Signal Representation/Calculating with Complex Numbers - Revision history

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	* The following applies to both series:		* The following applies to both series:
	:$$A(x) = 1 - \frac{x^2}{2!}		:$$A(x) = 1 - \frac{x^2}{2!}+ \frac{x^4}{4!} - \frac{x^6}{6!}+\text{ ...} \hspace{0.1cm}=\cos(x),\hspace{0.5cm}B(x) = \frac{x}{1!}- \frac{x^3}{3!}+ \frac{x^5}{5!} - \frac{x^7}{7!}+\text{ ...}=\sin(x).$$*From this the  $\text{Euler Theorem}$  follows directly::$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = \cos (x) + {\rm j} \cdot \sin (x) \hspace{2cm}\rm q.e.d.$$}}
	+ \frac{x^4}{4!} - \frac{x^6}{6!}+
	\text{ ...} \hspace{0.1cm}=
	\cos(x),\hspace{0.5cm}
	B(x) = \frac{x}{1!}- \frac{x^3}{3!}
	+ \frac{x^5}{5!} - \frac{x^7}{7!}+
	\text{ ...}=
	\sin(x).$$*From this the  $\text{Euler Theorem}$  follows directly::$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = \cos (x) + {\rm j} \cdot \sin (x) \hspace{2cm}
	\rm q.e.d.$$}}

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	*The series development of the exponential function is:		*The series development of the exponential function is:

	:$${\rm e}^{x} = 1 + \frac{x}{1!}+ \frac{x^2}{2!}+ \frac{x^3}{3!}		:$${\rm e}^{x} = 1 + \frac{x}{1!}+ \frac{x^2}{2!}+ \frac{x^3}{3!}+ \frac{x^4}{4!} +\text{ ...} \hspace{0.15cm}.$$
	+ \frac{x^4}{4!} +\text{ ...} \hspace{0.15cm}.$$

	*With an imaginary argument you can also write:		*With an imaginary argument you can also write:

	:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = 1 + {\rm j} \cdot \frac{x}{1!}+ {\rm j}^2 \cdot \frac{x^2}{2!}+ {\rm j}^3 \cdot \frac{x^3}{3!}		:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = 1 + {\rm j} \cdot \frac{x}{1!}+ {\rm j}^2 \cdot \frac{x^2}{2!}+ {\rm j}^3 \cdot \frac{x^3}{3!}+ {\rm j}^4 \cdot \frac{x^4}{4!} + \text{ ...} \hspace{0.15cm}.$$
	+ {\rm j}^4 \cdot \frac{x^4}{4!} + \text{ ...} \hspace{0.15cm}.$$

	*Considering  <math>{\rm j}^{2}=-1, \ \ {\rm j}^{3} = -{\rm j},\ \ {\rm j}^{4} = 1, \ \ {\rm j}^{5} = {\rm j}, \text{ ...} \hspace{0.15cm}</math>  and combining the real and the imaginary terms,  one obtains		*Considering  <math>{\rm j}^{2}=-1, \ \ {\rm j}^{3} = -{\rm j},\ \ {\rm j}^{4} = 1, \ \ {\rm j}^{5} = {\rm j}, \text{ ...} \hspace{0.15cm}</math>  and combining the real and the imaginary terms,  one obtains
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	+ \frac{x^5}{5!} - \frac{x^7}{7!}+		+ \frac{x^5}{5!} - \frac{x^7}{7!}+
	\text{ ...}=		\text{ ...}=
	\sin(x).$$		\sin(x).$$*From this the  $\text{Euler Theorem}$  follows directly::$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = \cos (x) + {\rm j} \cdot \sin (x) \hspace{2cm}

	*From this the  $\text{Euler Theorem}$  follows directly:

	:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = \cos (x) + {\rm j} \cdot \sin (x) \hspace{2cm}
	\rm q.e.d.$$}}		\rm q.e.d.$$}}

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	The arithmetic laws for two complex numbers		The arithmetic laws for two complex numbers

