Glanz: aiMOOC über GPT aiMOOC Action erstellt

2026-08-28T06:53:44Z

aiMOOC über GPT aiMOOC Action erstellt

Neue Seite

{{T}}
[[Category:English]]
[[Category:Complex Numbers]]

{{BR}}
= Introduction =

Complex numbers extend the real number system so that equations such as <math>x^2+1=0</math> have solutions. A '''complex number''' has the form <math>z=a+bi</math>, where <math>a</math> and <math>b</math> are real numbers and the '''imaginary unit''' <math>i</math> satisfies <math>i^2=-1</math>. The real numbers are therefore contained inside the complex numbers: a real number <math>a</math> can be written as <math>a+0i</math>.

In this aiMOOC for Grades 11–13, you will connect algebra, geometry, trigonometry, and applications. You will learn to calculate with complex numbers, represent them on the [[English:Complex plane|complex plane]], use [[English:Complex conjugate|conjugates]], move between rectangular and polar forms, interpret multiplication as scaling and rotation, apply [[English:Euler's formula|Euler's formula]], and find roots using [[English:De Moivre's formula|De Moivre's formula]]. You will also explore why complex numbers are important in polynomial equations, oscillations, electrical engineering, and signal analysis.

[[File:Complex Numbers Plane.svg|500px|frameless|center]]

The diagram above shows how real, imaginary, and genuinely complex numbers can be located on a two-dimensional number plane.

{{#ev:youtube|https://www.youtube.com/watch?v=SP-YJe7Vldo|500|center}}

The Khan Academy introduction above is useful for reviewing the transition from real to complex numbers before working with the more advanced representations in this course.

{{BR}}
== Learning Goals ==

By the end of the course, you should be able to explain the meaning of <math>a+bi</math>, perform the four arithmetic operations, use conjugates and modulus, plot complex numbers, convert between rectangular and polar forms, calculate powers and roots, interpret geometric transformations caused by multiplication, and justify how complex numbers complete important parts of algebra.

{{BR}}
= Foundations of Complex Numbers =

{{BR}}
== The Imaginary Unit and Rectangular Form ==

The defining relation <math>i^2=-1</math> gives a consistent way to work with square roots of negative real numbers. Powers of <math>i</math> repeat in a cycle:
<math>i^0=1</math>, <math>i^1=i</math>, <math>i^2=-1</math>, <math>i^3=-i</math>, and <math>i^4=1</math>. After that, the pattern repeats every four powers.

A complex number in '''rectangular form''' or '''Cartesian form''' is written
<math>z=a+bi</math>.
The number <math>a</math> is the '''real part''', written <math>\operatorname{Re}(z)=a</math>, and <math>b</math> is the '''imaginary part''', written <math>\operatorname{Im}(z)=b</math>. Notice that the imaginary part is the real coefficient <math>b</math>, not <math>bi</math>.

Two complex numbers are equal exactly when their corresponding real and imaginary parts are equal. Thus <math>a+bi=c+di</math> implies <math>a=c</math> and <math>b=d</math>.

{{BR}}
== Addition and Subtraction ==

Addition and subtraction are performed componentwise:
<math>(a+bi)+(c+di)=(a+c)+(b+d)i</math>,
and
<math>(a+bi)-(c+di)=(a-c)+(b-d)i</math>.

For example,
<math>(3+2i)+(-5+4i)=-2+6i</math>.
Geometrically, this works like adding two-dimensional vectors.

{{BR}}
== Multiplication ==

Use the distributive law and replace <math>i^2</math> by <math>-1</math>:
<math>(a+bi)(c+di)=ac+adi+bci+bdi^2=(ac-bd)+(ad+bc)i</math>.

For example,
<math>(2+3i)(1-4i)=2-8i+3i-12i^2=14-5i</math>.

Multiplication is more than an algebraic rule. In polar form, you will see that multiplying complex numbers multiplies their distances from the origin and adds their angles.

