John Vince Imaginary Mathematics for Computer Science 123 JohnVince BournemouthUniversity Poole, UK ISBN978-3-319-94636-8 ISBN978-3-319-94637-5 (eBook) https://doi.org/10.1007/978-3-319-94637-5 LibraryofCongressControlNumber:2018949636 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Preface pffiffiffiffiffiffiffi I first came across the (cid:1)1 in the complex roots of quadratic equations. i, or its doppelgänger j, popped up in my electrical engineering studies, where it separates phase differences in voltages and currents. In mathematics, I learned about Euler’s equation eipþ1¼0 and how i creates totally new subjects, such as complex function theory. In industry, I came across quaternions, which are complex num- bersinfourdimensions.Morerecently,Ihavediscoveredoctonionsandgeometric algebra. This journey of discovery has been long and arduous, but exciting. During my youth, I questioned the meaning of i, but I no longer worry about suchmatters.However,IwishthatIhaddiscoveredallthatInowunderstandabout imaginary mathematics from one source, which is the reason behind this book. I remember trying to understand an internal document on quaternions during my time in flight simulation. I felt that the author had written the document to delib- eratelyhidethecontentsfromme.Ilearnednothingfromthiscommunication,apart from a determination to understand the subject. AlthoughIamfarfrombeinganexpertinmathematics,Iwouldliketopasson whatIhavediscoveredaboutcomplexnumbersinthefollowingchapters.Isuppose I had to include an obligatory introductory chapter tracing the history of i’s rise to fame. Chapter 2 on Complex Numbers places them in a numerical context and describes topics such as the complex plane, complex exponentials, logarithms, hyperbolicfunctionsandsimplederivatives.Ihaveincludedmanyillustrationsand worked examples to reinforce the mathematical ideas. Chapter 3 is on Matrix Algebra and describes topics such as complex eigen- values and eigenvectors, representing complex numbers as matrices, complex matrix algebra, and the complex inner and outer products. I also include many worked examples. QuaternionsarethesubjectofChap.4,andtakemyword,thatifyouunderstand complex numbers, then quaternions are just as easy. The chapter starts with Hamilton’s struggle to develop a 3D form of complex numbers, describes the various forms and associated algebra, and concludes with some worked examples. Octonionsare new tome, and Chap. 5revealswhat I havediscovered from my research.TheCayley–Dicksonconstructionshowsthatanoctonioncanberegarded as an ordered pair of quaternions; a quaternion is an ordered pair of complex numbers,and acomplex number isan ordered pair of reals. Even if you never use theminyourwork,atleastyouknowwheretheybelonginimaginarymathematics. Chapter 6 describes geometric algebra, which was not developed to exploit the imaginary unit, but turns out to possess imaginary qualities. I have previously written about the subject and believe that it will play an important role in future descriptionsofscienceandphysics.Idescribethevariousproductsassociatedwith different geometric elements and their relationship to quaternions. The rest of the book deals with applications of the above algebras. Chapter 7 showshowcomplexnumberssimplifytherepresentationofcompoundangles,and Chap.8describeshowcomplexexponentialnotationsimplifiesthecombinationof waves. This chapter shows the importance of complex numbers in dealing with wave phenomena, be they simple water waves or waves in quantum fields. Chapter9coversCircuitAnalysisUsingComplexNumbers.Theobjectiveisnot toturnyouintoanelectricalengineer,buttoreinforcetheroleofcomplexnotation in representing out-of-phase electrical waves. Chapter 10 is on Geometry Using Geometric Algebra and may inspire you to write software using GA’s constructs. Still on a geometric theme, Chap. 11 shows how quaternions are used to rotate vectors about an arbitrary 3D axis. I have always been fascinated by prime numbers, especially the Riemann hypothesis.Entirebookshavebeenwrittenonthesubject,andinChap.12,Ihave attemptedtocondensetheexplanationtohalf-a-dozenpages. Chapter 13describes the simple algorithm behind the Mandelbrot set, using some beautiful images provided by Dr. Wolfgang Beyer and Dr. Dominic Ford. The last chapter concludes the book and reminds the reader how complex numbers have found their way into quantum physics, by including references to Pauli matrices, Dirac matrices, the Dirac equation, and the Schrödinger equation. Ihavereallyenjoyedwritingandresearchingthisbook.Duringthistime,Ihave discovered some extremely well-written books and articles on the Internet. As always, Wikipedia is an amazing resource, and long may it continue as an inde- pendent agency. I thank Dr. Tony Crilly for reading the final manuscript and making some important suggestions. Naturally, if I have included any mistakes, they are of my own doing! As always, I thank Beverley Ford, Editorial Director—Computer Science, and Helen Desmond, Editor—Computer Science for Springer-Verlag, for the support and guidance they have provided throughout the book’s development. Finally, enjoy this fascinating subject. Breinton, Herefordshire, UK John Vince August 2018 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Why i is Necessary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Language of Mathematics. . