# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 5/latex

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\zwischenueberschrift{Ordering properties of the real numbers}
It is known that the real number may be identified with the points on a line. If on the number line, a point is further on the right than another point, then its value is larger. We describe now these ordering properties of the real numbers.
\inputdefinition { } {
A field $K$ is called an \definitionswort {ordered field}{,} if there is a relation $>$ \zusatzklammer {\stichwort {larger than}} between the elements of $K$, fulfilling the following properties \zusatzklammer { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } means \mavergleichskette {\vergleichskette {a } { > }{b } { }{ } { }{ } { }{ } } or \mavergleichskette {\vergleichskette {a } { = }{b } { }{ } { }{ } { }{ } }} {.} \aufzaehlungvier {For two elements \mavergleichskette {\vergleichskette {a,b } { \in }{K } { }{ } { }{ } { }{ } }{,} we have either \mavergleichskette {\vergleichskette {a } { > }{b } { }{ } { }{ } { }{ } } or \mavergleichskette {\vergleichskette {a } { = }{b } { }{ } { }{ } { }{ } } or \mavergleichskette {\vergleichskette {b } { > }{a } { }{ } { }{ } { }{ } }{.} } {From \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {b } { \geq }{c } { }{ } { }{ } { }{ } }{,} one may deduce \mavergleichskette {\vergleichskette {a } { \geq }{c } { }{ } { }{ } { }{ } } \zusatzklammer {for any \mavergleichskette {\vergleichskette { a , b , c } { \in }{ K } { }{ } { }{ } { }{ } }} {.} } { \mavergleichskette {\vergleichskette { a } { \geq }{ b } { }{ } { }{ } { }{ } } implies \mavergleichskette {\vergleichskette { a + c } { \geq }{ b + c } { }{ } { }{ } { }{ } } \zusatzklammer {for any \mavergleichskette {\vergleichskette { a , b , c } { \in }{ K } { }{ } { }{ } { }{ } }} {.} } {From \mavergleichskette {\vergleichskette { a } { \geq }{ 0 } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette { b } { \geq }{ 0 } { }{ } { }{ } { }{ } }{,} one may deduce \mavergleichskette {\vergleichskette { a b } { \geq }{ 0 } { }{ } { }{ } { }{ } } \zusatzklammer {for any \mavergleichskette {\vergleichskette { a, b } { \in }{ K } { }{ } { }{ } { }{ } }} {.}
\mathkor {\Q} {and} {\R} are, with their natural orderings, ordered fields.
In an ordered field, the following properties hold. \aufzaehlungzehn { \mavergleichskette {\vergleichskette {1 } { \geq }{0 } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \geq }{0 } { }{ } { }{ } { }{ } } holds if and only if \mavergleichskette {\vergleichskette {-a } { \leq }{0 } { }{ } { }{ } { }{ } } holds. } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } holds if and only if \mavergleichskette {\vergleichskette {a-b } { \geq }{0 } { }{ } { }{ } { }{ } } holds. } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } holds if and only if \mavergleichskette {\vergleichskette {-a } { \leq }{-b } { }{ } { }{ } { }{ } } holds. } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {c } { \geq }{d } { }{ } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {a +c } { \geq }{b+d } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {c } { \geq }{0 } { }{ } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {ac } { \geq }{bc } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {c } { \leq }{0 } { }{ } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {ac } { \leq }{bc } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \geq }{b } { \geq }{0 } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {c } { \geq }{d } { \geq }{0 } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {ac } { \geq }{bd } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \geq }{0 } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {b } { \leq }{0 } { }{ } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {ab } { \leq }{0 } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {a } { \leq }{0 } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette {b } { \leq }{0 } { }{ } { }{ } { }{ } } imply \mavergleichskette {\vergleichskette {ab } { \geq }{0 } { }{ } { }{ } { }{ } }{.} }
In an ordered field, the following properties holds. \aufzaehlungfuenf {From \mavergleichskette {\vergleichskette {x } { > }{ 0 } { }{ } { }{ } { }{ } } one can deduce \mavergleichskette {\vergleichskette {x^{-1} } { > }{0 } { }{ } { }{ } { }{ } }{.} } {From \mavergleichskette {\vergleichskette {x } { < }{ 0 } { }{ } { }{ } { }{ } } one can deduce \mavergleichskette {\vergleichskette {x^{-1} } { < }{0 } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {x } { > }{0 } { }{ } { }{ } { }{ } } we have \mavergleichskette {\vergleichskette {x } { \geq }{1 } { }{ } { }{ } { }{ } } if and only if \mavergleichskette {\vergleichskette {x^{-1} } { \leq }{1 } { }{ } { }{ } { }{ } }{.} } {From \mavergleichskette {\vergleichskette {x } { \geq }{y } { > }{0 } { }{ } { }{ } } one can deduce \mavergleichskette {\vergleichskette {x^{-1} } { \leq }{ y^{-1} } { }{ } { }{ } { }{ } }{.} } {For positive elements $x,y$ the relation \mavergleichskette {\vergleichskette {x } { \geq }{y } { }{ } { }{ } { }{ } } is equivalent with \mavergleichskette {\vergleichskette { { \frac{ x }{ y } } } { \geq }{1 } { }{ } { }{ } { }{ } }{.} } } {
See Exercise 5.8 , Exercise 5.37 , Exercise 5.9 , Exercise 5.10
and Exercise 5.11 .
