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Chapter 11: Problem 2

Simplify the following complex numbers and diagram: show them on an Argand diagram: (a) \(3+3 \mathrm{j}\) (b) \(2=4\) (e) \(-\mathrm{i}\) (f) \(-5=1 \mathrm{j}\) (c) \(-0.5\) (d) \(6 \mathrm{j}\) (a) \(j^{2}\) (b) \(-\mathrm{j}^{2}\) (c) \((-\mathrm{j})^{2}\) (d) \(j^{3}\left(\right.\) e) \(j^{4}\)

Short Answer

Expert verified
Question: Simplify the given complex numbers, compute and simplify the expressions with j raised to different powers, and plot the results on an Argand diagram. Answer: The complex numbers are already in their simplest form. The simplified expressions for j raised to different powers are: (a) j^2 = -1, (b) -j^2 = 1, (c) (-j)^2 = -1, (d) j^3 = -j, (e) j^4 = 1. The corresponding Argand diagram represents these complex numbers as points or vectors in a 2-dimensional plane, with real parts along the x-axis and imaginary parts along the y-axis.

Step by step solution

01

Convert the given complex numbers into the form a + bj

All the given complex numbers are already in the simplest form, where a and b are real numbers and j is the imaginary unit.
02

Plot the complex numbers on an Argand diagram

An Argand diagram is a plot of complex numbers in a 2-dimensional plane showing the real part on the x-axis and imaginary part on the y-axis. Identify the real and imaginary parts of each complex number and plot them as points or vectors on the plane.
03

Compute and simplify j raised to different powers

Calculate j raised to various powers based on the properties and cyclicality of imaginary unit: - j^2 = -1 - j^3 = j * j^2 = j * -1 = -j - j^4 = j^2 * j^2 = -1 * -1 = 1 - (-j)^2 = (-1 * j)^2 = (-1)^2 * j^2 = 1 * -1 = -1 The simplified expressions are: (a) j^2 = -1 (b) -j^2 = 1 (c) (-j)^2 = -1 (d) j^3 = -j (e) j^4 = 1
04

Plot results of Step 3 on Argand diagram

For the results found in Step 3, plot the points on Argand diagram using their real and imaginary parts as coordinates: (a) -1 + 0j (b) 1 + 0j (c) -1 + 0j (d) 0 - 1j (e) 1 + 0j

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