The Complex Numbers

The Complex Numbers
0.1. Definition and Simple Properties
The number system is a tool devised by humans to aid in the description of quantities of the various things humans have to deal with. It has evolved as human culture has evolved, beginning with something very primitive like: 1, 2, 3, many, moving on to the natural numbers, then the integers, the rational numbers, the real numbers and then the complex numbers.
At each stage of development, the number system was expanded in response to the need to describe quantities that the old number system could not. For example, the negative numbers were introduced in order to be able to describe a loss as opposed to a gain, or moving backward rather than forward. The rational numbers were introduced because we do not always deal with whole numbers of things (we have of a pie left). The real number system evolved from the rational number system out of a need to be able to describe such things as the length of the hypotenuse of a right triangle (this involves square roots) and the area or circumference of a circle (this involves ).
In this course, we will assume students are familiar with the real number system and its properties. We will define the complex number system as a needed extension of the real number system and develop its properties. We will then go on to study functions of a complex variable.
The Real Numbers are Insufficient. The complex number system was developed in response to the need for solutions to polynomial equations. The simplest polynomial equation that does not have a solution in the real number system is the equation

which has no real solution because -1 has no real square root. More generally, a quadratic equation

where and are real numbers, formally has two solutions given by the quadratic formula

but these will not be real numbers if 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 . To make this possible, we expand the real number system in the following way, thus creating the complex number system .
Constructing . We begin by adjoining a single new number to our old number system . We will denote it by and declare it to be a square root of -1 . Thus,

Our new number system is to contain both and the new number and it should be closed under addition and multiplication. If it is to be closed under multiplication, we need a number for every real number . Likewise, if it is to be closed under addition, there should be a number in our new number system for each pair of real numbers . It turns out that this is enough. If we define the set of complex numbers to be the set of all symbols of the form where 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 , we begin by noting that, as a set, may be identified with - the set of all pairs of real numbers. Obviously, each pair determines a symbol and vice versa. This identification makes into a vector space over 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

It remains to define a product on .
We have already declared that . 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 and must be

We formalize this conclusion in the following definition.
Definition 1.1.1. We define the system of complex numbers to be the set of all symbols of the form with , with addition and multiplication defined by

and

A complex number of the form , with will be denoted simply as . This identifies as a subset of . Similarly, a complex number of the form with real will be denoted simply as . The numbers of this form are traditionally called the imaginary numbers.
Note that, from the above definition, if , then

and so and 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 and , the latter seems to look best. Thus, we usually write rather than .
Example 1.1.2. If and , find and .
Solution:

Example 1.1.3. Show that the quadratic equation (1.1.1) has solutions which are complex numbers.
Solution: If , the quadratic formula (1.1.2) tells us the solutions are

On the other hand, if , then is positive and has real square roots. By squaring both sides, and using , it is easy to see that

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

That these two numbers are, indeed, solutions to the quadratic equation may be verified by directly substituting them in for 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 with the operations as defined above? The following theorem says they do.
Theorem 1.1.4. If are complex numbers, then
(a) commutative law of addition;
(b) associative law of addition;
(c) commutative law of multiplication;
(d) associative law of multiplication;

Figure 1.1.1. Plot of the Complex Number .
(e) distributive law.
Proof. As far as the operation of addition is concerned, is just , which is a vector space over . Parts (a) and (b) of the theorem follow directly from this. Part (c) is obvious from the definition of multiplication. We will prove part (d) and leave part (e) as an exercise (Exercise 1.1.8).

while

Since the results are the same, the proof of is complete.
The properties described in the above theorem are some of the properties that must hold in a field. A field must also have additive and multiplicative identities that is, elements 0 and 1 which satisfy

and

for every element in the field. That this holds for follows immediately from the fact that is a vector space over .
A field must also have the properties that every element has an additive inverse, that is, an element such that

and every non-zero element has a mutiplicative inverse, that is, an element such that

The first of these follows immediately from the fact that is a vector space over . The second is nearly as easy. If , then a direct calculation shows that

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.