parts of ( ), we get x = , x – y = – , which, on solving simultaneously, give x = and y = . Algebra of Complex Numbers In this Section, we shall develop the algebra of complex numbers. . .
Addition of two complex numbers Let z = a + ib and z = c + id be any two complex numbers. Then, the sum z + z is defined as follows: z + z = ( a + c ) + i ( b + d ), which is again a complex number. For example, ( + i ) + (– + i ) = ( – ) + i ( + ) = – + i The addition of complex numbers satisfy the following properties: The closure law The sum of two complex numbers is a complex number, i.e., z + z is a complex number for all complex numbers z and z . The commutative law For any two complex numbers z and z , z + z = z + z (iii) The associative law For any three complex numbers z , z , z , ( z + z ) + z = z + ( z + z ).
(iv) The existence of additive identity There exists the complex number + i (denoted as ), called the additive identity or the zero complex number, such that, for every complex number z , z + = z. (v) The existence of additive inverse To every complex number z = a + ib , we have the complex number – a + i (– b ) (denoted as – z ), called the additive inverse or negative of z. We observe that z + (– z ) = (the additive identity). .
. Difference of two complex numbers Given any two complex numbers z and z , the difference