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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

