## Geometrical Representation of Fundamental Operations of Complex Numbers:

Consider two complex numbers z_{1} = a + *i* b and z_{2} = c + *i* d which are denoted by points P (a, b) and Q (c, d) on the complex plane. OP = | z_{1}| and OQ = | z_{2} |. The parallelogram OPRQ is completed.

PL, QM and RN are drawn perpendiculars on OX. Also, PS is drawn perpendicular to RN (Above Figure).

△OMQ ≅ △PSR (by AAS congruence) ∴ OM = PS = c QM = RS = d ∴ ON = OL + LN = OL + PS = a + c RN = RS + SN = RS + PL = b + d |

∴ R denotes the complex number whose real part is a + c and whose imaginary part is b + d. Thus, R represents the complex number-

a + c + i (b + d) = (a + i b) + (c + i d) = z_{1} + z_{2} ∴ | z _{1} + z_{2} | = ORNow in △ ORP, OP + PR ≥ OR (equality holds only when O, P and R are collinear) ∴ OP + OQ ≥ OR ⇒ | z _{1} | + | z_{2} | ≥ | z_{1} + z_{2} | ………….(i) |

Thus, the sum of the moduli of two complex numbers is greater than or equal to the modulus of their sum.

Now, let z and z_{2} be two complex numbers. Thus, from (i), we have | z | + | z_{2} | ≥ | z + z_{2} |.Replacing z + z _{2} by z_{1} i.e., z by z_{1} – z_{2},We get, | z _{1} – z_{2} | + | z_{2} | ≥ | z_{1} |∴ | z _{1} – z_{2} | ≥ | z_{1} | – | z_{2} |⇒ | z _{1} | – | z_{2} | ≤ | z_{1} – z_{2} |We now consider z _{1} and z_{2} in polar form, i.e., z_{1} = r_{1} (cos θ_{1} + i sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} + i sin θ_{2})∴ z _{1} . z_{2} = r_{1}r_{2} (cos θ_{1} + i sin θ_{1}) (cos θ_{2} + i sin θ_{2})⇒ z _{1} . z_{2} = r_{1}r_{2} {(cos θ_{1} cos θ_{2} – sin θ_{1} sin θ_{2}) + i (sin θ_{1} cos θ_{2} + cos θ_{1} sin θ_{2})}⇒ z _{1} . z_{2} = r_{1}r_{2} {cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})}, which denotes a complex number whose modulus is r_{1}r_{2} and whose argument is (θ_{1} + θ_{2}). |

Thus, the product of two complex numbers is a complex number whose modulus is equal to the product of the moduli of the individual complex numbers and whose argument is the sum of the arguments of those complex numbers.

Let P and Q denote the two complex numbers z_{1} = r_{1} (cos θ_{1} + *i *sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} + *i *sin θ_{2}) on the Argand plane such that | z_{1} | = OP = r_{1} and | z_{2} | = OQ= r_{2}. The point R would then denote the complex number z_{1} . z_{2} such that OR = | z_{1} z_{2} | = r_{1}r_{2} and arg (z_{1} z_{2}) = arg (z_{1}) + arg (z_{2}) = θ_{1} + θ_{2} = ∠ROX.

Thus, R must have polar coordinates (r_{1}r_{2}, θ_{1} + θ_{2}) (Fig. (a)).

Now, consider the case of the division of two complex numbers z_{1} = r_{1} (cos θ_{1} + *i *sin θ_{1}) and z_{2} = r_{2} (cos θ_{2} + *i *sin θ_{2}).

Obviously, the quotient z_{1}/z_{2} is a complex number whose modulus is equal to the quotient of the moduli of the individual complex numbers z_{1} and z_{2} and whose argument is the difference of the two arguments θ_{1} and θ_{2}. Thus, the quotient z_{1}/z_{2} would be denoted by the point R on the complex plane having polar coordinates (r_{1}/r_{2}, θ_{1} – θ_{2}) (Below Figure).

It has already been mentioned that the two complex numbers z = a + *i* b and z̄ = a – *i* b are conjugate to each other. On the complex plane if P denotes the complex number z(a, b) then P’ would denote z̄ (a, -b). (Below Figure)