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The various fundamental laws can be represented graphically. The graphical representation makes it easier to understand the laws. The four fundamental laws of geometry have been explained below:
(i) Geometrical Representation of Addition:
If two points P and Q represent complex numbers z_{1} and z_{2} respectively in the Argand Plane, then the sum z_{1}+z_{2 }is represented by the extremity R of the diagonal OR of parallelogram OPRQ having OP and OQ as two adjacent sides. The addition of the complex numbers z_{1} and z_{2} can be assumed to be the addition of the vectors by utilizing the parallelogram law.
(ii) Geometrical representation of subtraction:
The representation of the difference of two complex numbers is slightly complicated than the addition of the complex numbers. The easiest way of presenting the subtraction is to think of addition of a negative vector. If we want to represent z_{1}-z_{2}, then the easiest way of representing it would be to think of adding a negative vector z_{1 }+ (-z_{2}). The negative vector is the same as the positive one, the only difference being that the negative vector points in the opposite direction.
The difference vector z_{1} - z_{2} is represented below
Remark: It is important to note here that the vector representing the difference of the vectors z_{1} - z_{2} may also be drawn joining the end point of z_{2} to the tip of z_{1} instead of the origin. This kind of representation does not alter the meaning or interpretation of the difference operator. The difference operator joining the tips of z_{1} and z_{2} is represented below:
(iii) Modulus and argument of multiplication of two complex numbers: Modulus and argument are two extremely important concepts associated with complex numbers.
Modulus of a complex number: The modulus of a complex number is defined to be the distance of the number from the origin. It is denoted by |z| and the value of the modulus of z = x+iy is √x^{2}+y^{2}.
Argument of a complex Number: Argument of a complex number is basically the angle that explains the direction of the complex number. It is measured in radians. If we have the complex number in polar form i.e. z = r(cos θ + i sin θ), then its argument is θ.
Modulus of product of two complex numbers: Let z_{1} and z_{2} be two complex numbers in polar form
z_{1} = r_{1}(cos θ_{1} + i sin θ_{1})
z_{2 }= r_{2}(cos θ_{2}_{ }+ i sin θ_{2})
Hence their product is given by
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})].
This expression can be further simplified as
z_{1}z_{2} = r_{1}r_{2} [(cos (θ_{1} + θ_{2}) + i sin (θ_{1} + θ_{2})].
The product z_{1}z_{2} has the modulus r_{1}r_{2} and the argument (θ_{1} + θ_{2}).
Theorem: For any two complex numbers z_{1}, z_{2}
We have, |z_{1}, z_{2}| = |z_{1}| |z_{2}| and arg(z_{1}, z_{2}) = arg(z_{1}) + arg(z_{2})
Proof: z_{1} = r_{1} e^{iθ1}, z_{2} = r_{2} e^{iθ2 }
z_{1}z_{2} = r_{1} r_{2} e^{i(θ1+θ2)}
=> |z_{1} z_{2}|=|z_{1}|=|z_{2}|
arg (z_{1}z_{2}) = arg(z_{1}) + arg(z_{2})
i.e., to multiply two complex numbers, we multiply their absolute values and add their arguments. View the following video for more on argument and modulus
(i) P.V. arg(z_{1}z_{2}) ≠ P.V. arg(z_{1}) + P.V. arg(z_{2})
(ii) |z_{1}z_{2} ...... z_{n}| = |z_{1}| |z_{2}| ......... |z_{n}|
(iii) arg(z_{1}.z_{2} ......... z_{n}) = argz_{1}+argz_{2}+ ........ + argz_{n}
(iv) Geometrical representation of multiplication of complex numbers -
Let P, Q be represented by z_{1}= r_{1} e^{iθ1}, z_{2 }= r_{2} e^{iθ2 }respectively. To find point R representing complex number z_{1} z_{2}, we take a point L on real axis such that OL=1 and draw triangle OQR similar to triangle OLP. Therefore, OR/OQ = OP/OL = OR = OP*OQ
i.e., OR = r_{1} r_{2} and ∠ QOR = θ,
R(z_{1}z_{2}) Q(z_{2}) P(z_{1})
∠ LOR = ∠ LOP + ∠ POQ + ∠ QOR
= Q_{1} + Q_{2} - Q_{1} + Q_{1}
= Q_{1} + Q_{2}
Hence, R is represented by z_{1} z_{2} = r_{1} r_{2} e^{i(Q1+Q2)}
(v) Modulus and argument of division of two complex numbers:
Theorem: If z_{1} and z_{2} ( ≠0 ) are two complex numbers, then |z_{1}/z_{2}| =
|z_{1}|/|z_{2}| ^{ } and arg(z_{1}/z_{2}) = arg z_{1} - arg z_{2}
Note:
P.V. arg(z_{1}/z_{2}) ≠ P.V. arg (z_{1}) - P.V. arg (z_{2})
(vi) Geometrical representation of the division of complex numbers-
Let P, Q be represented by z_{1}= r_{1} e^{iθ1}, z_{2 }= r_{2} e^{iθ2 }respectively. To find point R representing complex number z_{1}/z_{2}, we tale a point L on real axis such that OL=1 and draw a triangle OPR similar to OQL.
Therefore, OP/OQ = OR/OL
=> OR = r_{1}/r_{2}
and ∠LOR = ∠LOP - ∠ROP
= θ_{1} - θ_{2}
Hence, R is represented by z_{1}/z_{2} =r_{1}/r_{2} e^{i(}^{θ}^{1 - }^{θ}^{2}^{)}
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