Complex Numbers
•
Section 6.3
139
6.3 Complex Numbers
There is no real number
x
such that
x
2
= −
1. However, it turns out to be useful
6
to invent
such a number, called the
imaginary unit
and denoted by the letter
i
. Thus,
i
2
= −
1, and
hence
i
=
r
−
1. If
a
and
b
are real numbers, then a number of the form
a
+
bi
is called a
complex number
, and if
b
n=
0 then it is called an
imaginary number
(and
pure imag
inary
if
a
=
0 and
b
n=
0). The real number
a
is called the
imaginary part
of the complex
number
a
+
bi
, and
bi
is called its
imaginary part
.
What does it mean to add
a
to
bi
in the deFnition
a
+
bi
of a complex number, i.e. adding
a real number and an imaginary number? You can think of it as a way of
extending
the set
of real numbers. If
b
=
0 then
a
+
bi
=
a
+
0
i
=
a
(since 0
i
is deFned as 0), so that every real
number is a complex number. The imaginary part
bi
in
a
+
bi
can be thought of as a way
of taking the
onedimensional
set of all real numbers and extending it to a
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Complex Numbers

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