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).