Complex Numbers


   
 
Geometrical representation of a Complex number, Argand diagram
Since every complex number z = x + iy is an order pair of real numbers (x, y), it can therefore be represented by a point P(x,y) in the xy plane by taking the real part along the x-axis and the imaginary part along the y-axis.
 
 
This representation of a complex number in a plane is called the argand diagram.
 
 
Let points P and Q in the complex plane represent two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2, respectively, as shown in figure.
 
Join the origin O with the points P and Q and complete the parallelogram OPRQ.
 
It is clear from the figure that the coordinates of R are
 
(x1 + x2, y1 + y2) and it represents the complex number
 
(x1 + x2) + i (y1 + y2), i.e., z1 + z2.
 
The absolute values of z1, z2 and z1 + z2 are geometrically given by
 
 
We know that the sum of any two sides of a triangle is greater than the third. Hence, in DORP, we have
 
 
with the equality holding only when O, P, Q are lying in a straight line. That is why this inequality for the absolute values of complex numbers is called the triangle inequality.
 
The triangle inequality can be extended to n complex numbers by finite induction, i.e., for any n complex numbers z1, z2…..., zn, we obtain
 
 
 
     
   
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