2.2.3 Algebraic operations on complex numbers

. . Algebraic operations on complex numbers In this section, we study the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known. (i) Scalar multiplication of complex numbers: If z iy and k ∈  , then we define k z kx ky i In particular z = , z and ( z z . Fig. . Fig. . Fig. . (ii) Addition of complex numbers : If z iy and z iy , where x x ,and ∈  , then we define = x iy iy = x i y = x i y We have already seen that vectors are characterized by length and direction, and that a given vector remains unchanged under translation. When z iy and z iy then by the parallelogram law of addition, the sum z = x i y corresponds to the point x . It also corresponds to a vector with those coordinates as its components. Hence the points , and z in complex plane may be obtained vectorially as shown in the adjacent Fig. . . Re z Im z O or Fig. . - - - - - - - - Re Im - i - -3i - +2i +i Complex numbers as points - - - - - - - - Re Im - i - -3i - +2i +i Complex numbers as position vectors O O Complex Numbers (iii) Subtraction of complex numbers Similarly the difference z can also be drawn as a position vector whose initial point is the origin and terminal point is x . We define