Complex Numbers
The complex numbers are an extension of the real numbers containing all roots of quadratic
equations. If we define i to be a solution of the equation x
2
= 1, then the set
C
of complex
numbers is represented in
standard form
as
{ a+bi

a,b
∈
R}.
We often use the variable z = a+bi to represent a complex number. The number a is called the
real part
of z (Re z) while b is called the
imaginary part
of z (Im z). Two complex numbers are
equal
if and only if their real parts are equal and their imaginary parts are equal.
We represent complex numbers graphically by associating z
= a+bi with the point (a,b) on the
complex plane
.
Basic Operations
The basic operations on complex numbers are defined as follows:
=
(a+c) + (b+d)i
=
(ac) + (bd)i
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(acbd) + (bc+ad)i
a+bi
c+di
=
a+bi
c+di
·
cdi
cdi
=
ac+bd
c
2
+d
2
+
bcad
c
2
+d
2
i
In dividing a+bi by c+di, we rationalized the denominator using the fact that
(c+di)(cdi) = c
2
cdi +cdi d
2
i
2
= c
2
+ d
2
. The complex numbers c+di and cdi are called
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 Spring '10
 Coluos
 Real Numbers, Equations, Complex Numbers, Complex number, polar form

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