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Chabot Mathematics
§7.7 Complex
Numbers
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
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Review § 7.6
MTH 55
 Any QUESTIONS About
• §7.6 → Radical Equations
 Any QUESTIONS About HomeWork
• §7.6 → HW-35
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Imaginary & Complex Numbers
 Negative numbers do not have square
roots in the real-number system.
 A larger number system that contains the
real-number system is designed so that
negative numbers do have square roots.
That system is called the
complex-number system.
 The complex-number system makes
use of i, a number that with the
property (i)2 = −1
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The “Number” i
 i is the unique number for which
i2 = −1 and so i  1
 Thus for any positive number p we can
now define the square root of a negative
number using the product-rule as follows
.
 p  1 p  i p or
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pi,
Imaginary Numbers
 An imaginary number is a number
that can be written in the form bi,
where b is a real number that is not
equal to zero
 Some
Examples
37i
i
5
i  29
 73
 i is called the “imaginary unit”
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Example  Imaginary Numbers
 Write each imaginary number as a
product of a real number and i
a) 16
b) 21
c)
32
 SOLUTION
a)
16
21
c)
32
 116
 1 21
 1 32
 1  16
 i4
 1  21
 1  32
 i 16  2
 4i 2
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b)
 i 21
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ReWriting Imaginary Numbers
 To write an imaginary number n
in terms of the imaginary unit i:
1. Separate the radical into two
factors 1  n .
2. Replace 1 with
n. i
3. Simplify
1  n .
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Example  Imaginary Numbers
a) i: 9
 Express in terms of
b)  48
b)
a)a) 9
 SOLUTION
b)  48
a)
9  1  9
 1 9  i  3, or 3i.
b)b)  48   1  16  3
  1 16 3  i  4  3  4i 3
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Complex Numbers
 The union of the set of all imaginary
numbers and the set of all real numbers
is the set of all complex numbers
 A complex number is any number that
can be written in the form a + bi, where
a and b are real numbers.
• Note that a and b both can be 0
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Complex Number Examples
 The following are examples of Complex
numbers
7  2i
1
2 i
3
11i
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Here a = 7, b =2.
Here a  2, b  1/ 3.
Here a  0, b  11.
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Add/Subtract Complex No.s
 Complex numbers obey the
commutative, associative, and
distributive laws.
 Thus we can add and subtract them as
we do binomials; i.e.,
• Add Reals-to-Reals
• Add Imaginaries-to-Imaginaries
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Example  Complex Add & Sub
 Add or subtract and simplify a+bi
(−3 + 4i) − (4 − 12i)
 SOLUTION: We subtract complex
numbers just like we subtract
polynomials. That is, add/sub LIKE Terms
→ Add Reals & Imag’s Separately
• (−3 + 4i) − (4 − 12i) = (−3 + 4i) + (−4 + 12i)
•
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= −7 + 16i
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Example  Complex Add & Sub
a) simplify
(3  2i)  to
(7 a+bi
8i)
 Add or subtract and
a)a) (3  2i)  (7  8i) b)b) (10  2i)  (9  i)
b) (10  2i)  (9  i) Combining real and
imaginary parts
 SOLUTION
a)a) (3  2i)  (7  8i)  (3  7)  (2i  8i)
 10  (2  8)i  10  10i
b)b) (10  2i)  (9  i)  (10  9)  (2i  i)
 1  3i
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Complex Multiplication
 To multiply square roots of negative
real numbers, we first express them
in terms of i. For example,
6  5  1 6  1 5
 i 6 i 5
i
2
30
 1 30   30.
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Caveat Complex-Multiplication
 CAUTION
 With complex numbers, simply
multiplying radicands is incorrect
when both radicands are negative:
3  5  15.
 The Correct Multiplicative Operation
  3    5    1 3   1 5    1  3  1  5 
2
   1   1 3  5     1  3  5    1 15   15
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Example  Complex Multiply
a) 2  10
b) 2i  5  3i 
a) to
2  a+bi
10 form
 Multiply & Simplify
a)a) 2  10 b)b) 2i  5  3i  c)c)  2  i  4  3i 
b) 2i  5  3i  c)  2  i  4  3i 
 c)
SOLUTION
 2  i  4  3i 
a)a) 2  10  1  2  1  10
 i  2  i  10
2
 i  20  1  2 5  2 5
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Example  Complex Multiply
a) to
2  a+bi
10 form
 Multiply & Simplify
a)
c)
b)b) 2i  5  3i 
c)  2  i  4  3i 
 SOLUTION: Perform Distribution
b)b) 2i  5  3i   2i  5  2i  3i
 10i  6i
2
 10i  6  6  10i
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Example  Complex Multiply
a) 2  10
b) 2i  5  3i 
 Multiply & Simplify to a+bi form
a)
b)
c)c)  2  i  4  3i 
 SOLUTION : Use F.O.I.L.
c)c)
 2  i  4  3i   8  6i  4i  3i
 8  2i  3
 11  2i
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2
Complex Number CONJUGATE
 The CONJUGATE of a complex
number a + bi is a – bi, and the
conjugate of a – bi is a + bi
 Some Examples
   13i Conjugate     13i
31  i 2 Conjugate  31  i 2
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Example  Complex Conjugate
 Find the conjugate of each number
a) 4 + 3i
b) −6 − 9i
c) i
 SOLUTION:
a) The conjugate is 4 − 3i
b) The conjugate is −6 + 9i
c) The conjugate is −i (think: 0 + i)
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Conjugates and Division
 Conjugates are used when dividing
complex numbers. The procedure is much
like that used to rationalize denominators.
 Note the Standard Form for Complex
Numbers does NOT permit i to appear in
the DENOMINATOR
• To put a complex division into Std Form,
Multiply the Numerator and Denominator by
the Conjugate of the DENOMINATOR
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Example  Complex Division
3  2i
 Divide & Simplify to a+bi form
a)
i
4i
 SOLUTION: Eliminate i fromb)DeNom by
2  3i
multiplying the Numer & DeNom by the
Conjugate of i
3  2i 3  2i  i  3i  2i
 2 1  3i
a)