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 A Brief History of i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Laws of Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Commutative Law . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Associative Law. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Distributive Law. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Types of Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Rational Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.4 Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.5 Real Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.6 Algebraic and Transcendental Numbers . . . . . . . . . . 14 2.3.7 Imaginary Numbers . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.8 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Representing Complex Numbers . . . . . . . . . . . . . . . . . . . . . . 16 2.4.1 Real and Imaginary Parts . . . . . . . . . . . . . . . . . . . . 16 2.4.2 The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Complex Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.1 Algebraic Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.2 Complex Conjugate . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5.3 Complex Division. . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.4 Powers of i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.5 Rotational Qualities of i . . . . . . . . . . . . . . . . . . . . . 22 2.5.6 Modulus and Argument. . . . . . . . . . . . . . . . . . . . . . 24 2.5.7 Complex Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.8 Complex Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.9 Complex Exponentials . . . . . . . . . . . . . . . . . . . . . . 28 2.5.10 de Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.11 nth Root of Unity. . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.12 nth Roots of a Complex Number. . . . . . . . . . . . . . . 35 2.5.13 Logarithm of a Complex Number . . . . . . . . . . . . . . 36 2.5.14 Raising a Complex Number to a Complex Power. . . 37 2.5.15 Visualising Simple Complex Functions . . . . . . . . . . 39 2.5.16 The Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . 41 2.5.17 Derivative of a Complex Number . . . . . . . . . . . . . . 42 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6.1 Summary of Complex Formulae . . . . . . . . . . . . . . . 44 2.7 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.1 Complex Addition . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.2 Complex Products. . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.3 Complex Division. . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.4 Complex Rotation. . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7.5 Polar Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.7.6 Real and Imaginary Parts . . . . . . . . . . . . . . . . . . . . 49 2.7.7 Magnitude of a Complex Number . . . . . . . . . . . . . . 49 2.7.8 Complex Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.9 Complex Inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.10 de Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.11 nth Root of Unity. . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.7.12 Roots of a Complex Number. . . . . . . . . . . . . . . . . . 52 2.7.13 Logarithm of a Complex Number . . . . . . . . . . . . . . 53 2.7.14 Raising a Number to a Complex Power . . . . . . . . . . 53 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Matrix Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Complex Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.1 Matrix Addition and Subtraction . . . . . . . . . . . . . . . 55 3.2.2 Matrix Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.3 Zero Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.4 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.5 Negative Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.6 Determinant of a Matrix . . . . . . . . . . . . . . . . . . . . . 58 3.2.7 Diagonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2.8 Identity Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.9 Transpose Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.10 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.11 Symmetric Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.12 Anti-symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . 63 3.2.13 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.14 Cofactor Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.15 Conjugate Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.16 Normal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.17 Conjugate Transpose. . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.18 Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.19 Orthogonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2.20 Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Real Eigenvectors and Eigenvalues . . . . . . . . . . . . . 75 3.3.2 Complex Eigenvectors and Eigenvalues . . . . . . . . . . 79 3.3.3 Eigenvectors of a Rotation Matrix . . . . . . . . . . . . . . 81 3.4 Representing a Complex Number as a Matrix. . . . . . . . . . . . . 83 3.5 Complex Algebra Using Matrices . . . . . . . . . . . . . . . . . . . . . 86 3.6 Complex Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.6.1 Cartesian Vector Space . . . . . . . . . . . . . . . . . . . . . . 87 3.6.2 Complex Vector Space . . . . . . . . . . . . . . . . . . . . . . 88 3.6.3 Inner Product inIRn. . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6.4 Inner Product in Cn . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6.5 Outer Product inIRn . . . . . . . . . . . . . . . . . . . . . . . . 91 3.6.6 Outer Product in Cn . . . . . . . . . . . . . . . . . . . . . . . . 92 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7.1 Summary of Formulae . . . . . . . . . . . . . . . . . . . . . . 93 3.8 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.8.1 Matrix Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.8.2 Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.8.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 97 3.8.4 Determinant of a Matrix . . . . . . . . . . . . . . . . . . . . . 97 3.8.5 Transpose Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.8.6 Symmetric Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.8.7 Anti-symmetric Matrix . . . . . . . . . . . . . . . . . . . . . . 98 3.8.8 Cofactor Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.8.9 Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.8.10 Conjugate Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.8.11 Complex Eigenvectors and Eigenvalues . . . . . . . . . . 102 3.8.12 Conjugate Transpose Matrix . . . . . . . . . . . . . . . . . . 104 3.8.13 Hermitian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.8.14 Orthogonal Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.8.15 Unitary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.8.16 Complex Vector Addition . . . . . . . . . . . . . . . . . . . . 106 3.8.17 Complex Inner Product. . . . . . . . . . . . . . . . . . . . . . 107 3.8.18 Complex Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.8.19 Distance Between Complex Vectors. . . . . . . . . . . . . 108 3.8.20 Complex Outer Product. . . . . . . . . . . . . . . . . . . . . . 108 Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.1.1 History of Quaternions . . . . . . . . . . . . . . . . . . . . . . 111 4.2 Some Algebraic History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3 Defining a Quaternion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.1 The Quaternion Units . . . . . . . . . . . . . . . . . . . . . . . 119 4.4 Algebraic Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.5 Adding and Subtracting Quaternions . . . . . . . . . . . . . . . . . . . 122 4.6 Real Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7 Scaling a Quaternion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.8 Pure Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.9 Unit Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.10 Additive Form of a Quaternion . . . . . . . . . . . . . . . . . . . . . . . 125 4.11 Binary Form of a Quaternion. . . . . . . . . . . . . . . . . . . . . . . . . 126 4.12 Quaternion Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.13 Norm of a Quaternion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.14 Normalised Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.15 Quaternion Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.15.1 Pure Quaternion Product . . . . . . . . . . . . . . . . . . . . . 129 4.15.2 Unit-Norm Quaternion Product . . . . . . . . . . . . . . . . 129 4.15.3 Square of a Quaternion . . . . . . . . . . . . . . . . . . . . . . 131 4.15.4 Norm of the Quaternion Product . . . . . . . . . . . . . . . 132 4.16 Inverse Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.17 Quaternion Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.17.1 Orthogonal Quaternion Matrix. . . . . . . . . . . . . . . . . 134 4.18 Quaternion Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.19 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.19.1 Summary of Operations . . . . . . . . . . . . . . . . . . . . . 136 4.20 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.20.1 Adding and Subtracting Quaternions . . . . . . . . . . . . 137 4.20.2 Norm of a Quaternion. . . . . . . . . . . . . . . . . . . . . . . 138 4.20.3 Unit-Norm Form of a Quaternion . . . . . . . . . . . . . . 138 4.20.4 Quaternion Product. . . . . . . . . . . . . . . . . . . . . . . . . 138 4.20.5 Square of a Quaternion . . . . . . . . . . . . . . . . . . . . . . 139 4.20.6 Inverse of a Quaternion. . