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          Geometry of complex numbers

          Starter                              5   4    3    2
          1.     (Review of last lesson)  Express x − x + x − x + x − 1 as the product of linear and 
                 quadratic factors with integer coefficients.
          2.     Let z =   3 + 1i and z = 1+i.  By drawing z  and z z  on an Argand diagram 
                     1    2    2      2                    1    1 2
                 describe the geometrical effect of multiplying by z .  
                                                          2              iθ
                 Hint: you may find it useful to express the complex numbers in re  form.
          Notes                                                         Im
          To obtain the line representing z z  we enlarge z  by the scale 

                                     1 2           1                                    z z
          factor |z | and rotate z  through arg z anticlockwise about O.
                1 2
                  2            1           2 
          Alternatively, we could enlarge z  by the scale factor | z  | 

                                      2                  1                   r r
          and rotate z  through arg z  anticlockwise about O.
                1 2
                    2            1
          This combination of an enlargement followed by a rotation 
                  z
                                                                               r       1
          is called a spiral dilation.
                                     θ   1
                                                                             2 θ           Re
                                                                                1
          
                                                          O
          Raising a complex number to a positive integer power leads 
  Im        z3
          to a repeated enlargement and rotation.

                                                                           r3          z2
                                                                               r2
                                                                           θ θ r       z
                                                                                 θ         Re
                                                                     O
          In general, multiplication by r(cosθ + isinθ) corresponds to enlargement of scale factor r 
          with anticlockwise rotation of θ about the origin.

          E.g. 1 What does division by r(cosθ + isinθ) correspond to geometrically?
          E.g. 2 State the geometrical effect of multiplying a complex number z by:
                 (a)   −3i                                  (b)   5−5i
                 Working:    (a)    | − 3i| = 3 and arg(−3i) = − π
                                                               2    π
                                   Enlargement by a scale factor 3 and a 2 clockwise rotation about the 
                                   origin.
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              E.g. 3 State the geometrical effect of dividing a complex number z by:
                       (a)      1+2i                                                 (b)     −24−7i
                                                  |        |         2     2                                       c
                       Working:          (a)       1+2i = 1 +2 = 5 and arg(1+2i)=1.11
                                                                                        1          5               c
                                                 Enlargement by a scale factor           5 = 5  and a 1.11  
                                                 clockwise rotation about the origin.
              E.g. 4 Write down the complex w in the form a + ib such that the product wz represents the 
                       following transformations of z:                                      π
                       (a)      an enlargement by scale factor 2 and a rotation of 3 anticlockwise about the origin.
                       (b)      an enlargement by scale factor 1 and a rotation of 2π clockwise about the origin.
                                                                      3                       3
                       Working:          (a)     w =2 cos π +isin π               =1+ 3i
                                                          (      3           3)
                                                 Alternatively:
                                                 Enlargement by scale factor 2:                 a2 +b2 = 2
                                                                                                 a2 + b2 = 4
                                                                 π                                                 −1 b      π
                                                 Rotation of 3 anticlockwise about the origin:                 tan     a = 3
                                                                                                                          b =   3
                                                                                                                       a
                                                                       2     2                 2       2                  b = a   3
                                                 Substitute into a + b = 4:                  a +3a =4  
                                                 Since angle is π anticlockwise:                       a = 1
                                                                    3                                  b =      3
                                                 w =1+ 3i
              E.g. 5 (a)        Given the point representing a complex number z  on an Argand diagram, explain 
                                how to find the following points geometrically:            1
                                (i)      3z                        (ii)     2iz                       (iii)    (3 + 2i)z  
                                           1                                   1                                          1
                       (b)      Sketch an Argand diagram to represent the points O, 3z , 2iz  and (3 + 2i)z  and 
                                state the geometrical connection between the points.              1     1                 1
                                              Video:           Geometrical effects of conjugating a complex number
                                                                  Explanation:              Geometry of complex numbers
                                                                                     Video:            Geometrical problem 1
                                                                                     Video:            Geometrical problem 2
                                                                                              Solutions to Starter and E.g.s 
              Exercise
              p49 2F Qu 1-14
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      Summary
      Multiplication by r(cosθ + isinθ) corresponds to enlargement of scale factor r with 
      anticlockwise rotation of θ about the origin.

      Division by r(cosθ + isinθ) corresponds to enlargement of scale factor 1 with clockwise 
                                        r
      rotation of θ about the origin.

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