Signal Representation/Calculating with Complex Numbers - Revision history

Guenter at 15:45, 7 June 2023

2023-06-07T15:45:29Z

Guenter at 15:43, 7 June 2023

2023-06-07T15:43:37Z

Show changes

Guenter at 15:07, 27 April 2023

2023-04-27T15:07:28Z

Guenter at 12:03, 8 February 2023

2023-02-08T12:03:35Z

Hwang at 22:10, 24 November 2022

2022-11-24T22:10:57Z

Hwang at 22:03, 24 November 2022

2022-11-24T22:03:15Z

Hwang at 22:00, 24 November 2022

2022-11-24T22:00:20Z

Guenter: Guenter moved page Signal Representation/Calculating With Complex Numbers to Signal Representation/Calculating with Complex Numbers

2021-06-15T10:38:28Z

Guenter moved page Signal Representation/Calculating With Complex Numbers to Signal Representation/Calculating with Complex Numbers

Guenter at 14:17, 31 May 2021

2021-05-31T14:17:33Z

Javier: Text replacement - "”" to """

2021-05-28T14:19:27Z

Text replacement - "”" to """

← Older revision		Revision as of 15:45, 7 June 2023
Line 187:		Line 187:


−	The following (German language) learning video summarizes the topic of this chapter in a compact way:<br>          [[Rechnen_mit_komplexen_Zahlen_(Lernvideo)\|"Rechnen mit komplexen Zahlen"]]   ⇒   "Arithmetic operations involving complex numbers".	+	The following  $($German language$)$  learning video summarizes the topic of this chapter in a compact way:<br>          [[Rechnen_mit_komplexen_Zahlen_(Lernvideo)\|»Rechnen mit komplexen Zahlen«]]   ⇒   "Arithmetic operations involving complex numbers".

@@ Line 172: / Line 172: @@
 [[File:P_ID825_Sig_T_1_3_S4_neu.png|right|frame|Sum, difference, product & quotient of complex numbers]]
 {{GraueBox|TEXT=
-$\text{Beispiel 2:}$&nbsp;
+$\text{Example 2:}$&nbsp;
 In the graphic are shown as points within the complex plane:

