraction Integer
axiom
approximate(s, 10^-15)::Float
 | (14) |
Type: Float
But we get the same without 'RECLOS':
axiom
t := sqrt(190)+sqrt(1751)-sqrt(208)-sqrt(1698)
 | (15) |
Type: AlgebraicNumber?
axiom
digits(30);
Type: PositiveInteger?
axiom
numeric t - approximate(s, 10^-30)::Float
 | (16) |
Type: Float