\inputdefinition { } {
Let $K$ be an ordered field. $K$ is called \definitionswort {Archimedean}{,} if the following \definitionswort {Archimedean axiom} holds, i.e. if for every \mavergleichskette {\vergleichskette {x } { \in }{K } { }{ } { }{ } { }{ } } there exists a natural number $n$ such that \mavergleichskettedisp {\vergleichskette {n } { \geq} {x } { } { } { } { } { } { } }
\faktsituation \faktfolgerung {\aufzaehlungdrei {For \mavergleichskette {\vergleichskette {x,y } { \in }{ \R } { }{ } { }{ } { }{ } } with \mavergleichskette {\vergleichskette {x } { > }{0 } { }{ } { }{ } { }{ } } there exists \mavergleichskette {\vergleichskette {n } { \in }{ \N } { }{ } { }{ } { }{ } } such that \mavergleichskette {\vergleichskette {nx } { \geq }{y } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {x } { > }{0 } { }{ } { }{ } { }{ } } there exists a natural number $n$ such that \mavergleichskette {\vergleichskette { { \frac{ 1 }{ n } } } { < }{ x } { }{ } { }{ } { }{ } }{.} } {For two real numbers \mavergleichskette {\vergleichskette {x } { < }{y } { }{ } { }{ } { }{ } } there exists a rational number \mathl{n/k} (with \mavergleichskette {\vergleichskette {n } { \in }{\Z } { }{ } { }{ } { }{ } }{,}
\mavergleichskette {\vergleichskette {k } { \in }{\N_+ } { }{ } { }{ } { }{ } }) such that \mavergleichskettedisp {\vergleichskette {x } { <} { { \frac{ n }{ k } } } { <} {y } { } { } { } { } }{.} }} \faktzusatz } {
(1). We consider \mathl{y/x}{.} Because of the Archimedean axiom there exists some natural number $n$ with \mavergleichskette {\vergleichskette {n } { \geq }{ y/x } { }{ } { }{ } { }{ } }{.} Since $x$ is positive, due to Lemma 5.2
also \mavergleichskette {\vergleichskette {nx } { \geq }{y } { }{ } { }{ } { }{ } } holds. For (2) and (3) see Exercise 5.21 .
\inputdefinition { } {
For real numbers \mathbed {a,b} {a \leq b} {,} we call \aufzaehlungvier { \mavergleichskette {\vergleichskette { [a,b] } { = }{ { \left\{ x \in \R \mid x \geq a \text{ and } x \leq b \right\} } } { }{ } { }{ } { }{ } } the \definitionswort {closed interval}{.} } { \mavergleichskette {\vergleichskette { ]a,b[ } { = }{ { \left\{ x \in \R \mid x >a \text{ and } x < b \right\} } } { }{ } { }{ } { }{ } } the \definitionswort {open interval}{.} } { \mavergleichskette {\vergleichskette { ]a,b] } { = }{ { \left\{ x \in \R \mid x > a \text{ and } x \leq b \right\} } } { }{ } { }{ } { }{ } } the \definitionswort {half-open interval} (closed on the right). } { \mavergleichskette {\vergleichskette { [a,b[ } { = }{ { \left\{ x \in \R \mid x \geq a \text{ and } x < b \right\} } } { }{ } { }{ } { }{ } } the \definitionswort {half-open interval} (closed on the left).
