z z z z z z = = z z z z z z z + z z z O z Re Im z z z + z z z z z z z Re Im O Fig. . Fig. .
- - = z z z z (by commutativity z z = z z ) Therefore, z z = z . Note It can be generalized by means of mathematical induction to any finite number of terms: z z z That is the modulus value of a product of complex numbers is equal to the product of the moduli of complex numbers. Similarly we can prove the other properties of modulus of a complex number. Example .
If z , and z , find z , z , and Using the given values for z , z and z we get z = z = z = = = = Note that the triangle inequality is satisfied in all the cases. (why?) Example . Find the following (i) (ii) ( )( )( (iii) i (i) i = z (ii) )( )( = ( z z z = z (iii) z - - Complex Numbers Example . Which one of the points i, i , and is farthest from the origin?
The distance between origin to z and are | z = | | i = | z = | | z = | | Since < < , the farthest point from the origin is . Example . If z