Abstract Algebra Theory and Applications - Computer Science ...

3.2 THE GROUP C ∗ 61zz 3 = –3 + 2iy1 = 2 + 3i0xz 2 = 1 – 2iFigure 3.2. Rectangular coordinates of a complex numberya + birθ0xFigure 3.3. Polar coordinates of a complex numberplane where a is the x (or real) coordinate and b is the y (or imaginary)coordinate. This is called the rectangular or Cartesian representation.The rectangular representations of z 1 = 2 + 3i, z 2 = 1 − 2i, and z 3 = −3 + 2iare depicted in Figure 3.2.Nonzero complex numbers can also be represented using polar coordinates.To specify any nonzero point on the plane, it suffices to give an angleθ from the positive x axis in the counterclockwise direction and a distancer from the origin, as in Figure 3.3. We can see thatz = a + bi = r(cos θ + i sin θ).Hence,r = |z| = √ a 2 + b 2