Real Numbers, Euclidean Domains, Complex Numbers

Real Numbers

📖 (Lindenmann.) There does not exist an equation of the form

$x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{0} = 0$

for $a_{i} \in Q$ , such that $x = π$ is a solution.

📖 For addition $+$ , multiplication $\cdot$ and an order relation $\leq$ :

Addition: $R \times R \to + R$
Multiplication: $R \times R \to \cdot R$
Order relation: $\leq$

$* R$ is a commutative, ordered field that is order-complete.*

💡 Since $R$ is a commutative field the field axioms hold:

Addition:

Associativity: $x + (y + z) = (x + y) + z, \forall x, y, z \in R$
Neutral Element: $x + 0 = x, \forall x \in R$
Inverse Element: $\forall x \exists y (x + y = 0), x, y \in R$

where $y$ is uniquely determined and is denoted $- x$ .
Commutativity: $x + z = z + x, \forall x, z \in R$

Multiplication:

Associativity: $x \cdot (y \cdot z) = (x \cdot y) \cdot z, \forall x, y, z \in R$
Neutral Element: $x \cdot 1 = x, \forall x \in R$
Inverse Element: $\forall x \exists y (x \cdot y = 1), x, y \in R, x \neq = 0$

where $y$ is uniquely determined and denoted $x^{- 1}$ .
Commutativity: $x \cdot z = z \cdot x, \forall x, z \in R$

From the axioms on addition and multiplication the distributive law can be defined:

Distributivity: $x \cdot (y + z) = x \cdot y + x \cdot z, \forall x, y, z \in R$

Order:

Reflexivity: $x \leq x, \forall x \in R$
Transitivity: $x \leq y \land y \leq z ⟹ x \leq z$
Antisymmetry: $x \leq y \land y \leq x ⟹ x = y$
Totally-Ordered: $\forall x, y \in R$ either $x \leq y$ or $y \leq x$ holds, i.e., all pairs of elements are comparable.

The order is compatible with the field axioms.

Compatibility:

$x \leq y ⟹ x + z \leq y + z, \forall x \forall y \forall z \in R$
$x \cdot y \geq 0, \forall x \geq 0, y \geq 0$

Recall the definitions for fields, partial orders and totally-ordered-ness from Algebra and Sets, Relations, and Functions:

💡 A field is a nontrivial commutative ring $F$ in which every nonzero element is a unit, i.e., $F^{*} = F ∖ {0}$ (in other words, $⟨ F ∖ {0}; \cdot,^{- 1}, 1 ⟩$ is an abelian group).

💡 A partial order (or simply an order relation) on a set $A$ is a relation that is reflexive, antisymmetric, and transitive. A set $A$ together with a partial order $⪯$ on $A$ is called a partially ordered set (or simply poset) and is denoted as $(A; ⪯)$ .

💡 If any two elements of a poset $(A; ⪯)$ are comparable, then $A$ is called totally ordered (or linearly ordered) by $⪯$ .

Also note that $Q$ , the rational numbers, also form a commutative field that is totally-ordered under the same operations and orders. The only property separating $R$ from $Q$ is order-completeness.

💡 Order-Completeness: Let $A, B \subseteq R$ such that

$A \neq = \emptyset$ , $B \neq = \emptyset$
$\forall a \in A$ and $\forall b \in B$ it holds that: $a \leq b$

Then there exists $c \in R$ such that $\forall a \in A a \leq c$ and $\forall b \in B c \leq b$ .

💡 Notation:

$a > b ⟺ a \geq b \land a \neq = b$

📎 Corollary:

Uniqueness of additive and multiplicative Inverses.
$0 \cdot x = 0, \forall x \in R$
$* (- 1) \cdot x = - x, \forall x \in R$ , especially $(- 1)^{2} = 1$ *
$y \geq 0 ⟺ (- y) \leq 0$
$* y^{2} \geq 0, \forall y \in R$ , especially $1 = 1 \cdot 1 \geq 0$ *
$x \leq y \land u \leq v ⟹ x + u \leq y + v$
$0 \leq x \leq y \land 0 \leq u \leq v ⟹ x \cdot u \leq y \cdot v$

📎 Archimedean Principle:

Let $x \in R$ with $x > 0$ and $y \in R$ . Then there exists $n \in N$ such that $y \leq n \cdot x$ .

