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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鈥檚 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

Bestimmen Sie den Betrag der folgenden komplexen Zahlen: $$ \begin{array}{lll} z_{1}=4-3 \mathrm{j} . & z_{2}=-2-6 \mathrm{j}, & z_{3}=3\left(\cos 60^{\circ}-\mathrm{j} \cdot \sin 60^{\circ}\right) \\ z_{4}=-3+4 \mathrm{j}, & z_{5}=-4 \mathrm{j}, & z_{6}=-3 \cdot \mathrm{e}^{j} 30^{\circ} \end{array} $$

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

Bestimmen Sie s?mtliche L枚sungen: 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 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?

Bringen Sie die in der Polarform vorliegenden komplexen Zahlen $$ \begin{array}{ll} z_{1}=4(\cos 1+\mathrm{j} \cdot \sin 1), & z_{2}=3 \cdot \mathrm{e}^{\mathrm{j} 30^{\circ}}, \quad z_{3}=5 \cdot \mathrm{e}^{j 135^{\circ}} \\ z_{4}=5\left[\cos \left(-60^{\circ}\right)+\mathrm{j} \cdot \sin \left(-60^{\circ}\right)\right], & z_{5}=2 \cdot \mathrm{e}^{-\mathrm{j} \frac{3}{2} \pi}, \quad z_{6}=\mathrm{e}^{\mathrm{j} 240^{\circ}} \\ z_{7}=2\left(\cos 210^{\circ}+\mathrm{j} \cdot \sin 210^{\circ}\right), & z_{8}=\cos (-0,5)+\mathrm{j} \cdot \sin (-0,5) \end{array} $$ in die kartesische Form und bestimmen Sie die jeweilige konjugiert komplexe Zahl.
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