61
L7
Complex Numbers and Quadratic Equations
Complex Numbers
The
imaginary unit
, which we denote by
i
, is a solution of
the equation
2
1
x
= −
,
that is,
2
1
i
=−
. Another solution of this equation is (
i
−
).
Complex numbers
are the numbers of the form
ab
i
+
,
where
a
and
b
are real.
The real numbers
a
and
b
are called respectively the
real
part
and
imaginary part
of the complex number
i
+
.
The form
ai
b
+
is the
standard form
of a complex
number.
A real number
a
can be written as
0
+
, thus, the set of
real numbers is a
subset
in the complex number system.
Complex number
bi
is called a
pure imaginary number
.
Equality of Complex Numbers
:
b
ci
d
+=
+
if and only if
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Sum of Complex Numbers
:
()
ai
b
ci
d
++
+
=
Product of Complex Numbers
:
b
d
=
Note:
The product of complex numbers is easy to
find by using FOIL and the fact that
2
1
i
=−
.
Complex Conjugates
:
The
complex conjugate
of a number
za
b
i
=
+
is
b
i
.
Example
: Write expressions in the standard form.
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 Fall '10
 Picklesimer
 Equations, Negative Numbers, Complex Numbers, Complex number, zz

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