z and z , we have z Proof Let z iy , z iy , and x x , and y ∈ , = x iy iy = x i y = x i y (since x x , and y ∈ ) = x iy iy = z Property Inverse Property under multiplication Prove that the multiplicative inverse of a nonzero complex number z iy is - - Proof The multiplicative inverse is less obvious than the additive one. Let z u iv be the inverse of z iy We have z z − = That is x iy u iv = xu yv i xv uy ) = + i Equating real and imaginary parts we get xu yv = 1and xv uy . Solving the above system of simultaneous equations in u and v we get u and v . ( z is non-zero ⇒ x If z iy , then z .
( z − is not defined when z = ). Note that the above example shows the existence of z − of the complex number z . To compute the inverse of a given complex number, we conveniently use z . If z and z are two complex numbers where z ¹ , then the product of z and z is denoted by z .
Other properties can be verified in a similar manner. In the next section, we define the conjugate of a complex number. It would help us to find the inverse of a complex number easily. Complex numbers obey the laws of indices (i) z z m m n (ii) z m m n , z ¹ (iii) z m mn (iv) z z z z m m m