2
i
i
i
  1
i
2
 2  3i
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3  2i
a)
Example  Complex Division
i
4i
b)
 Divide & Simplify to a+bi form
2  3i
 SOLUTION: Eliminate i from DeNom by
multiplying the Numer & DeNom by the
Conjugate of 2−3i
  NEXT SLIDE for Reduction
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Example  Complex Division
4i
4  i 2  3i
b)

 SOLN

2  3i 2  3i 2  3i
(4  i )(2  3i) 8  12i  2i  3i 2


(2  3i)(2  3i)
4  9i 2
8  14i  3 5  14i


49
13
5 14
  i
13 13
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Example  Complex Division
3  5i
 Divide & Simplify to a+bi form
5i
 SOLUTION: Rationalize DeNom
by Conjugate of 5−i
15  3i  25i  5

3  5i 3  5i 5  i


5i
5i 5i
15  3i  25i  5i

25  i 2
15  3i  25i  5(1)

25  (1)
2
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25  1
10  28i

26
10 28i


26 26
5 14i
 
13 13
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Powers of i → in
 Simplifying powers of i can be done by
using the fact that i2 = −1 and expressing
the given power of i in terms of i2.
 The First 12 Powers of i
i  1
i 5  1
i 9  1
i  1
i  1
i 10  1
i 3  i 2 • i  1 1
i 7  1 1
i 11  1 1
i 4  i 2 • i 2  1 • 1  1
i8  1
i 12  1
2
6
• Note that (i4)n = +1 for any integer n
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Example  Powers of i
 Simplify using Powers of i
a) i 40 a. i 40b. i33 b)
b. i33
a.
 SOLUTION : Use (i4)n = 1
a)
a. i
40
= i

4 10
= 110 = 1
Write i40 as (i4)10.
b)b. i 33 = i 32  i
= i
= 1 i
=i
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
4 8
i
Write i32 as (i4)8.
Replace i4 with 1.
Bruce Mayer, PE
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WhiteBoard Work
 Problems From §7.7 Exercise Set
• 32, 50, 62, 78, 100, 116

DC Ohm' s Law
Ohm’s Law of
Electrical Resistance
v  ir
in the Frequency
Domain uses
AC Ohm' s Law
Complex Numbers
(See ENGR43)
V  IZ
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All Done for Today
Electrical
Engrs Use
j instead
of i
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 1  i Math Def 
 1  j Engr Def 
Examples : j17 or  23 j
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Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
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Licensed Electrical & Mechanical Engineer
[email protected]
–
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Rational numbers:
Complex numbers that
are real numbers:
a + bi, b = 0
2
, 7,  18, 8.7...
3
Irrational numbers:
2,  ,  3 7,...
The complex
numbers:
a = bi
Complex numbers
(Imaginary)
a  bi , a  0, b  0 :
Complex numbers that
are not real numbers:
a + bi, b ≠ 0
 3i , 32 i , 17i ,...
Complex numbers
a  bi , a  0, b  0:
2  2i ,5  4i , 32  75 i
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Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
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4
5
6
5
5
y
4
4
3
3
2
2
1
1
0
-10
-8
-6
-4
-2
-2
-1
0
2
4
6
-1
0
-3
x
0
1
2
3
4
5
-2
-1
-3
-2
M55_§JBerland_Graphs_0806.xls
-3
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-4
M55_§JBerland_Graphs_0806.xls
-5
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