@@ Line 16: / Line 16: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*$\text{Natural Numbers}$&nbsp; $\mathbb{N} = \{1, 2, 3, \text{...}\hspace{0.05cm} \}$. &nbsp; Using these numbers, for&nbsp; $n, \ k \in \mathbb{N}$&nbsp; the arithmetic operations&nbsp; "addition"&nbsp; $(m = n +k)$,&nbsp; "multiplication"&nbsp; $(m = n \cdot k)$&nbsp; and&nbsp; "power formation"&nbsp; $(m = n^k)$&nbsp; are possible.&nbsp; The respective result of a calculation is again a natural number: &nbsp; $m \in \mathbb{N}$.
+*&raquo;'''Natural Numbers'''&laquo;&nbsp; $\mathbb{N} = \{1, 2, 3, \text{...}\hspace{0.05cm} \}$. &nbsp; Using these numbers, for&nbsp; $n, \ k \in \mathbb{N}$&nbsp; the arithmetic operations&nbsp; "addition"&nbsp; $(m = n +k)$,&nbsp; "multiplication"&nbsp; $(m = n \cdot k)$&nbsp; and&nbsp; "power formation"&nbsp; $(m = n^k)$&nbsp; are possible.&nbsp; The respective result of a calculation is again a natural number: &nbsp; $m \in \mathbb{N}$.
-*$\text{Integer Numbers}$&nbsp; $\mathbb{Z} = \{\text{...}\hspace{0.05cm} , -3, -2, -1, \ 0, +1, +2, +3, \text{...}\hspace{0.05cm}\}$. &nbsp; This set of numbers is an extension of the natural numbers&nbsp; $\mathbb{N}$.&nbsp; The introduction of the set&nbsp; $\mathbb{Z}$&nbsp; was necessary to capture the result set of a subtraction&nbsp; $(m = n -k$,&nbsp; for example&nbsp; $5 - 7 = - 2)$.
+*&raquo;'''Integer Numbers'''&laquo;&nbsp; $\mathbb{Z} = \{\text{...}\hspace{0.05cm} , -3, -2, -1, \ 0, +1, +2, +3, \text{...}\hspace{0.05cm}\}$. &nbsp; This set of numbers is an extension of the natural numbers&nbsp; $\mathbb{N}$.&nbsp; The introduction of the set&nbsp; $\mathbb{Z}$&nbsp; was necessary to capture the result set of a subtraction&nbsp; $(m = n -k$,&nbsp; for example&nbsp; $5 - 7 = - 2)$.
-*$\text{Rational Numbers}$&nbsp; $\mathbb{Q} = \{z/n\}$&nbsp; with&nbsp; $z \in \mathbb{Z}$&nbsp; and&nbsp; $n \in \mathbb{N}$. &nbsp; With this set of numbers, also known as fractions, there is a defined result for each division.&nbsp; If you write a rational number in decimal notation, only zeros appear after a certain decimal place&nbsp; $($Example:&nbsp; $-2/5 = -0.400\text{...}\hspace{0.05cm})$&nbsp; or periodicities&nbsp; $($Example:&nbsp; $2/7 = 0.285714285\text{...}\hspace{0.05cm})$.&nbsp; Since&nbsp; $n = 1$&nbsp; is allowed, the integers are a subset of the rational numbers: &nbsp; $\mathbb{Z} \subset \mathbb{Q}$.
+*&raquo;'''Rational Numbers'''&laquo;&nbsp; $\mathbb{Q} = \{z/n\}$&nbsp; with&nbsp; $z \in \mathbb{Z}$&nbsp; and&nbsp; $n \in \mathbb{N}$. &nbsp; With this set of numbers, also known as fractions, there is a defined result for each division.&nbsp; If you write a rational number in decimal notation, only zeros appear after a certain decimal place&nbsp; $($Example:&nbsp; $-2/5 = -0.400\text{...}\hspace{0.05cm})$&nbsp; or periodicities&nbsp; $($Example:&nbsp; $2/7 = 0.285714285\text{...}\hspace{0.05cm})$.&nbsp; Since&nbsp; $n = 1$&nbsp; is allowed, the integers are a subset of the rational numbers: &nbsp; $\mathbb{Z} \subset \mathbb{Q}$.
-*$\text{Irrational Numbers}$&nbsp; $\mathbb{I} \neq {z/n}$&nbsp; mit&nbsp; $z \in \mathbb{Z}$, $n \in \mathbb{N}$. &nbsp; Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.&nbsp; Examples are the number&nbsp;  $\pi = 3.141592654\text{...}\hspace{0.05cm}$&nbsp;  (where there are no periods even with more decimal places)&nbsp; or the result of the equation &nbsp; $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.&nbsp; This result is also irrational, which has already been proved by&nbsp; [https://en.wikipedia.org/wiki/Euclid $\text{Euclid}$]&nbsp; in antiquity.
+*&raquo;'''Irrational Numbers'''&laquo;&nbsp; $\mathbb{I} \neq {z/n}$&nbsp; mit&nbsp; $z \in \mathbb{Z}$, $n \in \mathbb{N}$. &nbsp; Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.&nbsp; Examples are the number&nbsp;  $\pi = 3.141592654\text{...}\hspace{0.05cm}$&nbsp;  (where there are no periods even with more decimal places)&nbsp; or the result of the equation &nbsp; $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.&nbsp; This result is also irrational, which has already been proved by&nbsp; [https://en.wikipedia.org/wiki/Euclid $\text{Euclid}$]&nbsp; in antiquity.
 [[File:EN_Sig_T_1_3_S1.png |right|frame|Real numbers on the number line]]
-*$\text{Real Numbers}$&nbsp; $\mathbb{R} = \mathbb{Q}  \cup  \mathbb{I}$ as the sum of all rational and irrational numbers.
+*&raquo;'''Real Numbers'''&laquo;&nbsp; $\mathbb{R} = \mathbb{Q}  \cup  \mathbb{I}$ as the sum of all rational and irrational numbers.
 :These can be ordered according to their numerical values and can be drawn on the so-called&nbsp; "number line"&nbsp; as shown in the adjacent graph.}}
@@ Line 48: / Line 48: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The&nbsp; $\text{complex number}$&nbsp; $z$&nbsp; is generally the sum of a real number&nbsp; $x$&nbsp; and an imaginary number&nbsp; ${\rm j} \cdot y$:
+The&nbsp; &raquo;'''complex number'''&laquo;&nbsp; $z$&nbsp; is generally the sum of a real number&nbsp; $x$&nbsp; and an imaginary number&nbsp; ${\rm j} \cdot y$:
 :$$z=x+{\rm j}\cdot y.$$

