MathAction Sqrt3Demo

Topics

FrontPage
- FriCAS Library
  - FriCAS Library Examples
    - Sqrt3Demo <-- You are here.

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?

fricas

tt1 := -1/sqrt(3)

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

(2)

Type: AlgebraicNumber?

fricas

t2 := sqrt(3)/6

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

(3)

Type: AlgebraicNumber?

fricas

t1+t2

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

(4)

Type: AlgebraicNumber?

fricas

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:

fricas

RAN ==> RECLOS FRAC INT

Type: Void

fricas

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))

fricas

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

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

(7)

Type: RealClosure(Fraction(Integer))

fricas

(x1=xx1)@Boolean

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

(8)

Type: Boolean

It's preferable to give names to the roots:

fricas

s3 := sqrt(3)$RAN

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

(9)

Type: RealClosure(Fraction(Integer))

fricas

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

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

(10)

Type: RealClosure(Fraction(Integer))

AlgebraicNumber? doesn't like the following:

fricas

f3 := sqrt(3,5)$RAN

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

(11)

Type: RealClosure(Fraction(Integer))

fricas

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

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

(12)

Type: RealClosure(Fraction(Integer))

fricas

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

Type: RealClosure(Fraction(Integer))

fricas

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

Type: RealClosure(Fraction(Integer))

fricas

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))

fricas

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

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

(14)

Type: RealClosure(Fraction(Integer))

fricas

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':

fricas

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))

fricas

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

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

(17)

Type: Float

But we get the same without 'RECLOS':

fricas

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

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

(18)

Type: AlgebraicNumber?

fricas

digits(30);

Type: PositiveInteger?

fricas

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

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

(19)

Type: Float

Subject: Be Bold !!

	( 15 subscribers )