For the real numbers, the intervals with bounds from $\Z$ \mathbed {[n,n+1 {n \in \Z} {,} form a disjoint union or a \stichwort {covering} of $\R$. Therefore, the following definition makes sense.
\inputdefinition { } {
For a real number $x$, the \definitionswort {floor} \mathl{\left \lfloor x \right \rfloor} is defined as
With the ordering properties, we can also define increasing and decreasing functions. \inputdefinition { } {
Let \mavergleichskette {\vergleichskette {I } { \subseteq }{ \R } { }{ } { }{ } { }{ } } denote an interval and let \mathdisp {f \colon I \longrightarrow \R} { } denote a function. Then $f$ is called \definitionswort {increasing} if \mathdisp {f(x') \geq f(x) \text { holds for all } x,x' \in I \text{ with } x' \geq x} { , } \definitionswort {strictly increasing} if \mathdisp {f(x') > f(x) \text { holds for all } x,x' \in I \text{ with } x'> x} { , } \definitionswort {decreasing} if \mathdisp {f(x') \leq f(x) \text { holds for all } x,x' \in I \text{ with } x' \geq x} { , } \definitionswort {strictly decreasing} if
\zwischenueberschrift{The modulus}
\inputdefinition { } {
For a real number \mavergleichskette {\vergleichskette {x } { \in }{ \R } { }{ } { }{ } { }{ } }{,} the \definitionswort {modulus} is defined in the following way. \mavergleichskettedisp {\vergleichskette { \betrag { x } } { =} {\begin{cases} x \, ,\text{ if } x \geq 0 \, , \\ -x,\, \text{ if } x < 0 \, . \end{cases} } { } { } { } { } { } { } }
So the modulus \zusatzklammer {also called the \stichwort {absolute value}} is never negative and has only at \mavergleichskette {\vergleichskette { x } { = }{ 0 } { }{ } { }{ } { }{ } } the value $0$, elsewhere it is always positive. The mapping \mathdisp {\R \longrightarrow \R , x \longmapsto \betrag { x }} { , } is called the \stichwort {modulus function} {.} Its graph consists of two half lines; such a function is called \stichwort {piecewisely linear} {.}
\faktsituation {The modulus function \mathdisp {\R \longrightarrow \R , x \longmapsto \betrag { x }} { , } fulfills the following properties ($x,y$ are arbitrary real numbers).} \faktfolgerung {\aufzaehlungacht { \mavergleichskette {\vergleichskette { \betrag { x } } { \geq }{ 0 } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { x } } { = }{ 0 } { }{ } { }{ } { }{ } } if and only if \mavergleichskette {\vergleichskette {x } { = }{ 0 } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { x } } { = }{ \betrag { y }
} { }{ } { }{ } { }{ } } if and only if \mavergleichskette {\vergleichskette {x } { = }{y } { }{ } { }{ } { }{ } } or \mavergleichskette {\vergleichskette {x } { = }{-y } { }{ } { }{ } { }{ } } holds. } { \mavergleichskette {\vergleichskette { \betrag { y-x } } { = }{ \betrag { x-y } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { xy } } { = }{ \betrag { x } \betrag { y } } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {x } { \neq }{0 } { }{ } { }{ } { }{ } } we have \mavergleichskette {\vergleichskette { \betrag { x^{-1} } } { = }{ \betrag { x }^{-1} } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \betrag { x+y } } { \leq }{ \betrag { x } + \betrag { y } } { }{ } { }{ } { }{ } } (\stichwort {triangle inequality for the modulus}). } { \mavergleichskette {\vergleichskette { \betrag { x+y } } { \geq }{ \betrag { x } - \betrag { y } } { }{ } { }{ } { }{ } }{.} }} \faktzusatz
\zwischenueberschrift{Bernoulli's inequality}
The following statement is called \stichwort {Bernoulli's inequality} {.}
\faktsituation {For every real number \mavergleichskette {\vergleichskette {x } { \geq }{ -1 } { }{ } { }{ } { }{ } }{,} and a natural number $n$,} \faktfolgerung {the estimate \mavergleichskettedisp {\vergleichskette { (1+x)^n} { \geq} { 1 +nx} { } { } { } } holds.} \faktzusatz } {
We do induction over $n$. Suppose that the statement is already known for $n$. Then \mavergleichskettealign {\vergleichskettealign { (1+x)^{n+1} } { =} { (1+x)^{n} (1+x) } { \geq} { (1+nx)(1+x) } { =} { 1+(n+1)x + nx^2 } { \geq} { 1+(n+1)x } }{,} since squares in an ordered field are nonnegative.
