. . 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