[[File:Complex multiplication.svg|500px|frameless|center]]

{{BR}}
= Conjugates, Modulus, and Division =

{{BR}}
== Complex Conjugates ==

For <math>z=a+bi</math>, the '''complex conjugate''' is
<math>\overline z=a-bi</math>.
On the complex plane, conjugation reflects a point across the real axis.

A central identity is
<math>z\overline z=(a+bi)(a-bi)=a^2+b^2</math>,
which is always a non-negative real number.

[[File:Complex conjugation.svg|500px|frameless|center]]

{{BR}}
== Modulus ==

The '''modulus''' or absolute value of <math>z=a+bi</math> is
<math>|z|=\sqrt{a^2+b^2}</math>.
Geometrically, this is the distance from the origin to the point <math>(a,b)</math> in the complex plane.

For example, if <math>z=3+4i</math>, then
<math>|z|=\sqrt{3^2+4^2}=5</math>.

The modulus satisfies useful properties such as <math>|zw|=|z||w|</math> and, for <math>w\ne 0</math>, <math>|z/w|=|z|/|w|</math>.

{{BR}}
== Division ==

To divide by a nonzero complex number, multiply numerator and denominator by the conjugate of the denominator:
<math>\frac{a+bi}{c+di}\cdot\frac{c-di}{c-di}</math>.
The denominator becomes the real number <math>c^2+d^2</math>.

For example,
<math>\frac{1+2i}{3-i}=\frac{(1+2i)(3+i)}{(3-i)(3+i)}=\frac{1+7i}{10}</math>.

This method works because a nonzero complex number has a reciprocal:
<math>\frac{1}{z}=\frac{\overline z}{|z|^2}</math>.

{{BR}}
= The Complex Plane =

{{BR}}
== From Coordinates to Geometry ==

The [[English:Complex plane|Complex plane]] places the real axis horizontally and the imaginary axis vertically. The complex number <math>z=a+bi</math> corresponds to the point <math>(a,b)</math> and also to the vector from the origin to that point. This geometric interpretation makes addition, subtraction, modulus, argument, and multiplication easier to visualize.

[[File:Complex plane examples 1.svg|500px|frameless|center]]

You can interpret <math>z+w</math> as vector addition. The difference <math>z-w</math> represents the displacement from <math>w</math> to <math>z</math>, so the distance between the points represented by <math>z</math> and <math>w</math> is <math>|z-w|</math>.

{{BR}}
== Loci in the Complex Plane ==

Equations and inequalities involving modulus often describe geometric sets. For example, <math>|z|=3</math> is a circle of radius 3 centered at the origin, while <math>|z-2|=1</math> is a circle of radius 1 centered at the point represented by the real number 2.

The condition <math>|z-a|=|z-b|</math> describes all points that are equally distant from the points <math>a</math> and <math>b</math>. Geometrically, this is the perpendicular bisector of the segment joining them.

{{BR}}
= Polar and Exponential Forms =

{{BR}}
== Argument and Polar Coordinates ==

For a nonzero complex number <math>z=a+bi</math>, let <math>r=|z|</math> and let <math>\theta</math> be an angle from the positive real axis to the vector representing <math>z</math>. Then
<math>a=r\cos\theta</math> and <math>b=r\sin\theta</math>,
so
<math>z=r(\cos\theta+i\sin\theta)</math>.

The angle <math>\theta</math> is an '''argument''' of <math>z</math>. Because adding a full turn does not change the point, all angles <math>\theta+2\pi k</math>, where <math>k</math> is an integer, represent the same direction. A common principal-value convention is <math>-\pi<\operatorname{Arg}(z)\le\pi</math>. The number zero has no defined argument.

[[File:Complex number illustration multiple arguments.svg|500px|frameless|center]]

{{#ev:youtube|https://www.youtube.com/watch?v=YZRzt_AsuVs|500|center}}

The polar form is especially efficient for multiplication, division, powers, and roots.

{{BR}}
== Multiplication as Scaling and Rotation ==

Suppose
<math>z_1=r_1(\cos\theta_1+i\sin\theta_1)</math>
and
<math>z_2=r_2(\cos\theta_2+i\sin\theta_2)</math>.
Their product is
<math>z_1z_2=r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))</math>.

