# Complex Analysis/Complex numbers

## The field of the complex numbers

Historically, it was observed that the equation ${\displaystyle x^{2}=-1}$  has no solution for a real ${\displaystyle x}$  (since ${\displaystyle x^{2}\geq 0}$  for ${\displaystyle x\in \mathbb {R} }$ ). Since mathematicians wanted to solve this equation, they just defined a number ${\displaystyle i}$ , called the imaginary unit, such that ${\displaystyle i^{2}=-1}$ . Of course, there exists no such number. But if we write a two-tuple ${\displaystyle (a,b)}$  with ${\displaystyle a,b\in \mathbb {R} }$  as ${\displaystyle a+ib}$  and calculate with these two-tuples using the calculation rule ${\displaystyle i^{2}=-1}$ , that is,

${\displaystyle (a+ib)+(c+id)=(a+c)+i(b+d)}$  and ${\displaystyle (a+ib)(c+id)=(ac-bd)+i(ad+bc)}$

(where we already wrote a two-tuple ${\displaystyle (x,y)}$  as ${\displaystyle x+iy}$ , 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 ${\displaystyle x+iy}$ , ${\displaystyle x}$  and ${\displaystyle y}$  not both zero, is given by

${\displaystyle (x+iy)^{-1}={\frac {x-iy}{x^{2}+y^{2}}}}$ ,

as can be checked by a direct computation.

Definition 1.1:

A complex number is a two-tuple of real numbers ${\displaystyle (a,b)}$ , written ${\displaystyle a+ib}$ , where ${\displaystyle a}$  is called the real part of the number and ${\displaystyle b}$  is called the imaginary part. The field of the complex numbers, denoted by ${\displaystyle \mathbb {C} }$ , is the set of all such numbers, together with addition

${\displaystyle (a+ib)+(c+id)=(a+c)+i(b+d)}$

and multiplication

${\displaystyle (a+ib)(c+id)=(ac-bd)+i(ad+bc)}$ .

## Absolute value, conjugation

To each complex number, we can assign an absolute value as follows: A complex number ${\displaystyle z=x+iy}$  (${\displaystyle x,y\in \mathbb {R} }$ ) is actually a two-tuple ${\displaystyle (x,y)}$ , which is as such an element of ${\displaystyle \mathbb {R} ^{2}}$ . Now in ${\displaystyle \mathbb {R} ^{2}}$ , we have the Euclidean absolute value, namely

${\displaystyle \|(x,y)\|_{2}={\sqrt {x^{2}+y^{2}}}}$ ,

and thus we just define:

Definition 1.2:

The absolute value of a complex number ${\displaystyle z=x+iy\in \mathbb {C} }$  is defined as

${\displaystyle |z|:={\sqrt {x^{2}+y^{2}}}}$ .

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 ${\displaystyle \mathbb {R} _{\geq 0}}$  to ${\displaystyle \mathbb {R} }$  (in fact to ${\displaystyle \mathbb {R} _{\geq 0}}$ ).

To each complex number ${\displaystyle z=x+iy}$  (${\displaystyle x,y\in \mathbb {R} }$ ), we also assign a different quantity, which is obtained by reflecting ${\displaystyle z}$  along the first axis:

Definition 1.3:

Let a complex number ${\displaystyle z=x+iy\in \mathbb {C} }$  be given. Then the conjugate of ${\displaystyle z}$ , denoted ${\displaystyle {\overline {z}}}$ , is defined as

${\displaystyle {\overline {z}}:=x-iy}$ .

That is, the second component changed sign; if, in precise terms, ${\displaystyle z=(x,y)}$ , then ${\displaystyle {\overline {z}}=(x,-y)}$ .

We observe:

Theorem 1.4:

Let two complex number ${\displaystyle z=x+iy,w=a+ib\in \mathbb {C} }$  be given. Then

${\displaystyle {\overline {zw}}={\overline {z}}{\overline {w}}}$ .

Proof:

${\displaystyle {\overline {zw}}={\overline {xa-yb+i(xb+ya)}}=xa-yb-i(xb+ya)}$

and

${\displaystyle {\overline {z}}{\overline {w}}=(x-iy)(a-ib)=ax-yb-i(xb+ya)}$ .${\displaystyle \Box }$

With this notation, we can write the absolute value of a complex ${\displaystyle z=x+iy}$  only in terms of ${\displaystyle z}$  without referring to ${\displaystyle x}$  or ${\displaystyle y}$ :

Theorem 1.5:

Let a complex number ${\displaystyle z=x+iy\in \mathbb {C} }$  be given. Then

${\displaystyle |z|={\sqrt {z{\overline {z}}}}}$ .

