Algebraic numbers

 

In Axiom

For comparison with other CAS choose from: Derive Macsyma Maple Mathematica Reduce

inputs outputs

    sqrt2:= rootOf(sqrt2**2-2)

    sqrt2
                                                        Type: AlgebraicNumber

    1/(sqrt2+1)

        1
    ---------
    sqrt2 + 1
                                                        Type: AlgebraicNumber

    (x**2+2*sqrt2*x+2)/(x+sqrt2)

    x + sqrt2
                                    Type: Fraction Polynomial AlgebraicNumber

    on gcd;

    (x**3+(sqrt2-2)*x**2-(2*sqrt2+3)*x-3*sqrt2)/(x**2-2);

       2
      x  - 2x - 3
    ---------------
        - sqrt2 - 4
    x + -----------
        2sqrt2 + 1
                                    Type: Fraction Polynomial AlgebraicNumber

    normalize 

       2                   2
    (2x  - 4x - 6)sqrt2 + x  - 2x - 3
    ---------------------------------
          (2x - 1)sqrt2 + x - 4
                                                     Type: Expression Integer

    sqrt(x**2-2*sqrt2*x*y+2*y**2)

     +-----------------------+
     |                 2    2
    \|- 2x y sqrt2 + 2y  + x
                                                     Type: Expression Integer

• multiple algebraic extensions

      sqrt5:= rootOf(sqrt5**2-5)
  

      sqrt5
                                                          Type: AlgebraicNumber

      cbrt3:= rootOf(cbrt3**3-3)
  

      cbrt3
                                                          Type: AlgebraicNumber

      cbrt3**3
  

      3
                                                          Type: AlgebraicNumber

      sqrt5**2;
  

      5
                                                          Type: AlgebraicNumber

  
      cbrt3;
  

      cbrt3
                                                          Type: AlgebraicNumber

  
      sqrt(x**2+2*(sqrt5-cbrt3)*x+5-2*sqrt5*cbrt3+cbrt3**2)
  

       +-------------------------------------------------+
       |                            2               2
      \|(- 2cbrt3 + 2x)sqrt5 + cbrt3  - 2x cbrt3 + x  + 5
                                                       Type: Expression Integer