Transcript No Slide Title

Complex Numbers – Add, Subtract, Multiply, and Divide
• Addition of complex numbers is given by:
(a  bi)  (c  di)  (a  c)  (b  d )i
• Example 1:
(3  2i)  (5  4i)
 (3  5)  (2  4)i
 8  2i
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• It is good to leave out the middle step and to work
the problem completely in your head.
(3  2i)  (5  4i)  8  2i
• Example 2:
(7i)  (2  2i)  2  5i
It is perfectly fine to think of addition of complex
numbers as adding binomials, but remember that i is
not a variable, but an imaginary number.
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• Subtraction of complex numbers is given by:
(a  bi)  (c  di)  (a  c)  (b  d )i
• Example 3:
(3  2i)  (5  4i)
 3  (5)   (2  (4))i
 3  5  (2  4)i
 8  2i
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• Again, not all the steps were necessary, and learning
to work the problem quickly in your head is good.
(3  2i)  (5  4i)  8  2i
• Example 4:
 2  3i   8i
 2  11i
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• Multiplication of complex numbers is given by:
(a  bi)(c  di)  (ac  bd )  (ad  bc)i
It is often easier to think of multiplication of complex
numbers using the foil pattern for binomials, even
though these are numbers and not true binomials.
Again, remember that i is not a variable, but an
imaginary number
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• Example 5:
5  3i  2  4i   10  20i  6i 12i
 10 14i 12  1
 10  14i  12
 22  14i
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• Example 6:
 4  2i 1 3i   4  12i  2i  6i
 4 14i  6  1
 4  14i  6
 2  14i
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• Consider the complex number
a  bi
• The Complex Conjugate of this number is given by:
a  bi
• Notice what happens when you multiply complex
conjugates.
 a  bi  a  bi   a
2
 abi  abi  b i
2 2
 a  b (1)
2
2
 a 2  b2
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• Notice the difference between multiplying complex
conjugates and multiplying binomials as in previous
work.
Binomials
Complex Conjugate
 a  b a  b
 a  bi  a  bi 
 a 2  b2
 a 2  b2
• When multiplying complex
conjugates, remember the
+ sign!
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• Example 7:
Complex Number
Complex Conjugate
3  5i
3  5i
2  3i
2  3i
7i
 7i
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• To compute the Division of complex numbers,
multiply both the numerator and the denominator by
the complex conjugate of the denominator.
a  bi c  di

c  di c  di
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• Example 8:
3  4i 2  3i 6  9i  8i  12i


2  3i 2  3i
22  32
2
6  17i  12  1

49
6  17i

13
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6  17i

13
• The problem is not complete at this point. Always
express complex number answers in a+bi form.
6  17i 6 17i


13
13
13
6 17
  i
13 13
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• Example 9:
2  4i
8
16  32i

 2
2
2  4i 2  4i
2 4
16  32i

20
16 32
  i
20 20
4 8
  i
5 5
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