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Chapter 3: Problem 4

Find the Cartesian representation \(z=x+j y\) for each of the following complex numbers: a. \(z=\sqrt{2} e^{j \pi / 2}\) b. \(z=\sqrt{2} e^{j \pi / 4}\) C. \(z=e^{j 3 \pi / 4}\) d. \(z=\sqrt{2} e^{j 3 \pi / 2}\) Plot the points on the complex plane.

Short Answer

Expert verified
(a) \(0 + j\sqrt{2}\), (b) \(1 + j1\), (c) \(-\frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}\), (d) \(0 - j\sqrt{2}\).

Step by step solution

01

- Convert Polar Form to Cartesian Form

Complex numbers in polar form can be written as \(z = re^{j\theta}\), where \(r\) is the magnitude, and \(\theta\) is the angle. To convert it to Cartesian form \(z = x + jy\), use the equations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
02

- Calculate Cartesian Form for Part (a)

Given \(z = \sqrt{2} e^{j \pi / 2}\): \[ x = r \cos(\theta) = \sqrt{2} \cos(\pi / 2) = \sqrt{2} \cdot 0 = 0 \]\[ y = r \sin(\theta) = \sqrt{2} \sin(\pi / 2) = \sqrt{2} \cdot 1 = \sqrt{2} \]Thus, the Cartesian representation is \(z = 0 + j\sqrt{2}\).
03

- Calculate Cartesian Form for Part (b)

Given \(z = \sqrt{2} e^{j \pi / 4}\): \[ x = r \cos(\theta) = \sqrt{2} \cos(\pi / 4) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 \]\[ y = r \sin(\theta) = \sqrt{2} \sin(\pi / 4) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1 \]Thus, the Cartesian representation is \(z = 1 + j1\).
04

- Calculate Cartesian Form for Part (c)

Given \(z = e^{j 3\pi / 4}\): \[ x = r \cos(\theta) = 1 \cdot \cos(3\pi / 4) = \cos(3\pi / 4) = -\frac{\sqrt{2}}{2} \]\[ y = r \sin(\theta) = 1 \cdot \sin(3\pi / 4) = \sin(3\pi / 4) = \frac{\sqrt{2}}{2} \]Thus, the Cartesian representation is \(z = -\frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}\).
05

- Calculate Cartesian Form for Part (d)

Given \(z = \sqrt{2} e^{j 3\pi / 2}\): \[ x = r \cos(\theta) = \sqrt{2} \cos(3\pi / 2) = \sqrt{2} \cdot 0 = 0 \]\[ y = r \sin(\theta) = \sqrt{2} \sin(3\pi / 2) = \sqrt{2} \cdot (-1) = -\sqrt{2} \]Thus, the Cartesian representation is \(z = 0 - j\sqrt{2}\).
06

- Plot the Points

Plot the points on the complex plane based on the Cartesian representations calculated:(a) \(0 + j\sqrt{2}\)(b) \(1 + j1\)(c) \(-\frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}\)(d) \(0 - j\sqrt{2}\).Each point is represented on the complex plane with the x-coordinate as the real part and the y-coordinate as the imaginary part.

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Key Concepts

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

Polar to Cartesian conversion
Converting a complex number from polar form to Cartesian form is an essential skill in mathematics and engineering. In polar form, a complex number is represented as \(z = re^{j\theta}\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the angle (or argument). To convert it to Cartesian form \(z = x + jy\), use the following equations:
\[ x = r\cos(\theta) \quad \text{and} \quad y = r\sin(\theta) \]
By calculating these values, we decode the real part \(x\) and the imaginary part \(y\). This conversion is straightforward when you correctly apply the sine and cosine functions to the angle.
Imaginary unit
The imaginary unit, denoted as \(j\) in electrical engineering (or \(i\) in pure mathematics), is a fundamental concept in complex numbers. Its primary property is that \(j^2 = -1\). In the complex number representation \(z = x + jy\), \(x\) is the real part, and \(jy\) is the imaginary part. Because of the imaginary unit:
  • It enables the representation of two-dimensional numbers.
  • It allows operations beyond real number constraints, greatly expanding algebraic functions.
Understanding the imaginary unit is crucial for grasping complex numbers and their applications in various fields such as physics and control systems.
Complex plane plotting
Plotting complex numbers on the complex plane helps visualize the behavior and relationship of these numbers. The complex plane consists of a horizontal axis (real part) and a vertical axis (imaginary part). Each complex number \(z = x + jy\) is a point on this plane.
Here are the steps to plot:
  • Identify the real part \(x\) which corresponds to the horizontal position.
  • Identify the imaginary part \(jy\) which corresponds to the vertical position.
For example, consider \(z = 1 + j1\). The plot will be at point (1, 1) on the complex plane. This visualization is useful for understanding operations like addition, subtraction, and complex conjugation.
Magnitude and angle in polar form
The magnitude and angle are the key components of a complex number in polar form. The magnitude \(r\) represents the distance from the origin to the point in the complex plane, calculated as:
\[ r = \sqrt{x^2 + y^2} \]
The angle \(\theta\), or argument, is the counterclockwise angle from the positive real axis to the line segment connecting the origin to the point, often calculated using:
\[ \theta = \text{atan2}(y, x) \]
These components allow for the efficient analysis of complex numbers, especially in fields like signal processing and electrical engineering. Converting between these forms often requires the use of trigonometric identities and functions.

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Most popular questions from this chapter

Call \(\mathrm{R}(\theta)\) the rotation matrix: $$ \mathrm{R}(\theta)=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] $$ Show that \(\mathrm{R}(-\theta)\) rotates by \((-\theta)\). What can you say about \(\mathrm{R}(-\theta) w\) when \(w=\mathrm{R}(\theta) z\) ?

Prove \(e^{j[(\pi / 2)+m 2 \pi]}=j, e^{j[(3 \pi / 2)+m 2 \pi]}=-j, e^{j(0+m 2 \pi)}=1\), and \(e^{j(\pi+m 2 \pi)}=-1\). Plot these identities on the complex plane. (Assume \(\mathrm{m}\) is an integer.)

Show that the additive inverse of \(z=r e^{j \theta}\) may be written as \(r e^{j(\theta+\pi)}\).

Compute and plot the roots of the following quadratic equations: a. \(z^{2}+2 z+1=0\) b. \(z^{2}-2 z+1=0\) c. \(z^{2}=0\) For each equation, check that \(z_{1}+z_{2}=-\frac{b}{a}\) and \(z_{1} z_{2}=\frac{c}{a}\)

Compute and plot the roots of the following quadratic equations: a. \(z^{2}+2 z+2=0\) b. \(z^{2}-2 z+2=0\) c. \(z^{2}+2=0\) For each equation, check that \(2 \operatorname{Re}\left[z_{1,2}\right]=-\frac{b}{a}\) and \(\left|z_{1,2}\right|^{2}=\frac{c}{a}\).
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