\zwischenueberschrift{The complex numbers}
We are going to introduce the complex numbers starting from the real numbers. It is not important for this construction that we have not completely listed all properties of the real numbers so far. We will use however that every positive real number has a unique real square root. With the construction of the complex numbers, we then have all number domains which are relevant for us.
\inputdefinition { } {
The set $\R^2$ with \mathkor {0 \defeq (0,0)} {and} {1 \defeq (1,0)} {,} with componentwise addition and the multiplication defined by \mavergleichskettedisp {\vergleichskette { (a,b) \cdot (c,d) } { \defeq} { (ac-bd, ad+bc) } { } { } { } { } { } { } }{,} is called the \definitionswort {field of complex numbers}{.} We denote it by
So the addition is just the vector addition in $\R^2$, but the multiplication is a new operation. We will see in Fact ***** a more geometric interpretation for the complex multiplication.
\faktsituation {The complex numbers} \faktfolgerung {form a field.} \faktzusatz
From now on we write \mavergleichskettedisp {\vergleichskette { a+b { \mathrm i} } { \defeq} { (a,b) } { } { } { } { } { } { } } and in particular we put \mavergleichskette {\vergleichskette { { \mathrm i} } { = }{ (0,1) } { }{ } { }{ } { }{ } }{,} this number is called the \stichwort {imaginary unit} {.} It has the important property \mavergleichskettedisp {\vergleichskette { { \mathrm i}^2 } { =} { -1 } { } { } { } { } { } { } }{.} From this property and the rules of a field, one can deduce all algebraic properties of the complex numbers. This also helps in memorizing the multiplication rule as we have \mavergleichskettedisp {\vergleichskette { (a+b { \mathrm i} )(c+d { \mathrm i} ) } { =} { ac+ad { \mathrm i} +b { \mathrm i} c+b { \mathrm i} d { \mathrm i} } { =} { ac+bd { \mathrm i}^2 +(ad+bc) { \mathrm i} } { =} { ac-bd +(ad+bc) { \mathrm i} } { } { } }{.} We consider a real number $a$ as the complex number \mavergleichskette {\vergleichskette { a+0 { \mathrm i} } { = }{ (a,0) } { }{ } { }{ } { }{ } }{.} Hence \mavergleichskette {\vergleichskette {\R } { \subset }{ \Complex } { }{ } { }{ } { }{ } }{.} It does not make a difference whether we add or multiply real numbers as real numbers or as complex numbers.
\inputdefinition { } {
For a complex number \mavergleichskettedisp {\vergleichskette {z } { =} { a+b { \mathrm i} } { } { } { } { } { } { } }{,} we call \mavergleichskettedisp {\vergleichskette { \operatorname{Re} \, { \left( z \right) } } { =} { a } { } { } { } { } { } { } } the \definitionswort {real part} of $z$ and \mavergleichskettedisp {\vergleichskette { \operatorname{Im} \, { \left( z \right) } } { =} { b } { } { } { } { } { } { } }
However, one should not think that complex numbers are less real than real numbers. The construction of the complex numbers starting from the reals is by far easier than the construction of the real numbers starting from the rational numbers. On the other hand, it was a long historic process until complex numbers were accepted as numbers; they form a field, but not an ordered field, and so at first glance they are numbers which do not measure anything.
