Complex numbers and vectors

177 Pages·2006·1.4 MB·English

by  Evans L.

Complex Numbers
and Vectors
Les Evans
MathsWorks for Teachers
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd ii 2222//66//0066 99::3300::5555 AAMM
First published 2006
by ACER Press
Australian Council for Educational Research Ltd
19 Prospect Hill Road, Camberwell, Victoria, 3124
Copyright © 2006 Les Evans and David Leigh-Lancaster
All rights reserved. Except under the conditions described in the Copyright Act 1968 of
Australia and subsequent amendments, no part of this publication may be reproduced, stored
in a retrieval system or transmitted in any form or by any means, electronic, mechanical,
photocopying, recording or otherwise, without the written permission of the publishers.
Edited by Marta Veroni and Ruth Siems Editorial Services
Cover design by FOUNDRY Typography, Design & Visual Dialogue
Text design by Robert Klinkhamer
Typeset by Desktop Concepts P/L, Melbourne
Printed by Shannon Books Pty Ltd
National Library of Australia Cataloguing-in-Publication data:
Evans, Les.
Complex numbers and vectors.
ISBN 0 86431 532 5.
1. Numbers, Complex. 2. Vector analysis. I. Title.
(Series: MathsWorks for teachers).
512.788
Visit our website: www.acerpress.com.au
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd iiii 2222//66//0066 99::3300::5577 AAMM
C O N T E N T S
Introduction vii
About the author viii
1
Complex numbers and vectors in the
secondary curriculum 1
Introduction 1
Complex numbers 2
2
A tale of intrigue and imagination 10
Thinking outside the square 10
All is number 11
Intrigue and 2 13
3
Secrecy, contrivance and inspiration 20
Renaissance Italy and mathematical duels 20
Two negatives make a positive ^-1#-1=1h 25
Quadratic equations and the cartesian plane 26
4
Form and structure: A careful exposition
on operating with complex numbers 34
Addition 35
Subtraction 36
Multiplication 37
Polar coordinates 40
Division 42
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd iiiiii 2222//66//0066 99::3300::5577 AAMM
Contents
5
The genius of Gauss 49
The Fundamental Theorem of Algebra 49
Conjugate Root Theorem 52
The Fundamental Theorem of Arithmetic 56
6
Mathematicians can read maps 62
Functions of a real variable 62
Functions of a complex variable 64
Mapping functions of a complex variable 66
z
The function f]zg=e 70
Intersecting tangents 73
Sets, curves and regions (Who are our neighbours?) 74
7
Plotting a course 82
Seeing through the fog 82
Ancient navigation and vectors 83
Sinking the Spanish Armada and vectors 87
8
Sailing against the wind 94
Tacking into the wind 94
The scalar or dot product of two vectors 97
Perpendicular vectors 99
Direction cosines of a vector 99
Parallel and perpendicular components for P 101
~
The cross product of two vectors 102
Vector components and the cross product 104
Measuring torque 104
Multiplying vectors three at a time 105
Volume of a parallelepiped 105
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd iivv 2222//66//0066 99::3300::5588 AAMM
Contents
9
It’s a circus 110
Plotting a path 110
Finding cartesian equations from parametric equations 111
Vector calculus 113
Supporting circus performers 114
Mathematics and the human cannonball 114
Bareback horse riding 118
10
It’s now possible to know where we are! 123
The importance of knowing where you are 123
Working out your location 124
Vectors and vector operations using matrices 126
Linear combination of vectors 128
Transposition of vectors 129
Using matrix multiplication 130
Solving simultaneous equations 131
Mathematics and the location of earthquakes 134
11
Pons asinorum—the asses’ bridge 138
Vector proofs 139
Using the scalar product 142
12
Curriculum connections 149
13
Solution notes to student activities 153

References and further reading 168
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd vv 2222//66//0066 99::3300::5588 AAMM
This page intentionally left blank
I N T R O D U C T I O N
MathsWorks is a series of teacher texts covering various areas of study and
topics relevant to senior secondary mathematics courses. The series has been
specifically developed for teachers to cover helpful mathematical background,
and is written in an informal discussion style.
The series consists of six titles:
(cid:127) An Introduction to Functional Equations
(cid:127) Contemporary Calculus
(cid:127) Matrices
(cid:127) Data Analysis Applications
(cid:127) Foundation Numeracy in Context
(cid:127) Complex Numbers and Vectors
Each text includes historical and background material; discussion of key
concepts, skills and processes; commentary on teaching and learning approaches;
comprehensive illustrative examples with related tables, graphs and diagrams
throughout; references for each chapter (text and web-based); student activities
and sample solution notes; and a bibliography.
The use of technology is incorporated as applicable in each text, and a gen-
eral curriculum link between chapters of each text and Australian state and
territory as well as and selected overseas courses is provided.
vii
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd vviiii 2222//66//0066 99::3300::5588 AAMM
A B O U T T H E A U T H O R
Les Evans is an experienced senior secondary mathematics teacher and school
director of curriculum, and has been a setter and assessor of formal system
extended assessment tasks. His mathematics and education interests are in
drawing together different areas of study within problem solving and model-
ling theoretical investigations and practical applications, and he has extensive
professional development experience in these areas.
viii
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd vviiiiii 2222//66//0066 99::3300::5599 AAMM
1
CHAPTER
C O M P L E X N U M B E R S A N D
V E C T O R S I N T H E S E C O N D A R Y
C U R R I C U L U M
INTRODUCTION
Complex numbers and vectors are both important areas of study within the
senior secondary mathematics curriculum. They are particularly significant
for those students wishing to undertake further study in mathematics or in
disciplines that require a strong background in mathematics. These students
typically study specialist or advanced mathematics subjects in their senior
mathematics curriculum. However, the advent of more sophisticated hand-
held technologies over the past decade or so has meant that students from
mainstream function, algebra and calculus courses also come across complex
numbers as roots to certain types of algebraic equations in the analysis of
polynomial functions.
The rationale for the inclusion of complex numbers in the curriculum is
often related to:
(cid:127) arguments for completeness of algebraic analysis of polynomial functions
and the solution of related equations
(cid:127) consideration of certain types of transformations of the (complex) plane, in
particular those involving combinations of dilations and rotations, as well as
some curves and regions in the complex plane
Students typically encounter complex numbers in the guise of some kind
of special number that enables one to extend certain algebraic manipulations
on quadratic functions of a real variable with real coefficients to ensure that
the rule of any quadratic function q]xg= ax2+bx+c can be expressed as a
product of two linear factors, and the equations q]xg= 0 always has two (not
necessarily distinct) roots. This is a restricted case of a more general result,
indeed, one of the major results of mathematics—the Fundamental Theorem
1
111100550011••CCoommpplleexx NNuummbbeerrss 44pppp..iinndddd SSeecc11::11 2222//66//0066 99::3300::5599 AAMM