1.1. Definition and Simple Properties 5

and every non-zero element z has a mutiplicative inverse, that is, an element z−1

such that

(1.1.6) z ·

z−1

= 1.

The first of these follows immediately from the fact that C is a vector space

over R. The second is nearly as easy. If z = x + iy = 0, then a direct calculation

shows that

z−1

=

x − iy

x2 + y2

=

x

x2 + y2

−

y

x2 + y2

i

satisfies (1.1.6). We conclude:

Theorem 1.1.5. With addition and multiplication defined as in Definition 1.1.1,

the complex numbers form a field.

Complex Conjugation and Modulus.

Definition 1.1.6. If z = x + iy is a complex number, then its complex conjugate,

denoted z, is defined by

z = x − iy,

while its modulus, denoted |z|, is defined by

|z| = x2 + y2.

Note that the modulus, as defined above, is just the usual Euclidean norm in

the vector space R2. Thus, if z1,z2 ∈ C, then |z1 − z2| is the Euclidean distance

from z1 to z2. The term modulus is traditional, but the terms norm and absolute

value are also commonly used to mean the same thing. We will use all three.

Note also that the two solutions of a quadratic equation with real coeﬃcients

given in Example 1.1.3 are complex conjugates of each other. Thus, the solutions

to a quadratic equation with real coeﬃcients occur in conjugate pairs. Quadratic

equations with complex coeﬃcients also have roots and they are also given by the

quadratic formula. However, we cannot prove this until we prove that every complex

number has a square root. In fact, in Section 1.4 we will prove that every complex

number has roots of all orders.

For a complex number z = x + iy, the real number x is called the real part

of z and is denoted Re(z), while the number y is called the imaginary part of z

and is denoted Im(z). In graphing complex numbers using a rectilinear coordinate

system, x determines the coordinate on the horizontal axis, while y determines the

coordinate on the vertical axis.

Note that a complex number z is real if and only if z = z, and it is purely

imaginary if and only if z = −z. Note also, that if z = x + iy, then

Re(z) = x =

z + z

2

and Im(z) = y =

z − z

2i

.

The elementary properties of conjugation and modulus are gathered together in the

next theorem.

Theorem 1.1.7. If z and w are complex numbers, then

(a) z = z;