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| Geometrical representation of a Complex number, Argand diagram |
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| 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. |
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| This representation of a complex number in a plane is called the argand diagram. |
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| 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. |
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| Join the origin O with the points P and Q and complete the parallelogram OPRQ. |
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| It is clear from the figure that the coordinates of R are |
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| (x1 + x2, y1 + y2) and it represents the complex number |
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| (x1 + x2) + i (y1 + y2), i.e., z1 + z2. |
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| The absolute values of z1, z2 and z1 + z2 are geometrically given by |
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| We know that the sum of any two sides of a triangle is greater than
the third. Hence, in DORP, we have |
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| 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. |
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| 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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