	:$$z_1 = x_1 + {\rm j} \cdot y_1 = \|z_1\| \cdot {\rm e}^{{\rm j}\hspace {0.05cm}\cdot		:$$z_1 = x_1 + {\rm j} \cdot y_1 = \|z_1\| \cdot {\rm e}^{{\rm j}\hspace {0.05cm}\cdot\hspace {0.05cm} \phi_1},$$
	\hspace {0.05cm} \phi_1},$$		:$$z_2 = x_2 + {\rm j} \cdot y_2 = \|z_2\| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot\hspace {0.05cm} \phi_2}$$
	:$$z_2 = x_2 + {\rm j} \cdot y_2 = \|z_2\| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot
	\hspace {0.05cm} \phi_2}$$

	are defined in such a way,  that for the special case of a vanishing imaginary part,  the calculation rules for real numbers are given.  This is called  "principle of permanence".  The following rules apply to the basic arithmetic operations:		are defined in such a way,  that for the special case of a vanishing imaginary part,  the calculation rules for real numbers are given.  This is called  "principle of permanence".  The following rules apply to the basic arithmetic operations:

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	{{Display}}		{{Display}}
			[[de:Signaldarstellung/Zum_Rechnen_mit_komplexen_Zahlen]]

Guenter at 15:45, 7 June 2023

2023-06-07T15:45:29Z

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	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".

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

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

Guenter at 12:03, 8 February 2023

2023-02-08T12:03:35Z

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definitions:}$		$\text{Definitions:}$
	*~~$\text{~~Natural Numbers}$  $\mathbb{N} = \{1, 2, 3, \text{...}\hspace{0.05cm} \}$.   Using these numbers, for  $n, \ k \in \mathbb{N}$  the arithmetic operations  "addition"  $(m = n +k)$,  "multiplication"  $(m = n \cdot k)$  and  "power formation"  $(m = n^k)$  are possible.  The respective result of a calculation is again a natural number:   $m \in \mathbb{N}$.		*»'''Natural Numbers'''«  $\mathbb{N} = \{1, 2, 3, \text{...}\hspace{0.05cm} \}$.   Using these numbers, for  $n, \ k \in \mathbb{N}$  the arithmetic operations  "addition"  $(m = n +k)$,  "multiplication"  $(m = n \cdot k)$  and  "power formation"  $(m = n^k)$  are possible.  The respective result of a calculation is again a natural number:   $m \in \mathbb{N}$.


	*~~$\text{~~Integer Numbers}$  $\mathbb{Z} = \{\text{...}\hspace{0.05cm} , -3, -2, -1, \ 0, +1, +2, +3, \text{...}\hspace{0.05cm}\}$.   This set of numbers is an extension of the natural numbers  $\mathbb{N}$.  The introduction of the set  $\mathbb{Z}$  was necessary to capture the result set of a subtraction  $(m = n -k$,  for example  $5 - 7 = - 2)$.		*»'''Integer Numbers'''«  $\mathbb{Z} = \{\text{...}\hspace{0.05cm} , -3, -2, -1, \ 0, +1, +2, +3, \text{...}\hspace{0.05cm}\}$.   This set of numbers is an extension of the natural numbers  $\mathbb{N}$.  The introduction of the set  $\mathbb{Z}$  was necessary to capture the result set of a subtraction  $(m = n -k$,  for example  $5 - 7 = - 2)$.