One should think of complex numbers as points of the plane; for the additive structure we have directly \mavergleichskette {\vergleichskette { \Complex } { = }{ \R^2 } { }{ } { }{ } { }{ } }{.} The horizontal axis is called the \stichwort {real axis} and the vertical axis is called the \stichwort {imaginary axis} {.}
\faktsituation \faktvoraussetzung {The real part and the imaginary part of complex numbers fulfill the following properties (for \mathkor {z} {and} {w} in $\Complex$).} \faktfolgerung {\aufzaehlungfuenf { \mavergleichskette {\vergleichskette { z } { = }{ \operatorname{Re} \, { \left( z \right) } + \operatorname{Im} \, { \left( z \right) } { \mathrm i}
} { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \operatorname{Re} \, { \left( z+w \right) } } { = }{ \operatorname{Re} \, { \left( z \right) } + \operatorname{Re} \, { \left( w \right) } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \operatorname{Im} \, { \left( z+w \right) } } { = }{ \operatorname{Im} \, { \left( z \right) } + \operatorname{Im} \, { \left( w \right) } } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {r } { \in }{\R } { }{ } { }{ } { }{ } } we have \mathdisp {\operatorname{Re} \, { \left( rz \right) } =r
\operatorname{Re} \, { \left( z \right) } \text{ and }
\operatorname{Im} \, { \left( rz \right) } =r
\operatorname{Im} \, { \left( z \right) }} { . } } { \mavergleichskette {\vergleichskette { z } { = }{ \operatorname{Re} \, { \left( z \right) } } { }{ } { }{ } { }{ } } holds if and only if \mavergleichskette {\vergleichskette {z } { \in }{\R } { }{ } { }{ } { }{ } } holds, and this holds if and only if \mavergleichskette {\vergleichskette { \operatorname{Im} \, { \left( z \right) } } { = }{ 0 } { }{ } { }{ } { }{ } } holds. }} \faktzusatz
\inputdefinition { } {
The mapping \mathdisp {\Complex \longrightarrow \Complex , z = a+b { \mathrm i} \longmapsto \overline{ z } \defeq a-b { \mathrm i}} { , }
For $z$, the number $\overline{ z }$ is called the \stichwort {complex-conjugated number} of $z$. Geometrically, the complex conjugation to \mavergleichskette {\vergleichskette {z } { \in }{ \Complex } { }{ } { }{ } { }{ } } is simple the reflection at the real axis.
\faktsituation {For the complex conjugation, the following rules hold \zusatzklammer {for arbitrary \mavergleichskette {\vergleichskette {z,w } { \in }{ \Complex } { }{ } { }{ } { }{ } }} {.}} \faktfolgerung {\aufzaehlungsechs { \mavergleichskette {\vergleichskette { \overline{ z+w } } { = }{ \overline{ z } + \overline{ w } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \overline{ -z } } { = }{ - \overline{ z } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \overline{ z \cdot w } } { = }{ \overline{ z } \cdot \overline{ w } } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {z } { \neq }{0 } { }{ } { }{ } { }{ } } we have \mavergleichskette {\vergleichskette { \overline{ 1/z } } { = }{ 1/\overline{ z } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \overline{ \overline{ z } } } { = }{ z } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \overline{ z } } { = }{ z } { }{ } { }{ } { }{ } } if and only if \mavergleichskette {\vergleichskette {z } { \in }{\R } { }{ } { }{ } { }{ } } holds. }} \faktzusatz
\faktsituation \faktvoraussetzung {For a complex number $z$ the following relations hold.} \faktfolgerung {\aufzaehlungdrei { \mavergleichskette {\vergleichskette { \overline{ z } } { = }{ \operatorname{Re} \, { \left( z \right) } - { \mathrm i} \operatorname{Im} \, { \left( z \right) }
} { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \operatorname{Re} \, { \left( z \right) } } { = }{ { \frac{ z+ \overline{ z } }{ 2 } } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette {
\operatorname{Im} \, { \left( z \right) } } { = }{ { \frac{ z - \overline{ z } }{ 2 { \mathrm i} } } } { }{ } { }{ } { }{ } }{.} }} \faktzusatz
