5. Alternative definitions of C 19
makes sense and satisfies the Cauchy-Riemann equations. Note that
(23)
a -b
b a
a
a2+b2
b
a2+b2
-b
a2+b2
a
a2+b2
=
1 0
0 1
= 1.
Thus (22) yields the formula
a-ib
a2+b2
for the reciprocal of the nonzero complex number
a + ib, expressed instead in matrix notation.
Thus C can be defined to be the set of two-by-two matrices satisfying the
Cauchy-Riemann equations. Addition and multiplication are defined as usual for
matrices. The additive identity 0 is given by (19) and the multiplicative identity
1 is given by (20). The resulting mathematical system is a field, and the element
i defined by (21) satisfies
i2
+ 1 = 0. This method of defining C should appease
readers who on philosophical grounds question the existence of complex numbers.
I
Exercise 1.24. Show that the square of the matrix in (21) is the negative of
the matrix in (20); in other words, show that
i2
= -1.
I
Exercise 1.25. Suppose a2 + b2 = 1 in (18). What is the geometric meaning of
multiplication by L?
I
Exercise 1.26. Suppose b = 0 in (18). What is the geometric meaning of
multiplication by L?
I
Exercise 1.27. Show that there are no real numbers x and y such that
1
x
+
1
y
=
1
x + y
.
Show on the other hand that there are complex numbers z and w such that
(24)
1
z
+
1
w
=
1
z + w
.
Describe all pairs (z,w) satisfying (24).
I
Exercise 1.28. Describe all pairs A and B of two-by-two matrices of real num-
bers for which A-1 and B-1 exist and
A-1
+
B-1
= (A +
B)-1.
Remark 5.1. Such pairs of n-by-n matrices exist if and only if n is even; the reason
is intimately connected with complex analysis.
An algebraic definition of C. We next describe C as a quotient space. This
approach allows us to regard a complex number as an expression a + ib, where
i2
= -1, as we wish to do. We will therefore define C in terms of the polynomial ring
divided by an ideal. The reader may skip this section without loss of understanding.
First we recall the general notion of an equivalence relation. Let S be a set.
We can think of an equivalence relation on S as being defined via a symbol
∼.
= We
decree that certain pairs of elements s,t S are equivalent; if so, we write s

= t.
The following three axioms must hold:
For all s S, s

=
s (reflexivity).
For all s,t S, s

=
t if and only if t

=
s (symmetry).
For all s,t,u S, s

=
t and t

=
u together imply s

=
u (transitivity).
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