Therefore multiplication by a nonzero complex number has a clear geometric effect: distances from the origin are scaled by its modulus and directions are rotated by its argument. Division reverses these operations by dividing moduli and subtracting arguments.

A particularly useful example is multiplication by <math>i</math>. Since <math>i=\cos(\pi/2)+i\sin(\pi/2)</math>, multiplying by <math>i</math> rotates a point by 90 degrees counterclockwise without changing its distance from the origin.

{{BR}}
== Euler's Formula ==

[[English:Euler's formula|Euler's formula]] connects the exponential function with trigonometry:
<math>e^{i\theta}=\cos\theta+i\sin\theta</math>.

This lets you write every nonzero complex number in '''exponential form''':
<math>z=re^{i\theta}</math>.

[[File:Euler's formula.svg|500px|frameless|center]]

Setting <math>\theta=\pi</math> gives
<math>e^{i\pi}=-1</math>,
or equivalently <math>e^{i\pi}+1=0</math>. This identity links five fundamental mathematical constants in one equation.

{{#ev:youtube|https://www.youtube.com/watch?v=sn3orkHWqUQ|500|center}}

The MIT OpenCourseWare video above develops complex numbers and Euler's formula in a context that connects naturally to differential equations and oscillations.

{{BR}}
= Powers and Roots =

{{BR}}
== De Moivre's Formula ==

For an integer <math>n</math>,
<math>(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)</math>.
More generally, if <math>z=r(\cos\theta+i\sin\theta)</math>, then
<math>z^n=r^n(\cos(n\theta)+i\sin(n\theta))</math>.

This is [[English:De Moivre's formula|De Moivre's formula]]. It turns repeated complex multiplication into two simpler operations: raise the modulus to a power and multiply the argument by the exponent.

For example, if <math>z=2(\cos 30^\circ+i\sin 30^\circ)</math>, then
<math>z^3=8(\cos 90^\circ+i\sin 90^\circ)=8i</math>.

{{BR}}
== Complex Roots ==

If
<math>z=r(\cos\theta+i\sin\theta)</math>
and you want all solutions of <math>w^n=z</math>, the roots are
<math>w_k=r^{1/n}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right)</math>
for <math>k=0,1,\ldots,n-1</math>.

These <math>n</math> roots have equal modulus and equally spaced arguments, so they form the vertices of a regular <math>n</math>-gon centered at the origin.

[[File:Visualisation complex number roots.svg|500px|frameless|center]]

{{BR}}
== Roots of Unity ==

The '''nth roots of unity''' are the solutions of <math>z^n=1</math>. They are
<math>e^{2\pi i k/n}</math> for <math>k=0,1,\ldots,n-1</math>.

For example, the fifth roots of unity lie at five equally spaced points on the unit circle.

[[File:One5Root.svg|500px|frameless|center]]

Roots of unity appear in polynomial factorization, discrete Fourier transforms, symmetry, and many areas of higher mathematics.

{{BR}}
= Polynomial Equations and Algebraic Closure =

A quadratic equation can have no real solutions but still have complex solutions. For example,
<math>x^2+4=0</math>
has solutions <math>x=\pm 2i</math>.

More generally, the [[English:Fundamental theorem of algebra|Fundamental theorem of algebra]] states that every nonconstant polynomial with complex coefficients has at least one complex root. Equivalently, a degree <math>n</math> polynomial has exactly <math>n</math> complex roots when multiplicities are counted. This is why the complex number system is described as '''algebraically closed'''.

If a polynomial has real coefficients and a non-real root <math>a+bi</math>, then its conjugate <math>a-bi</math> is also a root. This conjugate-pair property explains why non-real roots of real polynomials occur in pairs.

{{BR}}
= Applications and Connections =

{{BR}}
== Oscillations and Waves ==

Complex exponentials provide a compact language for sinusoidal behavior. By Euler's formula, a rotating complex quantity <math>e^{i\omega t}</math> contains both <math>\cos(\omega t)</math> and <math>\sin(\omega t)</math>. This makes calculations involving waves, harmonic motion, and differential equations more systematic.