📖 For every $t \geq 0, t \in R$ the equation $x^{2} = t$ has a solution in $R$ .

💡 For $t \geq 0$ there exists exactly one solution of $x^{2} = t$ such that $x \geq 0$ . It is denoted by $t$ .

💡 Let $x, y \in R$ :

$max (x, y) = {x y if if y \leq x x \leq y$
$min (x, y) = {y x if if y \leq x x \leq y$
The absolute value of a number $x \in R$ : $∣ x ∣ = max (x, - x)$

📖 Properties of the absolute value of a number:

$∣ x ∣ \geq 0, \forall x \in R$
$∣ x y ∣ = ∣ x ∣ ∣ y ∣, \forall x, y \in R$
$∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣, \forall x, y \in R$
$∣ x + y ∣ \geq ∣∣ x ∣ - ∣ y ∣∣, \forall x, y \in R$

📖 Young’s Inequality:

$\forall ϵ > 0, \forall x, y \in R$ it holds that:

$2 \cdot ∣ x y ∣ \leq ϵ x^{2} + \frac{1}{ϵ} y^{2} .$

💡 We introduce two new symbols:

$- \infty and + \infty$

with the convention that

$- \infty < x < + \infty$

Then, an interval is a subset of $R$ of the form:

for $a \leq b$ in $R$ :

$[a, b] = {x \in R ∣ a \leq x \leq b} [a, b) = {x \in R ∣ a \leq x < b} (a, b] = {x \in R ∣ a < x \leq b} (a, b) = {x \in R ∣ a < x < b}$
for $a \in R$ :

$[a, + \infty) = {x \in R ∣ a \leq x} (a, + \infty) = {x \in R ∣ a < x} (- \infty, a] = {x \in R ∣ a \geq x} (- \infty, a) = {x \in R ∣ a > x}$
$(- \infty, + \infty) = R$ .

💡 Let $A \subseteq R$ a subset of the reals.

$c \in R$ is an upper bound of $A$ if $\forall a \in A$ it holds that $a \leq c$ . The set $A$ is called bounded from above if there exists an upper bound of $A$ .
$c \in R$ is a lower bound of $A$ if $\forall a \in A$ it holds that $c \leq a$ . The set $A$ is called bounded from below if there exists a lower bound of $A$ ,
An element $m \in R$ is called a maximum of $A$ if $m \in A$ and $m$ is an upper bound of $A$ .
An element $m \in R$ is called a minimum of $A$ if $m \in A$ and $m$ is a lower bound of $A$ .

💡 Notation: If $A$ has a maximum (minimum) it is denoted $max A$ ( $min A$ ).

📖 Let $A \subseteq R$ , $A \neq = \emptyset$ .

Let $A$ be bounded from above. Then there exists a least upper bound of $A$ :

$c = sup A$

called the supremum of $A$ .
Let $A$ be bounded from below. Then there exists a greatest lower bound $A$ :

$c = in f A$

called the infimum of $A$ .

💡 It follows immediately that:

Let $A \subseteq R$ be bounded from above. Then the set of upper bounds of $A$ equals the interval $[sup A, + \infty)$ .
Let $A \subseteq R$ be bounded from below. Then the set of lower bounds of $A$ equals the interval $(- \infty, in f A]$ .

📎 Let $A \subseteq B \subseteq R$ be subsets of $R$ .

If $B$ is bounded from above, then $sup A \leq sup B$ .
If $B$ is bounded from below, then $in f B \leq in f A$ .

💡 Convention: If $A$ is not bounded from above (resp. not bounded from below), we define

$sup A = + \infty (resp. in f A = - \infty) .$

Cardinality

💡 Cardinality:

Two sets $X$ , $Y$ are called equinumerous, if there exists a bijection $f : X ⟶ Y$ .
A set $X$ is finite, if and only if $X = \emptyset$ or $\exists n \in N$ such that $X$ and ${1, 2, 3, \dots, n}$ are equinumerous. In the former case, the cardinality of $X$ , $card X = 0$ and in the latter it is $card X = n$ .
A set $X$ is called countable, if it is finite or if it is equinumerous to $N$ .

