Complex Analysis/Complex numbers

The field of the complex numbers

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

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

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

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

as can be checked by a direct computation.

Definition 1.1:

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

$(a+ib)+(c+id)=(a+c)+i(b+d)$

and multiplication

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

Absolute value, conjugation

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

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

and thus we just define:

Definition 1.2:

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

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

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

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

Definition 1.3:

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

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

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

We observe:

Theorem 1.4:

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

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

Proof:

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

and

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

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

Theorem 1.5:

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

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

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

Proof:

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

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

Corollary 1.6:

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

$|zw|=|z||w|$.

Proof:

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

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

The complex plane

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

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

Exercises

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