F08 - Further Complex Numbers - Lecture Notes (Teacher_s Version)

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Teacher’s Version) Page 1 of 24 National Junior College 2016 – 2017 H2 Further Mathematics Topic F8: Further Complex Numbers Key Questions to Answer: Complex Numbers in Polar & Exponential Form 1. How do we interpret geometrically the effects of conjugating a complex number, and adding/subtracting/multiplying/dividing two complex numbers? - What is the geometrical effect of multiplication by i? 2. What is de Moivre’s Theorem? - How do we apply de Moivre’s Theorem to find the powers and nth roots of a complex number, and to derive trigonometric identities? Complex Loci How do we sketch the loci of simple equations and inequalities involving a complex variable in the Argand diagram such as   , and arg ? z c r z a z b z a         §1 Geometrical Effects of Operations on Complex Numbers 1.1 Addition and Subtraction of Complex Numbers Recall that c omplex numbers can alternatively be represented by position vectors, i.e., a complex number iz x y  can be represented as a position vector xOP y       in the x-y plane. Techniques and operations used in coordinate geometry and vectors can be applied to adding and subtracting complex numbers , corresponding to the parallelogram law of vector addition and subtraction respectively. Let 1z = a + ib be represented by 1P . 2z = c + id be represented by 2P . 1 2z z z  = (a + c) + i(b + d) be represented by P. In terms of vectors, 1 2OP OP OP       . Re O z2 z1 Im P1 P2 P www.KiasuExamPaper.com 771

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Teacher’s Version) Page 2 of 24 The subtraction 2 1z z is the same as the addition of 2z and 1z . Thus, let 2P P   represents 1.z Then OP   represents 2 1z z . 2 1z z z  = (c – a) + i (d – b) is represented by P. In terms of vectors, 1 2 2 1OP P P OP OP           . What are some possible limitations of considering complex numbers as vectors on the Argand Diagram? Every complex number can only be represented by position vectors and not displacement vectors. So for example when finding the difference between two complex numbers (e.g. 2 1z z ), the vector representing the complex number 2 1z z must be translated to start from the origin first before the real and imaginary parts of the complex number can be obtained. Finding differences of vectors on the other hand, has no such restriction. 1.2 Conjugation of Complex Numbers Example 1.2.1 Represent the following complex numbers and their corresponding conjugates in an Argand diagram: 1 4ia   , 2 3ib  , 3 ic  , 2 4id   What is the geometrical effect of conjugating a complex number? Solution: z2 z1 Re Im O