go to www.bildungsgueter.de table of contents

First Steps with Axiom

To start Axiom, open a terminal window and enter this command:

axiom -noht

Now, the termial window acts as the Axiom dialog window.

Now that you have a dialog window, try to type in a simple formula, let's say a sum with five elements:

The most elementary simplifications are always performed. When possible, the elements of a sum are collected:

-> 2*x + 5*y - x + 2*z + 2*y
   2z + 7y + x
       Type: Polynomial(Integer)

Note that the variables are now alphabetically sorted. This does not seem to be a simplification, but it improves the readability of complicated formulae.

Note also that Axiom answers not only the simplified expression, but also its type. Types play a crucial role in Axiom, their importance will be discussed in a separate chapter later on.

The coloring of the input and the type information shown here follows the usage of the Texmacs frontend. When used in a terminal window, Axiom does not use text coloring.

If you do not want to see the type information, you can turn it off with this command:

-> )set messages type off

To turn the type information on again, enter

-> )set messages type on

The collection of similar elements of a sum works also when functions occur in a sum:

-> 2*sin(x) + cos(x) - sin(x)
   sin(x) + cos(x)
        Type: Expression(Integer)

Square roots are simplified by factoring of perfect squares

-> sqrt(150)
   

  5
  
  
    6
  

        AlgebraicNumber

Polynomial Arithmetic with Remainders

(1) -> poly := x**6 + 19*x**5 + x**4 - 14*x**3 - x**2 - 3*x + 1
(1) ->
         6      5    4      3    2
   (1)  x  + 19x  + x  - 14x  - x  - 3x + 1
                                                    Type: Polynomial(Integer)

(2) -> polym := poly :: Polynomial(PrimeField(11))
(2) ->
         6     5    4     3      2
   (2)  x  + 8x  + x  + 8x  + 10x  + 8x + 1
                                             Type: Polynomial(PrimeField(11))
(3) -> factor(%)
(3) ->
                 2            3     2
   (3)  (x + 1)(x  + 5x + 3)(x  + 2x  + 3x + 4)
                                   Type: Factored(Polynomial(PrimeField(11)))
(4) -> expand(%)
(4) ->
         6     5    4     3      2
   (4)  x  + 8x  + x  + 8x  + 10x  + 8x + 1
                                             Type: Polynomial(PrimeField(11))
(5) ->

Eine Besonderheit von Axiom ist, dass dieses Programm ein Typsystem verwendet, das stets angibt, mit welcher Art von Ausdrücken gerechnet wird. Dieses Typsystem bildet bis zu einem gewissen Grad die in der Algebra übliche Praxis ab, für einen Ausdruck eine Menge anzugeben, in der dieser Ausdruck enthalten ist. Diese Herangehensweise ist in der praktischen Mathematik eher unüblich und daher erklärungs- und gewöhnungsbedürftig.

table of contents