TheComplexNumber ManipulationofComplexNumbers GraphicalRepresentationusingtheArgandDiagram PolarForm Euler’sFormula DeMoivre’sTheorem
                        Engineering Analysis 2 : Complex Numbers
                                  P. Rees, O. Kryvchenkova and P.D. Ledger,
                                                engmaths@swansea.ac.uk
                                         College of Engineering, Swansea University, UK
                                       PDL (CoE)                                              SS2017   1/ 39
   TheComplexNumber ManipulationofComplexNumbers GraphicalRepresentationusingtheArgandDiagram PolarForm Euler’sFormula DeMoivre’sTheorem
   Outline
        1  TheComplexNumber
        2  Manipulation of Complex Numbers
        3  Graphical Representation using the Argand Diagram
        4  Polar Form
        5  Euler’s Formula
        6  DeMoivre’s Theorem
                                 PDL (CoE)                                      SS2017  2/ 39
   TheComplexNumber ManipulationofComplexNumbers GraphicalRepresentationusingtheArgandDiagram PolarForm Euler’sFormula DeMoivre’sTheorem
   TheNumberj
        Recall that a2 ≥ 0 for any real number a and that square root of a negative real number
        is not defined as a real number.
        In this part of the course we shall introduce a new set of numbers that allow us to make
        sense of numbers such as √−9.
        In particular we introduce a new number, j, for which
                                 2                       √
                                j =−1      so that   j =  −1
        j is not real and is instead an imaginary number. The symbol i is sometimes used in
        place of j.
                                 √      √ √
        Wecannowmakesenseof −9= −1 9=j3
                                PDL (CoE)                                   SS2017 3/ 39
   TheComplexNumber ManipulationofComplexNumbers GraphicalRepresentationusingtheArgandDiagram PolarForm Euler’sFormula DeMoivre’sTheorem
   TheComplexNumberz=a+jb
        Recall that from EG189 that the general roots of a x2 + a x + a = 0 are given by
                                                       2      1    0
                                                q2
                                          −a ± a −4a a
                                     x =    1       1    2 0
                                                 2a
                                                    2
        This result gives rise to the following implications
                 2
             For a > 4a a we have two real roots
                 1     2 0
                 2
             For a = 4a a we have one repeated root
                 1     2 0
                 2
             For a < 4a a we have no real roots.
                 1     2 0
                                            2
        Wecannowmakesenseofthecasea <4a a intermsofj. Wewillseeinthenext
                                            1     2 0
        slide that the result are two complex numbers each expressed in Cartesian form
                                            z = a +jb
        where Re(z) = a is called the real part of z and Im(z) = b is called the imaginary part of
        z. The set of all complex numbers is C.
                                 PDL (CoE)                                      SS2017  4/ 39