Some people think of complex numbers as the points in the plane, in real 2space. This is the approach Anton takes, which makes the definition of their multiplication a bit hard to motivate, and harder to remember.
Other people think of them as all formal expressions of the
form a + bi with real a and b,
subject to the familiar rules of arithmetic, along with the proviso that i^{2} = –1. This approach
seems palatable until you wonder if it isn’t all just chickenscratches on
paper.
Still others (yours
truly among them) prefer a more concrete approach, as follows.
We already know all
about matrix arithmetic for matrices of all shapes and sizes. Let’s reserve the
term complex number for any 2x2 matrix

[ 
a 
b 
] 
A = 




–b 
a 
with both diagonal
entries the same and each offdiagonal entry the negative of the other.
Quick checks (do
them!) reveal that, whenever A and B are two complex numbers, so
are:
(i) their ordinary
matrix sum A + B;
(ii) their ordinary
matrix difference A – B;
(iii) their ordinary
matrix product AB;
(iv) the transpose A^{T} of the complex number A; and
(v) all the real
multiples kA of the complex number A (here k may be any
real number).
Moreover, despite
the fact that, in general, the order of multiplication can make a big
difference when multiplying 2x2 matrices (AC need not
coincide with CA, as easy examples show – take, for example, any complex
number A as above having b ≠ 0, and test
with the matrix (not, in fact, a complex number)

[ 
0 
1 
] 
C = 




1 
0 
to discover just how
different CA can, in general, be from AC), nonetheless
(vi) for complex
numbers A and B one really will always have AB = BA (try it!).
Two more facts worth
noting:
(vii) for A
as above, det(A) = a^{2} + b^{2}, and
(viii) for A
as above, A^{T} A = det(A) Id_{2}_{x2 }.
From (vii) it
follows easily that
(ix) the only
complex number A with det(A) = 0 is the zero
matrix,
and from (viii) and
(ix) it becomes easy to check that, unless A is the zero matrix, the
only case in which det(A) has no real reciprocal,
(x) the matrix
(1/det(A)) A^{T} serves as (twosided) inverse for A.
How does this
approach jibe with the first two approaches mentioned? Here’s how:
(1) The top row of a
complex number “is” a point in the plane, and each point in the plane is
the top row of one of our matrixstyle complex numbers; and Anton’s seemingly ad
hoc definition of multiplication for his complex numbers (cf. §10.1)
is just what the top row of the matrix product of our matrix complex numbers
becomes (likewise for sums, differences, real multiples – all behave
similarly); and our transpose A^{T} simply has in its top row what Anton calls
the complex conjugate of the top row of A.
(2) If we write 1 for Id_{2}_{x2} , and i for the matrix given by

[ 
0 
1 
] 
i = 




1 
0 
then i^{2} = –1 and, for the complex number given by the
matrix A shown above, we have A = a1 + bi. So that alternate approach is not
mere chickenscratches on the paper.
———
Prepared 1920 Feb 2004.
Questions? Comments? Typos? Errata? Let me know, please.