The square \mathl{d^2} of a real number is always nonnegative, and the sum of two nonnegative real numbers is again nonnegative. For a nonnegative real number $c$ there exists a unique nonnegative \stichwort {square root} $\sqrt{c}$, see Exercise 8.9
Hence the following definition yields a well-defined real number. \inputdefinition { } {
For a complex number \mavergleichskettedisp {\vergleichskette {z } { =} {a+b { \mathrm i} } { } { } { } { } { } { } }{,} the \definitionswort {modulus} is defined by \mavergleichskettedisp {\vergleichskette { \betrag { z } } { =} { \sqrt{a^2+b^2} } { } { } { } { } { } { } }
The modulus of a complex number $z$ is, due to the \stichwort {Pythagorean theorem} {,} the distance of $z$ to the zero point \mavergleichskette {\vergleichskette {0 } { = }{ (0,0) } { }{ } { }{ } { }{ } }{.} The modulus is a mapping \mathdisp {\Complex \longrightarrow \R_{\geq 0} , z \longmapsto \betrag { z }} { . }
The set of all complex numbers with a certain modulus form a circle with the zero point as center and with the modulus as radius. In particular, all complex numbers with modulus $1$ form the \stichwort {complex unit circle} {.} The numbers of the complex unit circle are by the \stichwort {formula of Euler} in relation to the complex exponential function and to the trigonometric functions, see Fact ***** and Fact *****. We further mention that the product of two complex numbers of the unit circle may be computed by adding the angles starting from the positive real axis counter clockwise.
\faktsituation \faktvoraussetzung {The modulus of complex numbers fulfils the following properties.} \faktfolgerung {\aufzaehlungacht { \mavergleichskette {\vergleichskette { \betrag { z } } { = }{ \sqrt{ z \ \overline{ z } } } { }{ } { }{ } { }{ } }{.} } {For real $z$ the real and the complex modulus are the same. } {We have \mavergleichskette {\vergleichskette { \betrag { z } } { = }{ 0 } { }{ } { }{ } { }{ } } if and only if \mavergleichskette {\vergleichskette {z } { = }{ 0 } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { z } } { = }{ \betrag { \overline{ z } }
} { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { zw } } { = }{ \betrag { z } \cdot \betrag { w } } { }{ } { }{ } { }{ } }{.} } {For \mavergleichskette {\vergleichskette {z } { \neq }{0 } { }{ } { }{ } { }{ } } we have \mavergleichskette {\vergleichskette { \betrag { 1/z } } { = }{ 1/\betrag { z } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag {
\operatorname{Re} \, { \left( z \right) } }, \betrag {
\operatorname{Im} \, { \left( z \right) } } } { \leq }{ \betrag { z } } { }{ } { }{ } { }{ } }{.} } { \mavergleichskette {\vergleichskette { \betrag { z+w } } { \leq }{ \betrag { z } + \betrag { w } } { }{ } { }{ } { }{ } } \zusatzklammer {triangle inequality} {.} }} \faktzusatz } {
We only show the triangle inequality, for the other statements see Exercise 5.36 . Because of (7) we have for every complex number $u$ the estimate \mavergleichskette {\vergleichskette { \operatorname{Re} \, { \left( u \right) } } { \leq }{ \betrag { u } } { }{ } { }{ } { }{ } }{.} Therefore, \mavergleichskettedisp {\vergleichskette { \operatorname{Re} \, { \left( z \overline{ w } \right) } } { \leq} { \betrag { z } \betrag { w } } { } { } { } { } { } { } }{,} and hence \mavergleichskettealign {\vergleichskettealign { \betrag { z+w }^2 } { =} { (z+w) { \left( \overline{ z } + \overline{ w } \right) } } { =} { z \overline{ z } + z \overline{ w } + w \overline{ z } + w \overline{ w } } { =} { \betrag { z }^2 + 2 \operatorname{Re} \, { \left( z \overline{ w } \right) } + \betrag { w } ^2 } { \leq} { \betrag { z }^2 + 2 \betrag { z } \betrag { w } + \betrag { w } ^2 } } { \vergleichskettefortsetzungalign { =} { ( \betrag { z } + \betrag { w } )^2 } { } { } { } }{.} By taking the square root, we get the stated estimate.