# 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 ${\displaystyle \mathbb {C} }$.