{{BR}}
== Electrical Engineering ==

In alternating-current circuit analysis, sinusoidal voltages and currents can be represented by complex quantities called phasors. Resistance, capacitance, and inductance combine through complex impedance, allowing phase shifts and amplitudes to be handled algebraically.

{{BR}}
== Signal Processing ==

[[English:Fourier analysis|Fourier analysis]] represents signals using combinations of sinusoidal components. Complex exponentials are central because they encode amplitude and phase efficiently. The roots of unity also appear in the [[English:Discrete Fourier transform|discrete Fourier transform]], which is widely used in digital signal processing.

{{BR}}
== Two-Dimensional Rotations ==

If a point in the plane is represented by a complex number <math>z</math>, multiplication by <math>e^{i\theta}</math> rotates it through angle <math>\theta</math> around the origin. This provides a concise alternative to a two-dimensional rotation matrix in suitable problems.

{{BR}}
= Worked Examples =

{{BR}}
== Example: Solving a Quadratic ==

Solve <math>x^2-6x+13=0</math>. The quadratic formula gives
<math>x=\frac{6\pm\sqrt{36-52}}{2}=\frac{6\pm\sqrt{-16}}{2}=3\pm 2i</math>.
The two solutions are conjugates because the polynomial has real coefficients.

{{BR}}
== Example: Converting to Polar Form ==

For <math>z=-1+\sqrt{3}i</math>,
<math>|z|=\sqrt{(-1)^2+(\sqrt{3})^2}=2</math>.
The point lies in quadrant II with reference angle 60 degrees, so a principal argument is 120 degrees, or <math>2\pi/3</math> radians. Therefore
<math>z=2(\cos(2\pi/3)+i\sin(2\pi/3))=2e^{2\pi i/3}</math>.

{{BR}}
== Example: Geometric Multiplication ==

Let <math>z=2e^{i\pi/6}</math> and <math>w=3e^{i\pi/4}</math>. Then
<math>zw=6e^{i(5\pi/12)}</math>.
The modulus becomes <math>2\cdot 3=6</math>, while the arguments add: <math>\pi/6+\pi/4=5\pi/12</math>.

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''What value defines the square of the imaginary unit i?'''
(Negative one)
(!One)
(!Zero)
(!Two)

{{E}}
<br>

{{MC}}

'''What is the real part of three minus four i?'''
(Three)
(!Negative four)
(!Four)
(!Negative three)

{{E}}
<br>

{{MC}}

'''What is the modulus of three plus four i?'''
(Five)
(!Seven)
(!One)
(!Twenty five)

{{E}}
<br>

{{MC}}

'''What is the conjugate of two minus five i?'''
(Two plus five i)
(!Negative two plus five i)
(!Two minus five i)
(!Five plus two i)

{{E}}
<br>

{{MC}}

'''What is i multiplied by two plus three i?'''
(Negative three plus two i)
(!Three plus two i)
(!Negative two plus three i)
(!Two plus three i)

{{E}}
<br>

{{MC}}

'''What is the reciprocal of i?'''
(Negative i)
(!i)
(!One)
(!Negative one)

{{E}}
<br>

{{MC}}

'''What happens to arguments when two nonzero complex numbers are multiplied?'''
(Their arguments are added)
(!Their arguments are multiplied)
(!Their arguments are always equal)
(!Their arguments become zero)

{{E}}
<br>

{{MC}}

'''What does Euler's formula connect with the complex exponential?'''
(Sine and cosine)
(!Prime numbers and factors)
(!Matrices and determinants)
(!Logarithms and inequalities)

{{E}}
<br>

{{MC}}

'''Which equation is satisfied by every fourth root of unity?'''
(z to the fourth power equals one)
(!z squared equals two)
(!z cubed equals negative one)
(!z to the fifth power equals zero)

