2.3.1 Properties of complex numbers

. . Properties of complex numbers The complex numbers satisfy the following properties under addition . The complex numbers satisfy the following properties under multiplication.

(i) Closure property For any two complex numbers z and z , the sum z is also a complex number. (i) Closure property For any two complex numbers z and z , the product z z is also a complex number. Complex Numbers (ii) The commutative property For any two complex numbers z and z z + z = z + z (ii) The commutative property For any two complex numbers z and z z z = z z . (iii) The associative property For any three complex numbers z z , and z z + z + z = z + z + z (iii) The associative property For any three complex numbers z z , and z z z z = z z z (iv) The additive identity There exists a complex number i such that, for every complex number z , The complex number i is known as additive identity.

(iv) The multiplicative identity There exists a complex number = + i such that, for every complex number z , The complex number i is known as multiplicative identity. (v) The additive inverse For every complex number z there exists a complex number − z such that, . − z is called the additive inverse of z . (v) The multiplicative inverse For any nonzero complex number z , there exists a complex number w such that, w w = .

w is called the multiplicative inverse of z . w is denoted by z − . (vi) Distributive property (multiplication distributes over addition) For any three complex numbers z , and z z z z z z z and ( z z z z Let us now prove some of the properties. Property The commutative property under addition For any two complex numbers 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