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Chapter 3: Problem 2

Die folgenden komplexen Zahlen sind durch einen Zeiger in der GauBschen Zahlenebene bildlich darzustellen: $$ \begin{array}{llll} z_{1}=1+3 \mathrm{j}, & z_{2}=-2-4 \mathrm{j}, & z_{3}=1-\mathrm{j}, & z_{4}=5 \mathrm{j}, \\ z_{5}=-4+5 \mathrm{j}, & z_{6}=-3-2 \mathrm{j}, & z_{7}=6+2 \mathrm{j}, & z_{8}=-5 \end{array} $$

Short Answer

Expert verified
Plot each complex number as a point with its real part as the x-coordinate and its imaginary part as the y-coordinate.

Step by step solution

01

Understand Complex Numbers

Complex numbers are in the form of a + bj, where a is the real part and bj is the imaginary part. Each number can be represented as a point on the Gaussian plane with the real part (a) as the x-coordinate and the imaginary part (b) as the y-coordinate.
02

Plot the Point for \(z_1\)

The first complex number is \(z_1 = 1 + 3j\). It has a real part 1 and an imaginary part 3. Plot the point (1, 3) on the Gaussian plane.
03

Plot the Point for \(z_2\)

The second complex number is \(z_2 = -2 - 4j\). It has a real part -2 and an imaginary part -4. Plot the point (-2, -4) on the Gaussian plane.
04

Plot the Point for \(z_3\)

The third complex number is \(z_3 = 1 - j\). It has a real part 1 and an imaginary part -1. Plot the point (1, -1) on the Gaussian plane.
05

Plot the Point for \(z_4\)

The fourth complex number is \(z_4 = 5j\). It has a real part 0 and an imaginary part 5. Plot the point (0, 5) on the Gaussian plane.
06

Plot the Point for \(z_5\)

The fifth complex number is \(z_5 = -4 + 5j\). It has a real part -4 and an imaginary part 5. Plot the point (-4, 5) on the Gaussian plane.
07

Plot the Point for \(z_6\)

The sixth complex number is \(z_6 = -3 - 2j\). It has a real part -3 and an imaginary part -2. Plot the point (-3, -2) on the Gaussian plane.
08

Plot the Point for \(z_7\)

The seventh complex number is \(z_7 = 6 + 2j\). It has a real part 6 and an imaginary part 2. Plot the point (6, 2) on the Gaussian plane.
09

Plot the Point for \(z_8\)

The eighth complex number is \(z_8 = -5\). It has a real part -5 and an imaginary part 0. Plot the point (-5, 0) on the Gaussian plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gaussian plane
The Gaussian plane, also known as the complex plane, is a two-dimensional plane used to visually represent complex numbers. On this plane, the horizontal axis (x-axis) represents the real part of a complex number and the vertical axis (y-axis) represents the imaginary part. Each complex number corresponds to a unique point in this plane. For example, the complex number 1 + 3j would be represented by the point (1, 3). This graphical representation helps in understanding and visualizing the behavior of complex numbers, especially when performing operations like addition, subtraction, multiplication, and division.
Complex number representation
Complex numbers are expressed in the form of a + bj, where 'a' is the real part and 'b' is the imaginary part. The letter 'j' (or sometimes 'i') denotes the imaginary unit, which satisfies the equation \( j^2 = -1 \). To represent a complex number on the Gaussian plane, you plot 'a' on the x-axis and 'b' on the y-axis. For instance, in the complex number -2 - 4j, the real part is -2 and the imaginary part is -4. This number would be represented by the point (-2, -4) on the Gaussian plane.
Real and imaginary parts
Understanding the real and imaginary parts of a complex number is crucial for plotting it on the Gaussian plane. The real part is the component that lies on the x-axis, while the imaginary part lies on the y-axis. For example, for the complex number 1 - j, the real part is 1 and the imaginary part is -1. In mathematical terms: \( z = a + bj = \text{real part} + \text{imaginary part} \). Differentiating between these two parts can help in operations like addition and multiplication of complex numbers.
Plotting complex numbers
To plot a complex number on the Gaussian plane, follow these steps:
  • Identify the real part and imaginary part of the complex number.
  • Locate the real part on the x-axis.
  • Locate the imaginary part on the y-axis.
  • Mark the corresponding point on the plane.
For example, for the number 5j, the real part is 0 and the imaginary part is 5. You would plot this as the point (0, 5). This method works for any complex number, making it easier to visualize and understand their positions and relationships.

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Most popular questions from this chapter

Die in der kartesischen Form gegebenen komplexen Zahlen $$ \begin{array}{llll} z_{1}=2+\pi \mathrm{j} . & z_{2}=4,5-2,4 \mathrm{j}, & z_{3}=-3+5 \mathrm{j}, & z_{4}=-6 \\ z_{5}=-3-2 \mathrm{j}, & z_{6}=-1+\mathrm{j}, & z_{7}=-4 \mathrm{j}, & z_{8}=-3-\mathrm{j} \end{array} $$ sind in die Polarform umzurechnen. Wie lauten die konjugiert komplexen Zahlen?

Zeigen Sie: F?r jede komplexe Zahl \(z=x+j y\) gilt: a) \(z+z^{*}=2 \cdot \operatorname{Re}(z)\) b) \(z-z^{*}=2 \mathrm{j} \cdot \operatorname{Im}(z)\)

Mit dem komplexen Zeiger \(\underline{z}=1+2 \mathrm{j}\) werden folgende Operationen durchgeführt: a) \(j \cdot z\) b) \(\quad \underline{z}^{*}\) c) \(\frac{\pi}{j}\) d) \(2.2\) e) \(\mathrm{e}^{\mathrm{J} 30^{\circ}} \cdot z\) f) \(|z|\) g) \(z^{2}\) Stellen Sie diese Operationen in der GauBschen Zahlenebene bildlich dar. Was bedeuten sie geometrisch?

Zeichnen Sie die Ortskurven der folgenden komplexen Zeiger: a) \(z(t)=a \cdot \cos t+j \cdot b \cdot \sin t \quad(0 \leqslant t<2 \pi)\) b) \(\quad z(t)=2 \cdot \cos ^{2} t+j \cdot \sin (2 t) \quad\left(0 \leqslant t<\frac{\pi}{2}\right)\)

Stellen Sie die folgenden Kosinusschwingungen als komplexe Sinuszeiger dar: a) \(y_{1}=3 \cdot \cos (\omega t)\) b) \(y_{2}=-4 \cdot \cos \left(2 t-\frac{\pi}{4}\right)\) c) \(y_{3}=5 \cdot \cos (t+1)\) d) \(y_{4}=3 \cdot \cos \left(\pi t+\frac{2 \pi}{3}\right)\) Anleitung: Bringen Sie die Kosinusschwingungen zunãchst in die Sinusform.
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