COMPLEX NUMBERS
COMMON MISTAKES
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10/20/2009
Complex Numbers-Definition
How to Understand
Complex Numbers
Complex Numbers are
numbers of the form: a+bi,
where a is the Real part and
b is the Imaginary part.
Definition of i: i = − 1
Recall that − 9 is undefined
but if we state − 9 = 9 • − 1
and define i = − 1 , then we
say − 9 = ±3i .
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Common Mistakes
Forgetting the properties when i
Is raised to different powers.
i = −1
i = −1
i = −i
2
3
i =1
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Complex Numbers-Addition/Subtraction
How to Add/Subtract
Complex Numbers
Adding/Subtracting
Complex Numbers
requires that the real parts
are combined and then the
imaginary parts are
combined to form a new
complex number.
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Common Mistakes
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Incorrectly combining parts of the
complex numbers.
When subtracting, forgetting to
distribute the sign to both parts of
the complex number being
subtracted.
Incorrect: (4 + 3i ) − (17 − 2i )
= (4 − 17 ) + ( 3i − 2i )
Correct:
(4 + 3i ) − (17 − 2i )
= (4 − 17 ) + ( 3i + 2i )
= −13 + 5i
10/20/2009
Complex Numbers-Multiplication
How to correctly Multiply
Complex Numbers
The Process-Multiplying
complex numbers is
similar to “FOIL”-ing or
distributing except when i
is squared, cubed, etc. (See
the Definition Slide).
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Common Mistakes
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= 4 • 3 + 6i + 4i + 2i
= 12 + 10i + 2
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Distributing and combining
the wrong parts of the
complex numbers.
Incorrectly evaluating
Incorrect: (4 + 2i )( 3 + i )
Correct:
(4 + 2i )( 3 + i )
= 4 • 3 + 6i + 4i + 2i
= 12 + 10i + 2( −1)
= 10 + 10i
10/20/2009
2
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Complex Numbers-Rationalizing
How to correctly Rationalize
Complex Numbers
Common Mistakes
Rationalizing Complex
Numbers is similar to that
with radicals.
Complex Numbers also
have conjugates such
that, when multiplied, give
real number answers in
the denominator.
Recall: a + bi has
conjugate a − bi; and i = −1
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Incorrectly defining the
conjugate.
Not correctly multiplying
the complex numbers.
12
Rationalize: 2 + 3i
Incorrect: Conjugate = 2 + 3i
Correct: Conjugate = 2 − 3i ,
so verify 12 • 2 − 3i
2 + 3i 2 − 3i
2
5
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is
=
12( 2 − 3i )
13
10/20/2009