. . Properties of Modulus of a complex number ( ) z ( ) z ( ) z (Triangle inequality) ( ) z , where n is an integer ( ) z z ( ) Re z ( ) z ( ) Im z Let us prove some of the properties. Property Triangle inequality For any two complex numbers z , we have z Proof Using z z z z = = ( )( = z z z z z z z z = z z z z z z z Re Im P ( x, y ) M O Fig.
. z )( - - Complex Numbers = | Re( ) | z z Re( ) £ z z z Re( ) |) £ = z ( | | | || | | |) z z ⇒ z £ z £ z Geometrical interpretation Now consider the triangle shown in figure with vertices O z or z , and z .We know from geometry that the length of the side of the triangle corresponding to the vector z cannot be greater than the sum of the lengths of the remaining two sides. This is the reason for calling the property as "Triangle Inequality". It can be generalized by means of mathematical induction to finite number of terms: for n = , , .
Property The distance between the two points z and z in complex plane is | If z iy and z iy , then = x Remark The distance between the two points z and z in complex plane is z If we consider origin, z and z as vertices of a triangle, by the similar argument we have Property Modulus of the product is equal to product of the moduli. For any two complex numbers z , we have z z Proof We have z z = ( )( z