{{E}}
<br>

{{MC}}

'''How many complex roots does a degree six polynomial have when multiplicity is counted?'''
(Six)
(!Three)
(!Five)
(!Twelve)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Imaginary unit || Number whose square is negative one
|-
| Complex conjugate || Reflection of a complex number across the real axis
|-
| Modulus || Distance of a complex number from the origin
|-
| Argument || Angle from the positive real axis to a nonzero complex number
|-
| Polar form || Representation using a magnitude and an angle
|-
| Root of unity || Solution obtained from an equation in which a power equals one
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Real part'''
| Horizontal coordinate on the complex plane
|-
| '''Imaginary part'''
| Vertical coordinate on the complex plane
|-
| '''Conjugation'''
| Reflection across the real axis
|-
| '''Multiplication'''
| Scaling of modulus and addition of arguments
|-
| '''Division'''
| Division of moduli and subtraction of arguments
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Modulus || What word names the distance of a complex number from the origin?
|-
| Argument || What word names the direction angle of a nonzero complex number?
|-
| Conjugate || What is the reflected partner of a complex number across the real axis called?
|-
| Imaginary || Which word describes the part whose coefficient multiplies i?
|-
| Euler || Which mathematician is associated with the formula connecting complex exponentials and trigonometry?
|-
| Rotation || What geometric motion is produced by multiplication with a unit complex number?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Complex+Numbers </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
A complex number can be written in rectangular form using the imaginary unit { i }. The horizontal coordinate on the complex plane is the { real part }. The distance from a complex number to the origin is its { modulus }. Reflecting a point across the real axis produces its { conjugate }. A direction angle of a nonzero complex number is called an { argument }. In polar form, multiplication causes the moduli to be multiplied while the angles are { added }. Euler's formula links the complex exponential with sine and { cosine }. De Moivre's formula is especially useful for calculating powers and { roots }. The solutions of a power equation equal to one are called roots of { unity }. The fundamental theorem of algebra guarantees complex roots for every nonconstant { polynomial }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Complex plane poster|Complex plane poster]]: Create a labeled poster showing the real axis, imaginary axis, four complex numbers, their conjugates, and the modulus of one selected number.
# [[English:Imaginary unit pattern|Imaginary unit pattern]]: Build a visual or digital pattern showing at least twelve consecutive powers of i, then explain why the values repeat every four powers.
# [[English:Complex number story|Complex number story]]: Write a short explanation for a younger learner that shows why extending the real numbers makes an equation such as x squared plus one equals zero solvable.
# [[English:Geometry photo analogy|Geometry photo analogy]]: Find or create an image that illustrates reflection and rotation, then annotate it to compare those motions with conjugation and multiplication by a unit complex number.

{{BR}}
=== Standard ===
# [[English:Complex arithmetic tutorial|Complex arithmetic tutorial]]: Produce a one-page tutorial or short video that teaches addition, multiplication, conjugation, and division with one fully explained example of each.
# [[English:Polar form investigation|Polar form investigation]]: Choose six nonzero complex numbers in different quadrants, calculate their moduli and principal arguments, and compare rectangular, trigonometric, and exponential forms.
# [[English:STEM interview|STEM interview]]: Interview a mathematics, physics, computing, or engineering teacher or professional about where complex numbers appear in their work, then summarize the mathematical ideas in the interview.
# [[English:Roots of unity model|Roots of unity model]]: Draw or construct a digital model of the sixth roots of unity, explain the symmetry, and verify algebraically that each point satisfies the defining equation.