@@ Line 31: / Line 31: @@
 *$\text{Real Numbers}$&nbsp; $\mathbb{R} = \mathbb{Q}  \cup  \mathbb{I}$ as the sum of all rational and irrational numbers.
-:These can be ordered according to their numerical values and can be drawn on the so called&nbsp; "number line"&nbsp; as shown in the adjacent graph.}}
+:These can be ordered according to their numerical values and can be drawn on the so-called&nbsp; "number line"&nbsp; as shown in the adjacent graph.}}
@@ Line 142: / Line 142: @@
 \hspace {0.05cm} \phi_2}$$
-are defined in such a way, that for the special case of a vanishing imaginary part, the rules of calculation of real numbers are given. This is called the so called&nbsp; "principle of permanence".
+are defined in such a way, that for the special case of a vanishing imaginary part, the rules of calculation of real numbers are given. This is called the so-called&nbsp; "principle of permanence".
 The following rules apply to the basic arithmetic operations:&nbsp;

@@ Line 25: / Line 25: @@
-*$\text{Irrational Numbers}$&nbsp; $\mathbb{I} \neq {z/n}$&nbsp; mit&nbsp; $z \in \mathbb{Z}$, $n \in \mathbb{N}$. &nbsp; Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.&nbsp; Examples are the number&nbsp;  $\pi = 3.141592654\text{...}\hspace{0.05cm}$&nbsp;  (where there are no periods even with more decimal places)&nbsp; or the result of the equation &nbsp; $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.&nbsp; This result is also irrational, which has already been proved by&nbsp; [https://en.wikipedia.org/wiki/Euclid Euclid]&nbsp; in antiquity.
+*$\text{Irrational Numbers}$&nbsp; $\mathbb{I} \neq {z/n}$&nbsp; mit&nbsp; $z \in \mathbb{Z}$, $n \in \mathbb{N}$. &nbsp; Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.&nbsp; Examples are the number&nbsp;  $\pi = 3.141592654\text{...}\hspace{0.05cm}$&nbsp;  (where there are no periods even with more decimal places)&nbsp; or the result of the equation &nbsp; $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.&nbsp; This result is also irrational, which has already been proved by&nbsp; [https://en.wikipedia.org/wiki/Euclid $\text{Euclid}$]&nbsp; in antiquity.
 [[File:EN_Sig_T_1_3_S1.png |right|frame|Real numbers on the number line]]
@@ Line 40: / Line 40: @@
 With the introduction of the irrational numbers the solution of the equation&nbsp; $a^2-2=0$&nbsp; was possible, but not the solution of the equation&nbsp; $a^2+1=0$.
-The mathematician&nbsp;[https://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler]&nbsp; solved this problem by extending the set of real numbers by the&nbsp; "imaginary numbers"&nbsp;. He defined the&nbsp; '''imaginary unit'''&nbsp; as follows:
+The mathematician&nbsp;[https://en.wikipedia.org/wiki/Leonhard_Euler $\text{Leonhard Euler}$]&nbsp; solved this problem by extending the set of real numbers by the&nbsp; "imaginary numbers"&nbsp;. He defined the&nbsp; &raquo;'''imaginary unit'''&laquo;&nbsp; as follows:
 :$${\rm j}=\sqrt{-1} \ \Rightarrow \ {\rm j}^{2}=-1.$$
@@ Line 183: / Line 183: @@
-The following  (German language) learning video summarizes the topic of this chapter in a compact way:<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Rechnen_mit_komplexen_Zahlen_(Lernvideo)|Rechnen mit komplexen Zahlen]] &nbsp; &rArr; &nbsp; "Arithmetic operations involving complex numbers".
+The following  (German language) learning video summarizes the topic of this chapter in a compact way:<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Rechnen_mit_komplexen_Zahlen_(Lernvideo)|"Rechnen mit komplexen Zahlen"]] &nbsp; &rArr; &nbsp; "Arithmetic operations involving complex numbers".