💡 Countability of Sets:

Two sets $A$ amd $B$ are equinumerous, denoted $A \sim B$ , if there exists a bijection $A ⟶ B$ .
The set $B$ dominates the set $A$ , denoted $A ⪯ B$ , if $A \sim C$ for some subset $C \subseteq B$ or, equivalently, if there exists an injective function $A ⟶ B$ .
A set $A$ is called countable if $A ⪯ N$ , and uncountable otherwise.

📖 A set $A$ is countable if and only if it is finite or if $A \sim N$ .

📖 (Cantor.)

$R$ is not countable.

Euclidean Space

Let $n \geq 1$ and let

$R^{n} = {(x_{1}, \dots, x_{n}) ∣ \forall j x_{j} \in R}$

denote the $n$ -fold cartesian product of $R$ . Let $x = (x_{1}, \dots, x_{n})$ , $y = (y_{1}, \dots, y_{n})$ and $α \in R$ . Then we define:

$x + y = (x_{1} + y_{1}, \dots, x_{n} + y_{n}), α \cdot x = (α x_{1}, \dots, α x_{n})$

Linear algebra shows that $R^{n}$ forms a vector space with the operations:

$R \times R^{n} (α, x) ⟶ R ⟼ α \cdot x$

$R^{n} \times R^{n} (x, y) ⟶ R^{n} ⟼ x + y$

The scalar product of two vectors $x, y \in R^{n}$ is defined by

$⟨ x, y ⟩ = j = 1 \sum n x_{j} y_{j} .$

The following properties hold:

$\forall x, y \in R^{n} ⟨ x, y ⟩ = ⟨ y, x ⟩$ (symmetry)
$\forall α_{1}, α_{2} \in R, \forall x_{1}, x_{2}, y \in R^{n} ⟨ α_{1} x_{1} + α_{2} x_{2}, y ⟩ = α_{1} ⟨ x_{1}, y ⟩ + α_{2} ⟨ x_{2}, y ⟩$ (bilinearity)
$⟨ x, x ⟩ = \sum_{j = 1}^{n} x_{j}^{2} \geq 0$ with equality if and only if $x = 0$ (positive definiteness)

The Norm of the vector $x$ is $∣∣ x ∣∣ = ⟨ x, x ⟩$ .

📖 (Cauchy-Schwarz).

$∣⟨ x, y ⟩∣ \leq ∣∣ x ∣∣ \cdot ∣∣ y ∣∣ \forall x, y \in R^{n}$

📖 Statements for the norm:

$* ∣∣ x ∣∣ \geq 0$ with equality if and only if $x = 0$ .*
$∣∣ α \cdot x ∣∣ = ∣ α ∣ \cdot ∣∣ x ∣∣ \forall α \in R, \forall x \in R^{n}$
$∣∣ x + y ∣∣ \leq ∣∣ x ∣∣ + ∣∣ y ∣∣ \forall x, y \in R^{n}$

The cross product between two vectors $a, b \in R^{3}$ is defined by

$\times : R^{3} \times R^{3} (a, b) ⟶ R^{3} ⟼ a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}) .$

$a, b$ and $a \times b$ form a right-handed system. It holds that: $∣∣ a \times b ∣∣ =$ area of the parallelogram spanned by $a$ and $b$ .

The cross product has the following properties: $\forall a, b, c \in R^{3}$ :

$(a + b) \times c = a \times c + b \times c$ (distributivity)
$a \times b = - b \times a$ (antisymmetry)
$a \times (b \times c) + c \times (a \times b) + b \times (c \times a) = 0$ (Jacobi-Identity)

Complex Numbers

We define the following multiplication on $R^{2}$ :

$(x_{1}, y_{1}) \cdot (x_{2}, y_{2}) = (x_{1} x_{2} - y_{1} y_{2}, x_{1} y_{2} + x_{2} y_{1}) .$

Then, it holds that:

$(0, 0) (x, y) = (0, 0), (1, 0) (x, y) = (x, y) If (x, y) \neq = (0, 0) : (x, y) (\frac{x}{x ^{2} + y ^{2}}, \frac{- y}{x ^{2} + y ^{2}}) = (1, 0)$

📖 $* R^{2}$ with vector addition $+$ and the defined multiplication $\cdot$ is a commutative field with multiplicative neutral element $(1, 0)$ and additive neutral element $(0, 0)$ .*

$⟨ R^{2}, +, \cdot ⟩$ is called the field of the complex numbers and is denoted with $C$ .

