Complex Analysis/Complex numbers

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The field of the complex numbers[edit | edit source]

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, conjugation[edit | edit source]

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 plane[edit | edit source]

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:

Simple illustration of a complex number.svg

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

Exercises[edit | edit source]

  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 .