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Chapter 3: Problem 1

Stellen Sie die folgenden komplexen Zahlen durch Bildpunkte in der GauBschen Zahlenebene symbolisch dar: $$ \begin{array}{lll} z_{1}=3-4 j, & z_{2}=-2+3 j, \quad z_{3}=-5-4 j, & z_{4}=6. \\ z_{5}=3+5 j, & x_{6}=-1-2 j, \quad z_{7}=-4+j, & z_{8}=-3 j \end{array} $$

Short Answer

Expert verified
Plot the points (3, -4), (-2, 3), (-5, -4), (6, 0), (3, 5), (-1, -2), (-4, 1), (0, -3) on the Gaussian plane.

Step by step solution

01

Understanding the Gaussian Plane

The Gaussian plane (or complex plane) is a two-dimensional plane where the horizontal axis represents the real part of complex numbers and the vertical axis represents the imaginary part.
02

Plotting Point for z_1

Identify the real and imaginary parts of the complex number. For \( z_1 = 3 - 4j \), the real part is 3, and the imaginary part is -4. Plot the point (3, -4) on the plane.
03

Plotting Point for z_2

For \( z_2 = -2 + 3j \), the real part is -2, and the imaginary part is 3. Plot the point (-2, 3) on the plane.
04

Plotting Point for z_3

For \( z_3 = -5 - 4j \), the real part is -5, and the imaginary part is -4. Plot the point (-5, -4) on the plane.
05

Plotting Point for z_4

For \( z_4 = 6 \), there is no imaginary part, so it is on the real axis. Plot the point (6, 0) on the plane.
06

Plotting Point for z_5

For \( z_5 = 3 + 5j \), the real part is 3, and the imaginary part is 5. Plot the point (3, 5) on the plane.
07

Plotting Point for z_6

For \( z_6 = -1 - 2j \), the real part is -1, and the imaginary part is -2. Plot the point (-1, -2) on the plane.
08

Plotting Point for z_7

For \( z_7 = -4 + j \), the real part is -4, and the imaginary part is 1. Plot the point (-4, 1) on the plane.
09

Plotting Point for z_8

For \( z_8 = -3j \), there is no real part, so it is on the imaginary axis. Plot the point (0, -3) on the 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, popularly known as the complex plane, is a method to visualize complex numbers. Picture a two-dimensional grid where
  • The horizontal axis (x-axis) represents real numbers.
  • The vertical axis (y-axis) represents imaginary numbers.
Every complex number can be placed on this plane. For example, a complex number like \(3 - 4j\) would be placed at (3, -4). Real numbers lie on the real axis. Imaginary numbers lie on the imaginary axis.
Imaginary Part
In a complex number, the imaginary part is the coefficient of the imaginary unit \(j\) (or \(i\) in mathematical notation). If you have a complex number like \(z = a + bj\), '
  • 'a' is the real part.
  • 'b' is the imaginary part.
This imaginary part helps us understand the direction and position of the complex number on the Gaussian plane. For example:
  • In \(z_1 = 3 - 4j\), the imaginary part is -4.
  • In \(z_2 = -2 + 3j\), the imaginary part is 3.
Remember, the imaginary part is a real number that scales the imaginary unit, which is perpendicular to the real axis.
Real Part
The real part of a complex number is the non-imaginary component. For a complex number \(z = a + bj\):
  • 'a' is the real part.
  • 'b' is the imaginary part.
On the Gaussian plane, the real part determines how far left or right a point is on the horizontal axis. It shows the 'real' value of the complex number:
  • In \(z_1 = 3 - 4j\), the real part is 3.
  • In \(z_2 = -2 + 3j\), the real part is -2.
Without the imaginary part, this number would behave like a regular real number on the Cartesian plane.
Plotting Points
Plotting points on the Gaussian plane is like plotting any other pair of coordinates. The key steps are:
1. Identify the real part and plot it on the x-axis.
2. Identify the imaginary part and plot it on the y-axis.
Each complex number corresponds to a unique point on this plane. Let’s map some examples:
  • \(z_1 = 3 - 4j\): Real part is 3, imaginary part is -4 -> Point (3, -4)
  • \(z_2 = -2 + 3j\): Real part is -2, imaginary part is 3 -> Point (-2, 3)
For numbers without an imaginary part like \(z_4 = 6\), they fall exactly on the real axis as (6, 0). If there's no real part like \(z_8 = -3j\), it falls on the imaginary axis as (0, -3).

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Most popular questions from this chapter

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?

Bestimmen Sie s?mtliche Lösungen: a) \(\quad z^{3}=64\left[\cos \left(\frac{\pi}{4}\right)+j \cdot \sin \left(\frac{\pi}{4}\right)\right]\) b) \(z^{3}-2=5 \mathrm{j}\)

Die mechanischen Schwingungen $$ y_{1}(t)=20 \mathrm{~cm} \cdot \sin \left(\pi \mathrm{s}^{-1} \cdot t+\frac{\pi}{10}\right) \text { und } y_{2}(t)=15 \mathrm{~cm} \cdot \cos \left(\pi \mathrm{s}^{-1} \cdot t+\frac{\pi}{6}\right) $$ werden ungestört zur Uberiagerung gebracht. Wie lautet die resultierende Schwingung (in der Kosimusform dargestelit)?

Bestimmen Sie s?mtliche reellen und komplexen Lösungen der folgenden Gleichungen: a) \(x^{3}-x^{2}+4 x-4=0\) (Horner-Schema verwenden) b) \(x^{4}-2 x^{2}-3=0\)

Berechnen Sie mit den komplexen Zahlen \(z_{1}=-4 \mathrm{j}, \quad z_{2}=3-2 \mathrm{j}, \quad z_{3}=-1+\mathrm{j}\) die folgenden Terme: a) \(z_{1}-2 z_{2}+3 z_{3}\) b) \(\quad 2 z_{1} \cdot z_{2}^{*}\) c) \(\frac{z_{1}^{*} \cdot z_{2}}{z_{3}}\) d) \(z_{1}\left(2 z_{2}^{*}-z_{1}\right)+z_{3}^{*}\) e) \(\frac{z_{1}-z_{2}^{*}}{3 z_{3}^{*}}\) f) \(\frac{z_{1}+z_{3}^{*}}{z_{2}^{*} \cdot z_{3}}\)
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