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COMPLEX NUMBERS
The simple algebraic equation
2
10
x
+
=
has no solution in terms of real numbers. To overcome this limitation, we introduce
the
unit imaginary element
i
, defined as
1
i
=
−
In terms of this imaginary element, the solution of the above equation becomes
2
11
x
or
x
i
=−
=± − =±
Similarly, the solution of the equation
2
16
0
x
+
=
is given by:
2
16
16
4
x
or
x
i
=± −
=±
The above equation has two different solutions, given by
x
= 4
i
and
x
= –4
i
.
Numbers like
i
, 6
i
, –4
i
are called
purely imaginary
numbers, and they represent
special cases of the general
complex number
a
+
ib
, in which
a
and
b
are real
numbers.
The real numbers
a
and
b
are called the
real
and
imaginary
parts of the complex
number
z
=
a
+
ib
, and denoted by
R
e
()
Im
azbz
=
=
Basic Algebraic Rules for Complex Numbers
•
Sum and Difference
If
z
1
=
a
+
ib
and
z
2
=
c
+
id
are any two complex numbers, then their sum is
12
(
zz aci
bd
)
+
=++ +
and their difference is
(
)
−
=−+ −
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Product
If
z
1
=
a
+
ib
and
z
2
=
c
+
id
are any two complex numbers, then their product is
2
12
()
(
)
z z
a
ib
c
id
ac
i ad
bc
i bd
=+×+ =+
+ +
Taking into account that
i
2
= –1 gives
(
z z
ac
bd
i ad
bc
)
=
−+ +
The product of a real number
k
and a complex number
z
is given by
kz
k a
ib
ka
ikb
=
+=+
•
Complex conjugate
To any complex number
a
+
ib
there corresponds a complex number
a
–
ib
, obtained
by changing the sign of the imaginary part. The complex number
a
–
ib
is called the
complex conjugate
of
a
+
ib
. The usual notation for complex conjugate is
z
. It
follows that the product
z
z
is a real number, given by
22
(
)
zz
a
ib
a
ib
a
i ab
ab
i b
=+×− =+
− −
2
and since
i
2
= –1, the result is
a
b
=
+
•
Division
If
z
1
=
a
+
ib
and
z
2
=
c
+
id
are any two complex numbers, then their quotient is given
by
1
2
za
c
b
db
c
a
i
zcd cd
d
+
−
=+
++
The above result is easy to prove as follows:
( )
2
11
2
ai
bci
d
z
z
z
ac
bd
i bc
ad
c
i
d
c
i
d
cd
z
+−
++ −
=×=
=
+
•
Equality of complex numbers
If two complex numbers
z
1
=
a
+
ib
and
z
2
=
c
+
id
are equal, then
a
=
c
and
b
=
d
.
•
Zero complex number
The zero complex number is the number
0 + 0
i
, that is, both its real and imaginary
parts are zero.
Exercises
:
1. If
z
1
= 2 – 3
i
and
z
2
= – 1 +
i
, find:
a)
z
1
z
2
b)
z
1
(2
z
2
+1)
2. If
z
= 2 –
i
, find:
a)
z
b)
z
z
3. If
z
1
= 3 + 2
i
and
z
2
= 1 – 3
i
, find:
a)
z
1
/
z
2
b) Re(
z
1
/
z
2
) and Im(
z
1
/
z
2
)
4. If
z
1
= 1 – 2
i
and
z
2
= 2 +
i
, find:
a)
z
1
3
b) (
z
1
z
2
)
2
ModulusArgument Form of a Complex Number
An alternative and important representation of a complex number involves the use of
polar coordinates. This allows the complex number
zxi
y
=
+
to be written in the form
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cos
sin
zr
i
θ
=+
where the
modulus r
is given by
( )
1/2
22
rz x y
== +
and the
argument
θ
is such that
()
cos
sin
xx
yy
and
rr
xy
θθ
==
++
The convention for the argument
θ
is that it always lies in the interval
–
π
<
θ
≤
π
.
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This note was uploaded on 04/20/2011 for the course ME 1331 taught by Professor Zeshan during the Spring '11 term at American College of Computer & Information Sciences.
 Spring '11
 zeshan

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