🚀 go-pugleaf

     I don't mean to have kept people in suspense -- here is the basic idea
about using matrices to calcuate with complex numbers.  Consider the two
following 2x2 matrices  I  and  J :

                                          1    0                   0   -1
                                                        and
                                          0    1                   1     0

Of course  I  is just the 2x2 identity matrix.  Note that, using normal
matrix multiplication,  JJ = - I .  We can use the linear properties of
matrices to form the "cross matrix"   a I + b J, which looks like this:

        a     - b
                                    Basically, this matrix corresponds to the
        b        a                complex number  a + bj, or (a,b).

What we really have here is an _isomorphism_, under which various
operations in the two systems (complex number arithmetic and the
algebra of matrices) correspond to each other.  The beauty of it is that
the isomorphism is so natural!

Addition of cross-matrices corresponds to addition of the corresponding
complex numbers, and matrix multiplication to complex multiplication.
There are many further neat correspondences -- as one example, think
about the meaning of transposing a cross-matrix.  Also, the determinant
of a cross-matrix (or at least its square root) is especially significant.
This gives you a way to begin viewing this representation from a polar
viewpoint as well.  Oh yes, there's the matrix inverse...

There is just room on a TI-81 to get an interesting and useful collection
of cross-matrices in hand.  Probably the best strategy is to write a
program STUFF.XY to, say, create a cross-matrix [A] from the current
values of X and Y.  You can then copy [A] to [B] or [C], create an additional
 cross-matrix [A], and start performing calculations (including some big
ones).  There is much more room on the TI-82, and a lot of rich classroom
exploration can be performed "live".

For further details and ideas for exploration of mathematics in the
classroom at this level, see my paper "Complex Numbers in Matrix Form"
in the Proceedings of the Fifth Annual International Conference on
Technology in Collegiate Mathematics (Addison Wesley, 1993).  I would
be interested to hear what additional ideas others may come up with by
following this approach.

None of this applies to the TI-85, does it, since complex numbers are a
built-in data type.  On the other hand, if you are interested in playing
around with quaternions, consider creating an appropriate set (4) of
unit 4x4 real matrices (or 2x2 complex-valued matrices).  Hmmm....


||       RWW Taylor      rwtnts@rit.edu
||       National Technical Institute for the Deaf
||       Rochester Institute of Technology
||       Rochester NY 14623