The transformation ${R x ⟶ ⟼ R \times R (x, 0)$ follows the field axioms. Using this transformation, we can identify $R$ as a sub-field of $C$ .

Let $i = (0, 1)$ . Then, every element $z \in C$ has a unique representation

$z = x + y \cdot i with x, y \in R$

Note:

$i^{2} = (0, 1) (0, 1) = (- 1, 0) = - (1, 0)$

By applying the distributive laws we get:

$z_{1} z_{2} = (x_{1} + y_{1} i) \cdot (x_{2} + y_{2} i) = x_{1} x_{2} + x_{1} y_{2} i + y_{1} i x_{2} + y_{1} i y_{2} i = x_{1} x_{2} + (x_{1} y_{2} + y_{1} x_{2}) \cdot i + y_{1} y_{2} i^{2} = (x_{1} x_{2} - y_{1} y_{2}) + (x_{1} y_{2} + y_{1} x_{2}) \cdot i .$

This representation is also consistent with the definition of the above product.

Terminology: Let $z = x + y i :$

$x = Re z is called the real part of z, y = Im z is called the imaginary part of z$

Complex conjugate: For $z = x + y i$ we define

$\overset{z}{ˉ} = x - y i .$

📖 Complex conjugate rules:

$\overline{(z_{1} + z_{2})} = \overset{z_{1}}{ˉ} + \overset{z_{2}}{ˉ} \forall z_{1}, z_{2} \in C$
$\overline{(z_{1} z_{2})} = \overset{z_{1}}{ˉ} \overset{z_{2}}{ˉ} \forall z_{1}, z_{2} \in C$
$z \overset{z}{ˉ} = x^{2} + y^{2} = ∣∣ z ∣ ∣^{2}$

It follows from 3. that for $z \neq = 0$ the multiplicative inverse is given by

$z^{- 1} = \frac{z ˉ}{∣∣ z ∣ ∣ ^{2}} .$

Polar Form

Let $r = ∣∣ z ∣∣$ , then $x = r cos ϕ$ and $y = r sin ϕ$ where $θ mod 2 πk$ , $k \in Z$ .

In the representation

$z = r (cos ϕ + i sin ϕ)$

$r = ∣∣ z ∣∣$ denotes the absolute value and $ϕ$ the argument of the complex number $z$ .

Now let $z_{1} = r_{1} (cos ϕ_{1} + i sin ϕ_{1})$ and $z_{2} = r_{2} (cos ϕ_{2} + i sin ϕ_{2})$ . Then it follows:

$z_{1} z_{2} = r_{1} r_{2} ((cos ϕ_{1} cos ϕ_{2} - sin ϕ_{1} sin ϕ_{2}) + (cos ϕ_{1} sin ϕ_{2} + sin ϕ_{1} cos ϕ_{2}) \cdot i)$

From the addition theorems for sine and cosine it follows:

$z_{1} z_{2} = r_{1} r_{2} (cos (ϕ_{1} + ϕ_{2}) + sin (ϕ_{1} + ϕ_{2}) \cdot i)$

For $z = r (cos ϕ + i sin ϕ)$ it follows by induction that:

$z^{n} = r^{n} (cos (n \cdot ϕ) + i sin (n \cdot ϕ))$

📎 Let $n \in N, n \geq 1$ . Then the equation $z^{n} = 1$ has exactly $n$ solutions in $C : z_{1}, z_{2}, \dots, z_{n}$ , where:

$z_{j} = cos \frac{2 πj}{n} + i \cdot sin \frac{2 πj}{n}, 1 \leq j \leq n$

📖 (Fundamental Theorem of Algebra.)

Let $n \geq 1, n \in N$ and

$P (z) = z^{n} + a_{n - 1} z^{n - 1} + \dots + a_{0}, a_{j} \in C .$

Then there exist $z_{1}, \dots, z_{n}$ in $C$ such that

$P (z) = (z - z_{1}) (z - z_{2}) \dots (z - z_{n}) .$