A complex number: a complex number is an element (x,y) of the set. R^2={(x, y):x, y €R)}.

obeying the following rules of addition and multiplication. for Z1=(X1,y1), Z2=(X2, Y2), €R^2, we put. A1: Z1+Z2=(X1+X2, y1+y2). (addition). M1: Z1 Z2=(X1 X2 -y1y2 , X1 y2+y1 X2). (multiplication). equality of complex numbers: two complex numbers Z1=(X1, y1), Z2=(x2,y2) are equal if and only if X1=X2, y1=y2. we then write Z1=Z2. properties of complex numbers. some important consequence of the definitions of addition (A1) and multiplication (M1) of complex numbers are as follows; properties of multiplication defined by (M1): M2: for all Z1=(X1, y1), Z2=(x2, y2), Z3=(X3,y3)€R^2. (Z1.Z2). Z3= Z1.(z2.z3). (associative law of multiplication). here (Z1.Z2).Z3 =(X1 X2 -y1y2 , X1 y2+y1 X2)(X3, y3). (Z1.Z2).Z3={(X1 X2 -y1y2)X3 -(X1 y2+y1 X2)y3, (X1 X2 -y1y2 )y3+(X1 y2+y1 X2 ) X3 } (Z1 . Z2 ). Z3 ={X1 (. X2 X3 -y2y3)-y1(X2 y3+y2 y3), X1 (X2 y3+y2 y3)+y1(x2 y3-y2x3)}. (Z1.z2). Z3=(X1,y1)(x2 X3 -y2y3, x2 y3+y2 X3)}. (z1.z2). Z3=Z1.(Z2.Z3) M3: the complex number 1=(1,0)€R^2 and it satisfied the condition, _1.z=z. 1=z (multiplicative identity ). for all z=(x, y )€R^2 here_1.z=(1,0)(x, y) _1.z=(1.x-0.y ,1.y+0.x) _1.z=(x, y) _1.z=z )(x .