Geometrical Representation :- Consider a system of rectangular
co-ordinate 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 x-axis and N º
(O,b) is a point on y-axis. Hence X-axis is called the "real
axis" and Y-axis 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 | =
(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 = tan-1
= 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)
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
\ 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.
eiq = cos q
+ i sin q we get z = r (cos q
+ i sin q ) = r eiq
This is known as the exponential form of the complex number.
Now i4 = (i2)2 = (-1)2
= 1, i5 = i4´
i = (1) i = i
and i7 = i6´i
= (i2)3 . i = (-1)3 . i = - i ,
i13 = i12 ´
i = (i2)6 . i = (-1)6. i = i