Geometrical Representation : Consider a system of rectangular
coordinate axes X ' OX and Y ' OY. Then each complex number z =
(a,b) corresponds to a point P º
(a,b) and conversely to every point on XY plane there corresponds
a complex number. Any number of kind M º
(a,O) is a point on xaxis and N º
(O,b) is a point on yaxis. Hence Xaxis is called the "real
axis" and Yaxis is called the "imaginary axis".
Such a representation is called the "Argand's diagram"
due to J.R.Argand and XY plane is called Argand's plane.
(A) MODULUS : From the fig. OP = r =
If z = (a,b) is a complex number then r =  z 
=
_{ }
is called its modulus \  a + bi  =
For example,  3 + 4i  =
= 5,  5  12i  =
= 13
(B) Argument or Amplitude : If OP makes
angle q with ,
then q is called the "argument or amplitude" of
the complex number z = (a,b)
From the right triangle OPM, we get
OM = a = r cos q .....
(i) and ON = b = r sin q ....(ii)
So that r =
and tan q =
\ q = tan1
()
= argument z
Thus if q satisfies
simultaneously the relations
cos q =
=
and sin q =
=
and that  p £ q £
p is called the argument or the amplitude of the complex number
z = (a, b)
Polar Form
Using the above relations i.e. a = r cos q
and b = r sin q
then z = a + bi = r cos q + i (r sin
q)
\ z = r (cos q
+ i sin q) ...... This is called the
polar form of the complex number z.
Note that by Eular's formula i.e.
e^{iq} = cos q
+ i sin q we get z = r (cos q
+ i sin q ) = r e^{iq}
This is known as the exponential form of the complex number.
Example 1
Evaluate
Solution
Now i^{4} = (i^{2})^{2} = (1)^{2}
= 1, i^{5} = i^{4}´
i = (1) i = i
and i^{7} = i^{6}´i
= (i^{2})^{3} . i = (1)^{3} . i =  i ,
i^{13} = i^{12} ´
i = (i^{2})^{6} . i = (1)^{6}. i = i
\
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