{{BR}}
=== Advanced ===
# [[English:Complex transformation experiment|Complex transformation experiment]]: Use dynamic geometry or graphing software to multiply many points by one fixed complex number and infer the scaling-and-rotation rule from your observations.
# [[English:Polynomial roots project|Polynomial roots project]]: Investigate several real polynomials with non-real roots, test the conjugate-pair pattern, and explain how your evidence relates to the fundamental theorem of algebra.
# [[English:Phasor application study|Phasor application study]]: Research a simple alternating-current circuit example, represent a voltage or current with a complex phasor, and explain what modulus and argument mean in that context.
# [[English:Fourier connection project|Fourier connection project]]: Create a report, animation, or presentation showing how complex exponentials and roots of unity contribute to a discrete Fourier transform and why this is useful for signal analysis.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Representation and reasoning|Representation and reasoning]]: Given a complex number in rectangular form, represent it geometrically and in polar form, then justify every step and explain what each representation makes easier to see.
# [[English:Error analysis|Error analysis]]: Analyze a worked solution in which a student incorrectly divides two complex numbers, identify the exact mathematical error, correct it, and explain why the conjugate method works.
# [[English:Geometric transfer|Geometric transfer]]: Predict the image of a set of points after multiplication by a fixed nonzero complex number, then verify your prediction algebraically and describe the transformation in terms of scaling and rotation.
# [[English:Polynomial connection|Polynomial connection]]: Solve a polynomial problem with non-real roots and explain how conjugate pairs and the fundamental theorem of algebra help you check whether your answer is complete.
# [[English:Method comparison|Method comparison]]: Calculate a high power of a complex number using both rectangular multiplication and polar or exponential form, then compare the efficiency and reliability of the two approaches.
# [[English:Application interpretation|Application interpretation]]: Choose a context such as oscillations, alternating-current circuits, or signal processing and explain how modulus and argument can represent measurable features such as amplitude and phase.

<br>
<br>

{{BR}}
= Evidence of Learning =

'''Knowledge:''' You can define the imaginary unit, real and imaginary parts, conjugate, modulus, argument, rectangular form, polar form, exponential form, and roots of unity, and you can state the central ideas of Euler's formula, de Moivre's formula, and the fundamental theorem of algebra.

'''Skills:''' You can perform arithmetic with complex numbers, convert between representations, locate numbers on the complex plane, determine modulus and argument, calculate powers and roots, and interpret multiplication geometrically.

'''Products:''' Your diagrams, written explanations, calculations, digital models, investigations, interviews, videos, or presentations communicate correct mathematics and make your reasoning visible.

'''Reasoning:''' You can choose an efficient representation for a problem, justify why a method works, check results using geometric and algebraic relationships, and explain patterns such as conjugate root pairs and equally spaced roots of unity.

'''Transfer:''' You can apply complex-number ideas to unfamiliar polynomial, geometric, oscillation, circuit, or signal problems and explain how the mathematics models the new context.

<br>
<br>

{{BR}}
= OERs on the Topic =

The English Wikipedia article on [[English:Complex number|complex numbers]] provides a broad reference for definitions, history, representations, operations, and applications.

<iframe> https://en.m.wikipedia.org/wiki/Complex_number </iframe>

<br>

{{BR}}
= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Complex Numbers|Complex Numbers]]'''
# [[English:Imaginary unit|Imaginary unit]]
# [[English:Complex plane|Complex plane]]
# [[English:Complex conjugate|Complex conjugate]]
# [[English:Absolute value|Absolute value]]
# [[English:Polar coordinates|Polar coordinates]]
# [[English:Euler's formula|Euler's formula]]
# [[English:De Moivre's formula|De Moivre's formula]]
# [[English:Roots of unity|Roots of unity]]
# [[English:Fundamental theorem of algebra|Fundamental theorem of algebra]]
# [[English:Fourier analysis|Fourier analysis]]
|}

Complex numbers connect strongly with [[English:Algebra|Algebra]], [[English:Geometry|Geometry]], [[English:Trigonometry|Trigonometry]], [[English:Precalculus|Precalculus]], [[English:Polynomial|polynomials]], [[English:Differential equation|differential equations]], [[English:Electrical engineering|Electrical engineering]], and [[English:Signal processing|Signal processing]]. These connections make the topic a bridge between upper-secondary mathematics and university-level mathematics, physics, engineering, and computing.

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Complex Numbers]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Precalculus]]
[[Category:Grades 11-13]]
[[Category:Secondary education]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Complex Numbers - Versionsgeschichte

Glanz: aiMOOC über GPT aiMOOC Action erstellt