MathAction Sqrt3Demo

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Some demo involving the algebraic number $\sqrt{3}$ .

fricas

t1 := (sqrt(3)-3)*(sqrt(3)+1)/6

$\label{eq1}-{{\sqrt{3}}\over 3}$

(1)

Type: AlgebraicNumber?

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tt1 := -1/sqrt(3)

$\label{eq2}-{{\sqrt{3}}\over 3}$

(2)

Type: AlgebraicNumber?

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t2 := sqrt(3)/6

$\label{eq3}{\sqrt{3}}\over 6$

(3)

Type: AlgebraicNumber?

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t1+t2

$\label{eq4}-{{\sqrt{3}}\over 6}$

(4)

Type: AlgebraicNumber?

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tt1+t2

$\label{eq5}-{{\sqrt{3}}\over 6}$

(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:

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RAN ==> RECLOS FRAC INT

Type: Void

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x1 := (sqrt(3)$RAN-3)*(sqrt(3)$RAN+1)/6

$\label{eq6}{{\left({{1 \over 6}\ {\sqrt{3}}}-{1 \over 2}\right)}\ {\sqrt{3}}}+{{1 \over 6}\ {\sqrt{3}}}-{1 \over 2}$

(6)

Type: RealClosure(Fraction(Integer))

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xx1 := -1/sqrt(3)$RAN

$\label{eq7}-{{1 \over 3}\ {\sqrt{3}}}$

(7)

Type: RealClosure(Fraction(Integer))

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(x1=xx1)@Boolean

$\label{eq8} \mbox{\rm true}$

(8)

Type: Boolean

It's preferable to give names to the roots:

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s3 := sqrt(3)$RAN

$\label{eq9}\sqrt{3}$

(9)

Type: RealClosure(Fraction(Integer))

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(s3-3)*(s3+1)/6

$\label{eq10}-{{1 \over 3}\ {\sqrt{3}}}$

(10)

Type: RealClosure(Fraction(Integer))

AlgebraicNumber? doesn't like the following:

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f3 := sqrt(3,5)$RAN

$\label{eq11}\root{5}\of{3}$

(11)

Type: RealClosure(Fraction(Integer))

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f25 := sqrt(1/25,5)$RAN

$\label{eq12}\root{5}\of{1 \over{25}}$

(12)

Type: RealClosure(Fraction(Integer))

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f32 := sqrt(32/5,5)$RAN;

Type: RealClosure(Fraction(Integer))

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f27 := sqrt(27/5,5)$RAN;

Type: RealClosure(Fraction(Integer))

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expr1 := sqrt(f32-f27,3)

$\label{eq13}\root{3}\of{-{\root{5}\of{{27}\over 5}}+{\root{5}\of{{32}\over 5}}}$

(13)

Type: RealClosure(Fraction(Integer))

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expr2 := (1+f3-f3^2)

$\label{eq14}-{{\root{5}\of{3}}^{2}}+{\root{5}\of{3}}+ 1$

(14)

Type: RealClosure(Fraction(Integer))

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expr1 - f25*expr2

$\label{eq15}0$

(15)

Type: RealClosure(Fraction(Integer))

Although the main point of RECLOS is supposed do be mathematical ordering and approximation, I could not find a convincing example. From the "examples" section of 'RECLOS':

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s := sqrt(190)$RAN+sqrt(1751)$RAN-sqrt(208)$RAN-sqrt(1698)$RAN

$\label{eq16}-{\sqrt{1698}}-{\sqrt{208}}+{\sqrt{1751}}+{\sqrt{190}}$

(16)

Type: RealClosure(Fraction(Integer))

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approximate(s, 10^-15)::Float

$\label{eq17}-{0.2341060678 \<u> 6455900874 E - 10}$

(17)

Type: Float

But we get the same without 'RECLOS':

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t := sqrt(190)+sqrt(1751)-sqrt(208)-sqrt(1698)

$\label{eq18}{\sqrt{1751}}-{\sqrt{1698}}+{\sqrt{190}}-{4 \ {\sqrt{13}}}$

(18)

Type: AlgebraicNumber?

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digits(30);

Type: PositiveInteger?

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numeric t - approximate(s, 10^-30)::Float

$\label{eq19}-{0.5522026336 \<u> 5 E - 29}$

(19)

Type: Float

Subject: Be Bold !!

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