Complex Numbers Worksheet - Appendix F, Cengage
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11APPENDIX F
Complex Numbers
Operations with Complex Numbers • Complex Solutions of Quadratic Equations •
Polar Form of a Complex Number • Powers and Roots of Complex Numbers
Operations with Complex Numbers
Some equations have no real solutions. For instance, the quadratic equation
2
x
1
0
Equation with no real solution
has no real solution because there is no real number x that can be squared to produce
1.
To overcome this deficiency, mathematicians created an expanded system of
numbers using the imaginary unit i, defined as
i
1
Imaginary unit
2
where
i
1.
By adding real numbers to real multiples of this imaginary unit, you
obtain the set of complex numbers. Each complex number can be written in the
standard form
a
bi.
Definition of a Complex Number
For real numbers a and b, the number
a
bi
is a complex number. If
b
0,
a
bi
is called an imaginary number, and bi
is called a pure imaginary number.
To add (or subtract) two complex numbers, you add (or subtract) the real and
imaginary parts of the numbers separately.
Addition and Subtraction of Complex Numbers
If
a
bi
and
c
di
are two complex numbers written in standard form, their
sum and difference are defined as follows.
Sum:
a
bi
c
di
a
c
b
d i
Difference: a
bi
c
di
a
c
b
d i
F1
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