Complex Analysis/Complex numbers

The field of the complex numbersEdit

Historically, it was observed that the equation   has no solution for a real   (since   for  ). Since mathematicians wanted to solve this equation, they just defined a number  , called the imaginary unit, such that  . Of course, there exists no such number. But if we write a two-tuple   with   as   and calculate with these two-tuples using the calculation rule  , that is,

  and  

(where we already wrote a two-tuple   as  , which we will continue to do throughout this book), then the set of all two-tuples with this addition and multiplication forms a field. Indeed, the required axioms for a commutative ring are easy to check, and an inverse of  ,   and   not both zero, is given by

 ,

as can be checked by a direct computation.

Definition 1.1:

A complex number is a two-tuple of real numbers  , written  , where   is called the real part of the number and   is called the imaginary part. The field of the complex numbers, denoted by  , is the set of all such numbers, together with addition

 

and multiplication

 .

Absolute value, conjugationEdit

To each complex number, we can assign an absolute value as follows: A complex number   ( ) is actually a two-tuple  , which is as such an element of  . Now in  , we have the Euclidean absolute value, namely

 ,

and thus we just define:

Definition 1.2:

The absolute value of a complex number   is defined as

 .

Note that the absolute value of the absolute value of a complex number is always a real number, since the root function maps everything in   to   (in fact to  ).

To each complex number   ( ), we also assign a different quantity, which is obtained by reflecting   along the first axis:

Definition 1.3:

Let a complex number   be given. Then the conjugate of  , denoted  , is defined as

 .

That is, the second component changed sign; if, in precise terms,  , then  .

We observe:

Theorem 1.4:

Let two complex number   be given. Then

 .

Proof:

 

and

 . 

With this notation, we can write the absolute value of a complex   only in terms of   without referring to   or  :

Theorem 1.5:

Let a complex number   be given. Then

 .

Here juxtaposition denotes multiplication, as usual (albeit complex multiplication in this case).

Proof:

 . 

From this follows that the absolute value has the following crucial property:

Corollary 1.6:

Let   be complex numbers. Then

 .

Proof:

 

by theorems 1.4 and 1.5. Note that the argument of the square roots was always real. 

The complex planeEdit

Since each complex number is in fact a two-tuple  ,  , the set of all complex numbers   can be visualized as the plane, where   is the first coordinate and   the second coordinate. The situation is indicated in the following picture:

 

The horizontal axis (or  -axis) indicates the real part and the vertical (or  -) axis indicates the imaginary part.

ExercisesEdit

  1. Compute the absolute value of the following complex numbers:  ,  ,  .
  2. Assume that   and   are natural numbers which can be written as the sum of two squares of natural numbers:   and   for some  . Prove that the product   can also be written as the sum of two squares. Hint: Plug in that   (and similarly for  ) and use the rules of computation for complex numbers.
  3. Prove the following relation connecting complex multiplication and the standard scalar product of  :  .
  4. This exercise introduces central concepts in algebra. Make yourself familiar with the concept of a field in algebra. If   is a field, a subfield   is defined to be a subset of   which is closed under the addition, multiplication, subtraction and division inherited from   and contains the elements   and   (ie. the neutral elements of addition and multiplication) of  . Prove:
    1. Let   be a family of subfields of a field  . Prove that the intersection   is also a subfield of  .
    2. Make yourself familiar with the concept of a partially ordered set, and prove that the set of subfields of a given field   is partially ordered by inclusion (ie.  ). Prove that with regard to that order, any family of subfields   has a greatest lower bound.
    3. Prove that a field   has a smallest subfield, called the prime field, and identify the prime field of  .