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5 Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.3 The Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.3.2 Cayley–Dickson Construction . . . . . . . . . . . . . . . . . 144 5.4 Octonion Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.4.1 Octonion Addition and Subtraction . . . . . . . . . . . . . 146 5.4.2 Octonion Multiplication . . . . . . . . . . . . . . . . . . . . . 147 5.4.3 Octonion Conjugate . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4.4 Norm of an Octonion . . . . . . . . . . . . . . . . . . . . . . . 148 5.4.5 Inverse of an Octonion . . . . . . . . . . . . . . . . . . . . . . 149 5.5 Summary of Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.6 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.6.1 Adding and Subtracting Octonions. . . . . . . . . . . . . . 150 5.6.2 Multiplying Two Octonions. . . . . . . . . . . . . . . . . . . 150 5.6.3 Conjugate of an Octonion . . . . . . . . . . . . . . . . . . . . 151 5.6.4 Norm of an Octonion . . . . . . . . . . . . . . . . . . . . . . . 151 5.6.5 Inverse of an Octonion . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6 Geometric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.3 Symmetric and Anti-symmetric Functions. . . . . . . . . . . . . . . . 154 6.4 Trigonometric Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.5 Vectorial Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.6 Inner and Outer Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.7 The Geometric Product in 2D . . . . . . . . . . . . . . . . . . . . . . . . 159 6.8 The Geometric Product in 3D . . . . . . . . . . . . . . . . . . . . . . . . 161 6.9 The Outer Product of Three 3D Vectors. . . . . . . . . . . . . . . . . 164 6.10 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.11 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.12 Grades, Pseudoscalars and Multivectors . . . . . . . . . . . . . . . . . 166 6.13 Redefining the Inner and Outer Products . . . . . . . . . . . . . . . . 168 6.14 The Inverse of a Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.15 The Imaginary Properties of the Outer Product . . . . . . . . . . . . 170 6.16 Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.17 The Relationship between the Vector Product and the Outer Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.18 The Relationship between Quaternions and Bivectors . . . . . . . 174 6.19 Reflections and Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.19.1 2D Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.19.2 3D Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.19.3 2D Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.20 Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.21 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.21.1 Summary of Formulae . . . . . . . . . . . . . . . . . . . . . . 182 6.22 Worked Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.22.1 2D Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.22.2 2D Outer Product . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.22.3 2D Geometric Product . . . . . . . . . . . . . . . . . . . . . . 184 6.22.4 3D Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.22.5 3D Outer Product . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.22.6 3D Geometric Product . . . . . . . . . . . . . . . . . . . . . . 185 6.22.7 Outer Product of Three Vectors. . . . . . . . . . . . . . . . 186 6.22.8 Inverse of a Vector. . . . . . . . . . . . . . . . . . . . . . . . . 186 6.22.9 Recovering a Vector from a Geometric Product . . . . 187 6.22.10 Reflecting a 2D Vector about a Line . . . . . . . . . . . . 187 6.22.11 Reflecting a 3D Vector about a Line . . . . . . . . . . . . 187 6.22.12 Rotating a 3D Vector . . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7 Trigonometric Identities Using Complex Numbers . . . . . . . . . . . . . 189 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.2 Compound Angle Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.3 de Moivre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8 Combining Waves Using Complex Numbers. . . . . . . . . . . . . . . . . . 195 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8.3 Combining Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.3.1 Using Trigonometric Identities. . . . . . . . . . . . . . . . . 197 8.4 Using Complex Exponentials. . . . . . . . . . . . . . . . . . . . . . . . . 200 8.4.1 Same Frequency and Amplitude, but no Phase Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 8.4.2 Same Frequency, Different Amplitudes, but no Phase Angle. . . . . . . . . . . . . . . . . . . . . . . . . 201 8.4.3 Same Frequency, Amplitude and Phase Angle . . . . . 203 8.4.4 Same Frequency and Amplitude, but Different Phase Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 8.4.5 Same Frequency and Amplitude, but One has a Phase Angle . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.4.6 Same Frequency, Different Amplitudes, and One has a Phase Angle. . . . . . . . . . . . . . . . . . . 205