1.1. Definition and Simple Properties 3

A complex number of the form x + i0, with x ∈ R will be denoted simply as x.

This identifies R as a subset of C. Similarly, a complex number of the form 0 + iy

with y real will be denoted simply as iy. The numbers of this form are traditionally

called the imaginary numbers.

Note that, from the above definition, if x, y ∈ R, then

yi = (y + i0)(0 + i) = 0 + iy = iy

and so x + iy and x + yi are the same complex number. Which form is used to

describe this number is usually dictated by which looks best typographically. When

specific numbers replace x and y, the latter seems to look best. Thus, we usually

write 2 + 3i rather than 2 + i3.

Example 1.1.2. If z1 = 5 + 2i and z2 = 3 − 4i, find z1 + z2 and z1z2.

Solution:

z1 + z2 = 5 + 3 + (2 − 4)i = 8 − 2i,

z1z2 = (5 · 3 − 2 · (−4)) + (5 · (−4) + 3 · 2)i = 23 − 14i.

Example 1.1.3. Show that the quadratic equation (1.1.1) has solutions which are

complex numbers.

Solution: If b2 − 4ac ≥ 0, the quadratic formula (1.1.2) tells us the solutions

are

−b +

√

b2 − 4ac

2a

and

−b −

√

b2 − 4ac

2a

.

On the other hand, if b2 − 4ac 0, then 4ac − b2 is positive and has real square

roots. By squaring both sides, and using i2 = −1, it is easy to see that

± b2 − 4ac = ±i 4ac − b2.

This suggests that the solutions to the quadratic equation in this case are the two

complex numbers

−

b

2a

+ i

√

4ac − b2

2a

and −

b

2a

− i

√

4ac − b2

2a

.

That these two numbers are, indeed, solutions to the quadratic equation may be

verified by directly substituting them in for x in (1.1.1). We leave this as an exercise

(Exercise 1.1.7).

Field Properties. In our definition of the product of two complex numbers,

we were guided by the desire to have the usual rules of arithmetic hold – that is,

the commutative and associative laws for addition and for multiplication and the

distributive law. Did we succeed? These are some of the properties of a field. Do

these laws actually hold in C with the operations as defined above? The following

theorem says they do.

Theorem 1.1.4. If z1,z2,z3 are complex numbers, then

(a) z1 + z2 = z2 + z1 – commutative law of addition;

(b) (z1 + z2) + z3 = z1 + (z2 + z3) – associative law of addition;

(c) z1z2 = z2z1 – commutative law of multiplication;

(d) (z1z2)z3 = z1(z2z3) – associative law of multiplication;