	*~~$\text{~~Rational Numbers}$  $\mathbb{Q} = \{z/n\}$  with  $z \in \mathbb{Z}$  and  $n \in \mathbb{N}$.   With this set of numbers, also known as fractions, there is a defined result for each division.  If you write a rational number in decimal notation, only zeros appear after a certain decimal place  $($Example:  $-2/5 = -0.400\text{...}\hspace{0.05cm})$  or periodicities  $($Example:  $2/7 = 0.285714285\text{...}\hspace{0.05cm})$.  Since  $n = 1$  is allowed, the integers are a subset of the rational numbers:   $\mathbb{Z} \subset \mathbb{Q}$.		*»'''Rational Numbers'''«  $\mathbb{Q} = \{z/n\}$  with  $z \in \mathbb{Z}$  and  $n \in \mathbb{N}$.   With this set of numbers, also known as fractions, there is a defined result for each division.  If you write a rational number in decimal notation, only zeros appear after a certain decimal place  $($Example:  $-2/5 = -0.400\text{...}\hspace{0.05cm})$  or periodicities  $($Example:  $2/7 = 0.285714285\text{...}\hspace{0.05cm})$.  Since  $n = 1$  is allowed, the integers are a subset of the rational numbers:   $\mathbb{Z} \subset \mathbb{Q}$.


	*~~$\text{~~Irrational Numbers}$  $\mathbb{I} \neq {z/n}$  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 $\text{Euclid}$]  in antiquity.		*»'''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 $\text{Euclid}$]  in antiquity.

	[[File:EN_Sig_T_1_3_S1.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]]


	*~~$\text{~~Real Numbers}$  $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ as the sum of all rational and irrational numbers.		*»'''Real Numbers'''«  $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ 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  "number line"  as shown in the adjacent graph.}}		:These can be ordered according to their numerical values and can be drawn on the so-called  "number line"  as shown in the adjacent graph.}}

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	The  ~~$\text{~~complex number}$  $z$  is generally the sum of a real number  $x$  and an imaginary number  ${\rm j} \cdot y$:		The  »'''complex number'''«  $z$  is generally the sum of a real number  $x$  and an imaginary number  ${\rm j} \cdot y$:

	:$$z=x+{\rm j}\cdot y.$$		:$$z=x+{\rm j}\cdot y.$$

Hwang at 22:10, 24 November 2022

2022-11-24T22:10:57Z

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	*$\text{Real Numbers}$  $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ as the sum of all rational and irrational numbers.		*$\text{Real Numbers}$  $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ 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  "number line"  as shown in the adjacent graph.}}		:These can be ordered according to their numerical values and can be drawn on the so-called  "number line"  as shown in the adjacent graph.}}


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	\hspace {0.05cm} \phi_2}$$		\hspace {0.05cm} \phi_2}$$

	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  "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  "principle of permanence".

	The following rules apply to the basic arithmetic operations:		The following rules apply to the basic arithmetic operations:

Hwang at 22:03, 24 November 2022

2022-11-24T22:03:15Z

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	*$\text{Irrational Numbers}$  $\mathbb{I} \neq {z/n}$  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 $\text{Euclid}$]  in antiquity.

	[[File:EN_Sig_T_1_3_S1.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]]
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	With the introduction of the irrational numbers the solution of the equation  $a^2-2=0$  was possible, but not the solution of the equation  $a^2+1=0$.		With the introduction of the irrational numbers the solution of the equation  $a^2-2=0$  was possible, but not the solution of the equation  $a^2+1=0$.

	The mathematician [https://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler]  solved this problem by extending the set of real numbers by the  "imaginary numbers" . He defined the  '''imaginary unit'''  as follows:		The mathematician [https://en.wikipedia.org/wiki/Leonhard_Euler $\text{Leonhard Euler}$]  solved this problem by extending the set of real numbers by the  "imaginary numbers" . He defined the  »'''imaginary unit'''«  as follows:

	:$${\rm j}=\sqrt{-1} \ \Rightarrow \ {\rm j}^{2}=-1.$$		:$${\rm j}=\sqrt{-1} \ \Rightarrow \ {\rm j}^{2}=-1.$$
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	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".

Hwang at 22:00, 24 November 2022

2022-11-24T22:00:20Z

← Older revision		Revision as of 00:00, 25 November 2022
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	==The ~~Set~~ of real numbers==		==The set of real numbers==
	<br>		<br>
	In the following chapters of this book, complex quantities always play an important role.  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.  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.		In the following chapters of this book, complex quantities always play an important role.  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.  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.