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

Proof:

${\displaystyle {\sqrt {z{\overline {z}}}}={\sqrt {(x+iy)(x-iy)}}={\sqrt {x^{2}-(iy)^{2}}}={\sqrt {x^{2}-i^{2}y^{2}}}={\sqrt {x^{2}+y^{2}}}=|z|}$ .${\displaystyle \Box }$

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

Corollary 1.6:

Let ${\displaystyle z,w\in \mathbb {C} }$  be complex numbers. Then

${\displaystyle |zw|=|z||w|}$ .

Proof:

${\displaystyle |zw|={\sqrt {zw{\overline {zw}}}}={\sqrt {zw{\overline {z}}{\overline {w}}}}={\sqrt {z{\overline {z}}}}{\sqrt {w{\overline {w}}}}=|z||w|}$

by theorems 1.4 and 1.5. Note that the argument of the square roots was always real.${\displaystyle \Box }$

## The complex plane

Since each complex number is in fact a two-tuple ${\displaystyle (x,y)}$ , ${\displaystyle x,y\in \mathbb {R} }$ , the set of all complex numbers ${\displaystyle x+iy}$  can be visualized as the plane, where ${\displaystyle x}$  is the first coordinate and ${\displaystyle y}$  the second coordinate. The situation is indicated in the following picture:

The horizontal axis (or ${\displaystyle x}$ -axis) indicates the real part and the vertical (or ${\displaystyle y}$ -) axis indicates the imaginary part.

## Exercises

1. Compute the absolute value of the following complex numbers: ${\displaystyle 3+4i}$ , ${\displaystyle 3+2i}$ , ${\displaystyle 1+{\frac {1}{2}}i}$ .
2. Assume that ${\displaystyle m}$  and ${\displaystyle n}$  are natural numbers which can be written as the sum of two squares of natural numbers: ${\displaystyle m=a^{2}+b^{2}}$  and ${\displaystyle n=c^{2}+d^{2}}$  for some ${\displaystyle a,b,c,d\in \mathbb {N} }$ . Prove that the product ${\displaystyle m\cdot n}$  can also be written as the sum of two squares. Hint: Plug in that ${\displaystyle a^{2}+b^{2}=|a+ib|^{2}}$  (and similarly for ${\displaystyle c,d}$ ) and use the rules of computation for complex numbers.
3. Prove the following relation connecting complex multiplication and the standard scalar product of ${\displaystyle \mathbb {R} ^{2}}$ : ${\displaystyle \langle (a,b),(x,y)\rangle =\operatorname {Re} \left[(a-ib)(x+iy)\right]}$ .
4. This exercise introduces central concepts in algebra. Make yourself familiar with the concept of a field in algebra. If ${\displaystyle \mathbb {F} }$  is a field, a subfield ${\displaystyle \mathbb {E} \subseteq \mathbb {F} }$  is defined to be a subset of ${\displaystyle \mathbb {F} }$  which is closed under the addition, multiplication, subtraction and division inherited from ${\displaystyle \mathbb {F} }$  and contains the elements ${\displaystyle 0}$  and ${\displaystyle 1}$  (ie. the neutral elements of addition and multiplication) of ${\displaystyle \mathbb {F} }$ . Prove:
1. Let ${\displaystyle (\mathbb {E} _{\alpha })_{\alpha \in A}}$  be a family of subfields of a field ${\displaystyle \mathbb {F} }$ . Prove that the intersection ${\displaystyle \bigcap _{\alpha \in A}\mathbb {E} _{\alpha }}$  is also a subfield of ${\displaystyle \mathbb {F} }$ .
2. Make yourself familiar with the concept of a partially ordered set, and prove that the set of subfields of a given field ${\displaystyle \mathbb {F} }$  is partially ordered by inclusion (ie. ${\displaystyle \mathbb {E} \leq \mathbb {E} ':\Leftrightarrow \mathbb {E} \subseteq \mathbb {E} '}$ ). Prove that with regard to that order, any family of subfields ${\displaystyle (\mathbb {E} _{\alpha })_{\alpha \in A}}$  has a greatest lower bound.
3. Prove that a field ${\displaystyle \mathbb {F} }$  has a smallest subfield, called the prime field, and identify the prime field of ${\displaystyle \mathbb {C} }$ .