@@ Line 6: / Line 6: @@
-==The Set of real numbers==
+==The set of real numbers==
 <br>
 In the following chapters of this book, complex quantities always play an important role.&nbsp; Although calculating with complex numbers is already treated and practiced in school mathematics, our experience has shown that even students of natural sciences and technical subjects have problems with it.&nbsp; Perhaps these difficulties are also related to the fact that "complex" is often used as a synonym for "complicated" in everyday life, while "real" stands for "reliable, honest and truthful" according to the Duden dictionary.

← Older revision		Revision as of 14:17, 31 May 2021
Line 27:		Line 27:
	*$\text{Irrational Numbers}$  $\mathbb{I} \neq {z/n}$  mit  $z \in \mathbb{Z}$, $n \in \mathbb{N}$.   Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.  Examples are the number  $\pi = 3.141592654\text{...}\hspace{0.05cm}$  (where there are no periods even with more decimal places)  or the result of the equation   $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.  This result is also irrational, which has already been proved by  [https://en.wikipedia.org/wiki/Euclid Euclid]  in antiquity.		*$\text{Irrational Numbers}$  $\mathbb{I} \neq {z/n}$  mit  $z \in \mathbb{Z}$, $n \in \mathbb{N}$.   Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.  Examples are the number  $\pi = 3.141592654\text{...}\hspace{0.05cm}$  (where there are no periods even with more decimal places)  or the result of the equation   $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.  This result is also irrational, which has already been proved by  [https://en.wikipedia.org/wiki/Euclid Euclid]  in antiquity.

−	[[File:~~P_ID821_Sig_T_1_3_S1_rah~~.png \|right\|frame\|Real numbers on the number line]]	+	[[File:EN_Sig_T_1_3_S1.png \|right\|frame\|Real numbers on the number line]]

@@ Line 44: / Line 44: @@
 :$${\rm j}=\sqrt{-1} \ \Rightarrow \ {\rm j}^{2}=-1.$$
-It should be noted that Euler called this quantity&nbsp; "$\rm i$&rdquo;&nbsp; and this is still common in mathematics today.&nbsp; In electrical engineering, on the other hand, the designation&nbsp; "$\rm j$"&nbsp; has become generally accepted since&nbsp; "$\rm i$"&nbsp; is already occupied by the time-dependent current.
+It should be noted that Euler called this quantity&nbsp; "$\rm i$"&nbsp; and this is still common in mathematics today.&nbsp; In electrical engineering, on the other hand, the designation&nbsp; "$\rm j$"&nbsp; has become generally accepted since&nbsp; "$\rm i$"&nbsp; is already occupied by the time-dependent current.
 {{BlaueBox|TEXT=