Complex Numbers
As early as 250 A.D., Greek algebraist, Diophantus, attempts to solve quadratic equations of the form
. He accepts only positive rational roots and ignores all others. He rejects; equations such
as
as insolvable. In the sixteenth

and seventeenth

century new numbers are obtained by
extending the arithmetic operation of square root to whatever numbers appeared in solving quadratic
equations by the usual method of completing the square. For instance, Cartan (1545) obtains the roots of
the equation
as
2
0
ax
bx
c
++
=
22
0
xa
+=
()
(
)
2
10
40
5
15
xx
x
−=
⇔−
=
−
51
5
+
−
and
5
−
−
. However he regarded them
as useless. By the eighteenth century, through the work of Euler, de Moivre, d’Alembert and Cauchy,
complex numbers were finally accepted by mathematicians. In here you will study the arithmetic
operations on complex numbers, geometrical representation and use of de Moivre’s theorem to simplify
complex expressions. Finally you will look at the solutions of the equation
.
0
0
n
zz
1
Basic idea of complex numbers
(p.52 – p. 53)
(a)
1
−
is defined as a number whose square is
1
−
and it is denoted by
i
and is called the (purely)
imaginary
unit. If
b
is real then
bi
is said to be a (purely) imaginary number.
(b) A complex
number is the sum of a real number and a (purely) imaginary number such that for real
a
,
b
,
c
,
d
,
ab
iff
and
icd
i
+=+
ac
=
bd
=
.
Thus a complex number can be uniquely written as
zab
i
=
+
where
a
,
b
are real and in this
representation,
a
is termed the real part of
z
, and
b
is termed the (purely) imaginary part of
z
.
(c)
A complex number cannot be said to be larger or smaller than the other.
(d)
The complex conjugate
of
is
i
+
iab
i
+
=−
.
2
Operations with complex numbers
(p.53 – p.54)
We operate with complex numbers in exactly the same way as we operate with real numbers; for
example:
(a)
(
)
(
i cd
i ac bd
i
+
+
+
)
)
)
(b)
(
)
(
i
+−
−
+
−
(c)
and
(
)
(
a
bi
c
di
ac
bd
bc
ad i
=
−
+
+
( )( )
2
i a b
2
+
+
, since
; also
2
1
i
(d)
( )
( ) ( )
a
bi
c
di
a
bi
c
di
ac
bd
bc
ad i
i
cd
i
c d
+
+
−
+
===
−
+
+
, provided
0.
i
+≠
Hence, the sum, difference, product and quotient of two complex numbers is a complex number; the
product of two conjugate complex numbers is real and so is their sum.
1
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Geometrical representation of complex numbers
(p.54)
The real numbers can be represented by a straight line in the sense that it is possible to set up a one

to

one
correspondence between the set of all real numbers and the set of all points on a straight line.
To represent the complex numbers, a straight line is not sufficient and it is necessary to make use of a
plane.
First, as shown in Figure 1, two perpendicular straight lines in the plane are drawn, one horizontal (called
the real axis
) and the other vertical (called the imaginary
axis), intersecting at a point usually denoted by
O and called the origin
. Then for each axis, a one

to

one correspondence with the real numbers is set up
in the ‘usual’ way so that the origin will correspond to the number zero in each case.
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 Spring '09
 GXXY
 Cos, Complex number

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