2 1. The Complex Numbers

where a, b and c are real numbers, formally has two solutions given by the quadratic

formula

(1.1.2) x =

−b ±

√

b2 − 4ac

2a

,

but these will not be real numbers if

b2

− 4ac is negative. If we could take square

roots of negative numbers, then the quadratic formula would give us solutions to

(1.1.1) for all choices of real coeﬃcients a, b, c. To make this possible, we expand

the real number system in the following way, thus creating the complex number

system C.

Constructing C. We begin by adjoining a single new number to our old number

system R. We will denote it by i and declare it to be a square root of −1. Thus,

i2

= −1.

Our new number system is to contain both R and the new number i and it should be

closed under addition and multiplication. If it is to be closed under multiplication,

we need a number iy for every real number y. Likewise, if it is to be closed under

addition, there should be a number x + iy in our new number system for each pair

of real numbers (x, y). It turns out that this is enough. If we define the set of

complex numbers C to be the set of all symbols of the form x + iy where (x, y) is

a pair of real numbers, and if we define addition and multiplication appropriately,

then the resulting number system is a field in which every polynomial equation has

a root. We will be a long time proving the latter half of this statement, but it is

not hard to prove the first part.

To define the operations of addition and multiplication in C, we begin by noting

that, as a set, C may be identified with

R2

– the set of all pairs (x, y) of real

numbers. Obviously, each pair (x, y) determines a symbol x + iy and vice versa.

This identification makes C into a vector space over R and gives us operations of

addition and scalar multiplication by reals which satisfy the usual associative and

distributive rules. The resulting operation of addition is

(x1

+ iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2).

It remains to define a product on C.

We have already declared that i2 = −1. If we also require that the associative

and distributive laws of multiplication should hold and that the multiplication of

real numbers should remain as before, then the product of two complex numbers

x1 + iy1 and x2 + iy2 must be

(x1 +iy1)(x2 +iy2) = x1x2 +ix1y2 +iy1x2

+i2y1y2

= (x1x2 −y1y2)+i(x1y2 +y1x2).

We formalize this conclusion in the following definition.

Definition 1.1.1. We define the system C of complex numbers to be the set of

all symbols of the form x + iy with (x, y) ∈

R2,

with addition and multiplication

defined by

(x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2)

and

(x1 + iy1)(x2 + iy2) = (x1x2 − y1y2) + i(x1y2 + y1x2).