18 1. From the Real Numbers to the Complex Numbers

introduced an object i whose square is -1. The ﬁrst approach uses matrices of real

numbers, and it conveys signiﬁcant geometric information. The second approach

fully justiﬁes starting with (15) and (16) and it provides a quintessential example

of what mathematicians call a quotient space.

A matrix approach to C. The matrix deﬁnition of C uses two-by-two matrices

of real numbers and some of the ideas are crucial to subsequent developments. In

this approach we think of C as the set of two-by-two matrices of the form (18),

thereby presaging the Cauchy-Riemann equations which will appear throughout

the book. In some sense we identify a complex number with the operation of mul-

tiplication by that complex number. This approach is especially useful in complex

geometry.

We can regard a complex number as a special kind of linear transformation

of R2. A general linear transformation (x,y) → (ax + cy,bx + dy) is given by a

two-by-two matrix M of real numbers:

(17) M =

a c

b d

.

A complex number will be a special kind of two-by-two matrix. Given a pair of

real numbers a,b and motivated by (13), we consider the mapping L :

R2

→

R2

deﬁned by

L(x,y) = (ax - by,bx + ay).

The matrix representation (in the standard basis) of this linear mapping L is the

two-by-two matrix

(18)

a -b

b a

.

We say that a two-by-two matrix of real numbers satisﬁes the Cauchy-Riemann

equations if it has the form (18). A real linear transformation from R2 to itself whose

matrix representation satisﬁes (18) corresponds to a complex linear transformation

from C to itself, namely multiplication by a + ib.

In this approach we deﬁne a complex number to be a two-by-two matrix (of real

numbers) satisfying the Cauchy-Riemann equations. We add and multiply matrices

in the usual manner. We then have an additive identity 0, a multiplicative identity

1, an analogue of i, and inverses of nonzero elements, deﬁned as follows:

(19) 0 =

0 0

0 0

,

(20) 1 =

1 0

0 1

,

(21) i =

0 -1

1 0

.

If a and b are not both 0, then

a2

+

b2

0. Hence in this case the matrix

(22)

a

a2+b2

b

a2+b2

-b

a2+b2

a

a2+b2