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ES193 - Engineering Mathematics - Week 3 Practise Questions and Solutions - University of Warwick
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Questions and Solutions for week 3 of the ES193 Engineering Mathematics module for the Engineering course.
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School of Engineering, University of WarwickES193 Engineering Mathematics
Briefing Sheet:
Topic: COMPLEX NUMBERS
1. Motivation
Complex numbers are widely used in the analysis of electrical circuits and mechanical vibrating
systems. An understanding of them is also a pre-requisite for the theory of complex analytical
functions, which has wide application in heat transfer, fluid mechanics and electrostatics.
2. Key Concepts
This week several concepts need to be grasped:
√
• that −1 can be called i (or j in some texts) an imaginary quantity
• that complex numbers have real and imaginary parts
• that complex numbers can be added, subtracted, multiplied and divided
• that there is a geometric representation of complex numbers
• that there is a modulus-argument form of a complex number Polar
• de Moivre’s theorem
• that numerous roots of complex numbers can be found.
3. Learning Outcomes
At the end of this week’s work you should be able to:
• add, subtract, multiply and divide complex numbers, be they in real-imaginary or
modulus-argument form
• draw the graphical representation of a complex number
• convert complex numbers between real-imaginary and modulus-argument forms
• apply de Moivre’s theorem
• find roots of complex numbers
4. This Week’s Reading
For his week’s work we jump to Chapter 11 of the book. This jump should not concern you as the
topic is self-contained. However, the polar coordinates studied previously will have prepared you
for the r6 θ form of complex numbers.
√
Important Throughout the text book j is used for −1. However, in the Data Book, the revision
questions below and, ultimately, the examination questions, i will be used. Whether i or j is used
seems to depend upon the text book. Here at Warwick, we have settled on the use of i because
this practice is most widespread.
As in previous weeks, I am suggesting certain questions from the block exercises which serve to
help you learn. You should do as many of these as you feel necessary to be confident that you have
understood the topic. The revision questions in §6 below should enable you to find out whether
you have ‘exercised enough’ !
1
, 5. Example Problems
Chapter 11
Pages 462-464: BLOCK 1: Arithmetic of complex numbers
(the imaginary number i (in book, j) such that i2 = −1; complex conjugate;
addition, subtraction, multiplication and division of complex numbers)
Exercises: Page 454, Q1, Q3, Q4 (a) & (b) only; Page 456, Q2, Q3; Page 457, Q1,
Q3; Page 459, Q1, Q2, Q6; Page 462, Q2, Q3, Q5
Pages 465-489: BLOCK 2: The Argand diagram and polar form of a complex number
(Plotting complex numbers - Argand diagram; r-θ form (use radians as opposed
to degrees); the notation r6 θ; the equivalent form z = r(cos θ + i sin θ))
Exercises: Page 473, Q2 (f) & (g) only, Q4, Q5; Page 480, Q2, Q3, Q6
Pages 490-495: BLOCK 3: The exponential form of a complex number
(Power series of cosx and sinx; Euler’s relation; exponential form of a complex
number
Exercises: Page 494, Q1, Q2, Q3; Page 494 (End-of-Block), Q1, Q2, Q3
Pages 497-503: BLOCK 4: De Moivre’s theorem
(use of De Moivre’s theorem - note that the formula is given in the Data Book
on Page 3)
Exercises: Page 499, Q1, Q2, Q3
Pages 504-511: BLOCK 5: Solving equations and finding roots of complex numbers
(General form of argument, θ + k2π, where k is an integer; finding complex roots,
1
eg. z 3 ; solving polynomial equations in (complex) z)
Exercises: Page 510, Q2, Q3 (a) & (c) only; Page 511 (End-of-block), Q5, Q6
6. Revision Questions for Tutorials
At the completion of the above learning, now attempt each of the following revision questions (not
from the book) which relate to all of this week’s topics. Your tutor will expect you to be familiar
with (either have done or attempted) these questions because they will be raised at your weekly
tutorial.
Q1 (*) Two complex numbers are given as z1 = 3 + 4i and z2 = −2 + 3i.
Find (a) z1 + z2 , (b) z1 − z2 , (c) -z1 + z2 , (d) 3z1 + 2z2 .
Draw z1 , z2 and the operations represented by (a) and (b) on an Argand diagram.
[Ans. (a) 1 + 7i, (b) 5 + i, (c) −5 − i, (d) 5 + 18i]
Q2 (*) Two complex numbers are given as z1 = 3 + i and z2 = 1 − 2i.
Find (a) z1 z2 , (b) zz21 ,
[Ans. (a) 5(1 − i), (b) 15 (1 + 7i)]
Q3 (*) Convert the complex number
1 − 2i
z=
4i
into the form z = a + bi.
[Ans. − 21 − 14 i]
2
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