• Some computations. An important thing: everything is mathematically typed in Axiom.


    (1) -> 1+1
    
       (1)  2
                                                            Type: PositiveInteger
    (2) -> integrate(1/x^(1/3),x)
    
             3+-+2
            3\|x
       (2)  ------
               2
                                              Type: Union(Expression Integer,...)
      

  • Axiom is capable of nesting a matrix within a matrix as can be seen from this screenshot:


  • Some more complicated computations:



    )cl all
     
       All user variables and function definitions have been cleared.
    
    Word := OrderedFreeMonoid(Symbol)
     
    
       (1)  OrderedFreeMonoid Symbol
                                                                     Type: Domain
    poly:= XPR(Integer,Word)
     
    
       (2)  XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
                                                                     Type: Domain
    p:poly := 2 * x - 3 * y + 1
     
    
       (3)  1 + 2x - 3y
                          Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
    q:poly := 2 * x + 1
     
    
       (4)  1 + 2x
                          Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
    p + q
     
    
       (5)  2 + 4x - 3y
                          Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
    p * q
     
    
                            2
       (6)  1 + 4x - 3y + 4x  - 6y x
                          Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
    (p +q)^2 -p^2 -q^2 - 2*p*q
     
    
       (7)  - 6x y + 6y x
                          Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
    M := SquareMatrix(2,Fraction Integer)
     
    
       (8)  SquareMatrix(2,Fraction Integer)
                                                                     Type: Domain
    poly1:= XPR(M,Word)
     
    
       (9)
       XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
                                                                     Type: Domain
    m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]
     
    
             +1  2+
       (10)  |    |
             +4  8+
                                           Type: SquareMatrix(2,Fraction Integer)
    m2:M := m1 - 5/4
     
    
             +  1    +
             |- -  2 |
             |  4    |
       (11)  |       |
             |     27|
             | 4   --|
             +      4+
                                           Type: SquareMatrix(2,Fraction Integer)
    m3: M := m2**2
     
    
             +129     +
             |---  13 |
             | 16     |
       (12)  |        |
             |     857|
             |26   ---|
             +      16+
                                           Type: SquareMatrix(2,Fraction Integer)
    pm:poly1   := m1*x + m2*y + m3*z - 2/3
     
    
             +  2     +             +  1    +    +129     +
             |- -   0 |             |- -  2 |    |---  13 |
             |  3     |   +1  2+    |  4    |    | 16     |
       (13)  |        | + |    |x + |       |y + |        |z
             |       2|   +4  8+    |     27|    |     857|
             | 0   - -|             | 4   --|    |26   ---|
             +       3+             +      4+    +      16+
    Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
    qm:poly1 := pm - m1*x
     
    
             +  2     +   +  1    +    +129     +
             |- -   0 |   |- -  2 |    |---  13 |
             |  3     |   |  4    |    | 16     |
       (14)  |        | + |       |y + |        |z
             |       2|   |     27|    |     857|
             | 0   - -|   | 4   --|    |26   ---|
             +       3+   +      4+    +      16+
    Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
    qm**3
     
    
       (15)
         +   8      +   +  1  8+    +43   52 +    +  129       +
         |- --   0  |   |- -  -|    |--   -- |    |- ---  - 26 |
         |  27      |   |  3  3|    | 4    3 |    |   8        | 2
         |          | + |      |y + |        |z + |            |y
         |         8|   |16    |    |104  857|    |         857|
         | 0    - --|   |--   9|    |---  ---|    |- 52   - ---|
         +        27+   + 3    +    + 3    12+    +          8 +
       + 
         +  3199     831 +      +  3199     831 +      +  103169     6409 +
         |- ----   - --- |      |- ----   - --- |      |- ------   - ---- |
         |   32       4  |      |   32       4  |      |    128        4  | 2
         |               |y z + |               |z y + |                  |z
         |  831     26467|      |  831     26467|      |   6409     820977|
         |- ---   - -----|      |- ---   - -----|      | - ----   - ------|
         +   2        32 +      +   2        32 +      +     2        128 +
       + 
         +3199   831 +     +103169   6409 +      +103169   6409 +
         |----   --- |     |------   ---- |      |------   ---- |
         | 64     8  | 3   |  256      8  | 2    |  256      8  |
         |           |y  + |              |y z + |              |y z y
         |831   26467|     | 6409   820977|      | 6409   820977|
         |---   -----|     | ----   ------|      | ----   ------|
         + 4      64 +     +   4      256 +      +   4      256 +
       + 
         +3178239   795341 +       +103169   6409 +       +3178239   795341 +
         |-------   ------ |       |------   ---- |       |-------   ------ |
         |  1024      128  |   2   |  256      8  |   2   |  1024      128  |
         |                 |y z  + |              |z y  + |                 |z y z
         |795341   25447787|       | 6409   820977|       |795341   25447787|
         |------   --------|       | ----   ------|       |------   --------|
         +  64       1024  +       +   4      256 +       +  64       1024  +
       + 
         +3178239   795341 +      +98625409  12326223 +
         |-------   ------ |      |--------  -------- |
         |  1024      128  | 2    |  4096       256   | 3
         |                 |z y + |                   |z
         |795341   25447787|      |12326223  788893897|
         |------   --------|      |--------  ---------|
         +  64       1024  +      +   128       4096  +
    Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
     

Axiom can do rule substitution to help simplify complex results as this screenshot shows:

Axiom has a new Firefox front end which will replace the Hyperdoc help system: