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Transcript
Complex Numbers
Section 2.4
Objectives:
• Use the imaginary unit i to write complex
numbers.
• Add, subtract, and multiply complex numbers.
• Use complex conjugates to write the quotient of
two complex numbers in standard form.
• Plot complex numbers in the complex plane.
The Complex-Number
System
• The complex-number system is used to find
zeros of functions that are not real numbers.
• When looking at a graph of a function, if
the graph does not cross the x-axis, it has no
real-number zeros.
The Complex-Number
System
Complex Numbers C
Real Numbers R
Integers Z
Whole numbers W
Natural
Numbers N
Rational Numbers Q
R
Irrational
Numbers
Q -bar
Imaginary
Numbers i
What is a imaginary
number?
• It is a tool to solve an equation.
– Invented to solve quadratic equations of the
form x 2  4 .
• It has been used to solve equations for the
last 200 years or so.
• “Imaginary” is just a name, imaginary do
indeed exist, they are numbers.
Imaginary Numbers
Previously, when we encountered square roots
of negative numbers in solving equations, we
would say “no real solution” or “not a real
number”.
Imaginary Unit
The imaginary unit i, is the number whose square is – 1.
That is,
i 2  1 and i  1
The Imaginary Unit, i
Example:
Write the following with the i notation.
 25 
25   1  5 i
 32 
32   1  16  2   1  4 2  i  4 i 2


 121   121   1   11i
Your Turn:
Write as an imaginary number.
(a)  16   116
 i 16
 4i
(b)  80  i 80
 i 16  5
 4i 5
Powers of i
ii
i  1
2
We could continue but notice that
they repeat every group of 4.
For every i 4 it will = 1
i  i i  1(i)  i
To simplify higher powers of i
4
2 2
then, we'll group all the i
and
i  i i   1 1  1 see what is left.
5
4
8
33
4 8
i  i i  1i   i
i  i  i  1 i  i
6
4 2
i  i i  1 1  1
4 will go into 33 8 times with 1 left.
7
4 3
i  i i  1 i   i
20 3
83
4 20 3
i  i  i  1 i  i
8
4 4
i  i i  11  1
3
2
4ths
4 will go into 83 20 times with 3 left.
Powers of i
• Always reduce powers of i to;
i, -1, 1, or –i.
• To reduce, divide the exponent by 4 and the
remainder determines the reduced value.
–
–
–
–
remainder
remainder
remainder
remainder
1⇒ i
2 ⇒ -1
3 ⇒ -i
0 (no remainder) ⇒1
• Negative exponent, the term goes in the
denominator and the exponent becomes positive.
Powers of i
For exponents larger than 4, divide the exponent by
4, then use the remainder as your exponent instead.
Example:
i ?
23
23
 5 with a remainder of 3
4
So, use i which  -i
3
i  i
23
Powers of i
Example:
Simplify each of the following powers.
i  i i  i
53
i
17
52
413
 i  1i  i
1
1
1 1 i i
1
  
 2 
 17  16 
i
i i 1 i i i  i  i
i
i
 i

1
 (1)
Your Turn:
• Simplify each of the following powers.
1) i22
-1
2)
i36
1
3)
i13
i
4)
i47
-i
Complex Numbers
Real numbers and imaginary numbers are
both subsets of a new set of numbers.
Complex Numbers
A complex number is a number that can be
written in the form a + bi, where a and b are
real numbers.
Equity of Complex Numbers
a+bi =c+di if and only if a=c and b=d.
Standard Form of
Complex Numbers
Complex numbers can be written in the form
a + bi (called standard form), with both a and
b as real numbers.
a is a real number and bi would be an
imaginary number.
If b = 0, a + bi is a real number.
If a = 0, a + bi is an imaginary number.
Standard Form of
Complex Numbers
• Since all numbers belong to the Complex number
system, C, all numbers can be classified as complex.
Real numbers, R, and Imaginary numbers, i, are subsets
of Complex Numbers.
Complex Numbers
a + bi
Real Numbers
a + 0i
Imaginary Numbers
0 + bi
Standard Form of
Complex Numbers
Zero is the only number that is both real and imaginary.
Standard Form of Complex
Numbers
Example:
Write each of the following in the form of a
complex number in standard form a + bi.
6 = 6 + 0i
8i = 0 + 8i
 24 
4  6  1  2 i 6  0  2 i 6
6   25  6  25   1  6 + 5i
Your turn:
• Write each of the following in the form of a
complex number in standard form a + bi.
•
9  5
•
 24
5  3i
0  2i 6
Adding and Subtracting
Complex Numbers
Sum or Difference of Complex Numbers
If a + bi and c + di are complex numbers, then
their sum is
(a + bi) + (c + di) = (a + c) + (b + d)i
Their difference is
(a + bi) – (c + di) = (a – c) + (b – d)i
Combine like Terms: add/sub. real parts and
add/sub. Imaginary parts.
Adding and Subtracting
Complex Numbers
1. Pretend that “i” is a variable and that a
complex number is a binomial
2. Add and subtract as you would binomials
Example:
(2 + i) – (-5 + 7i) + (4 – 3i)
2 + i + 5 – 7i + 4 – 3i
11 – 9i
Adding and Subtracting
Complex Numbers
Example:
Add or subtract the following complex numbers.
Write the answer in standard form a + bi.
(4 + 6i) + (3 – 2i) = (4 + 3) + (6 – 2)i = 7 + 4i
(8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i
Your Turn:
1) (7 + 3i) + (5 – 4i)
12 – i
2) (3 + 4i) – (5 – 2i)
-2 + 6i
3) 4i – (-3 + 5i) + (2 – 6i)
5 – 7i
4) (5 – i) + (3 + 4i) – (8 + 3i)
0
Multiplying Complex
Numbers
The technique for multiplying complex
numbers varies depending on whether the
numbers are written as single term (either the
real or imaginary component is missing) or
two terms.
Multiplying Complex
Numbers
Note that the product rule for radicals does NOT
apply for imaginary numbers.
2
 16   25  4i  5i  20i  20(1)   20
 16   25 
16  25 
400  20
Before performing operations with imaginary
numbers, be sure to rewrite the terms or factors in
i-form first and then proceed with the operations.
Euler’s Blunder
Multiplication of Complex
Numbers
1. Pretend that “i” is a variable and that a complex
number is a binomial
2. Multiply as you would binomials
3. Simplify by changing “i2” to “-1” and combining
like terms
Example:
(-4 + 3i)(5 – i)
-20 + 4i + 15i - 3i2
-20 + 4i + 15i + 3
-17 + 19i
Multiplying Complex
Numbers
Example:
Multiply the following complex numbers.
8i · 7i
56i2
56(1)
56
Multiplying Complex
Numbers
Example:
Multiply the following complex numbers.
Write the answer in standard form a + bi.
5i(4 – 7i)
20i – 35i2
20i – 35(–1)
20i + 35
35 + 20i
Multiplying Complex
Numbers
Example:
Multiply the following complex numbers.
Write the answer in standard form a + bi.
(6 – 3i)(7 + 4i)
42 + 24i – 21i – 12i2
42 + 3i – 12(–1)
42 + 3i + 12
54 + 3i
Your Turn:
1) 8i(4 – 3i)
24 + 32i
2) (3 – i)(5 + 4i)
19 + 7i
3) (2 + 4i)(2 - 4i)
20
4) (5 – 3i)2
16 – 30i
Complex Conjugate
Complex Conjugates
The complex numbers (a + bi) and (a – bi) are
complex conjugates of each other, and
(a + bi)(a – bi) = a2 + b2
Complex Conjugate
The conjugate of a + bi is a – bi.
The conjugate of a – bi is a + bi.
The product of (a + bi) and (a – bi) is
(a + bi)(a – bi)
a2 – abi + abi – b2i2
a2 – b2(–1)
a2 + b2, which is a real number.
Complex Conjugate
Examples:
6 + 7i and 6  7i
8  3i and 8 + 3i
14i and 14i
• The product of a complex number and its conjugate is a
real number.
Example:
(4 + 9i)(4  9i) = 42  (9i)2
= 16  81i2
= 16  81(1)
= 97
Example:
(7i)(7i) =  49i2
= 49(1)
= 49
Your Turn:
• Multiply each complex number by its
complex conjugate.
1) 3 + i
10
2) 5 – 4i
41
Division of Complex
Numbers
1. Write the division problem in fraction form.
2. The trick is to make the denominator real, this is
done using the complex conjugate.
3. Multiply the fraction by a special “1” where “1” is
the conjugate of the denominator over itself.
4. Simplify and write answer in standard form: a + bi.
Example: 4  3i   3  2i 
4  3i 3  2i 12  8i  9i  6i 2
4  3i




2
3  2i 3  2i 3  2i
9  4i
6 17
12  8i  9i  6 6  17i
 i


13 13
13
94
Dividing Complex
Numbers
Example:
Use complex conjugates to divide the
following complex numbers. Write the answer
in standard form.
24  18i  8i  6i 2
6  2i 6  2i 4  3i




2
4  3i 4  3i 4  3i 16  12i  12i  9i
18 26
24  26i  6(1) 18  26i

 i

25
25 25
16  9(1)
Dividing Complex
Numbers
Example:
Divide the following complex numbers.
5  6i
 30i
 30i
5
5
 30i





i

2
6i  6i  36i
36
6i
6
 36(1)
Your Turn:
• Divide 4  3i by 1  8i.
4  3i 4  3i 1  8i


1  8i 1  8i 1  8i
( 4  3i )(1  8i )

(1  8i )(1  8i )
4  29i  24i 2

1  64i 2
4  29i  24

65
28  29i

65
Your Turn:
2i
1)
2i
3 4
 i
5 5
3i
2)
i
1  3i
Your Turn:
3  11i
3)
1  2i
5  i
7  4i
4)
2  5i
6 43
  i
29 29
The Complex plane
Real Axis
Imaginary Axis
Graphing in the complex
plane
.
 2  5i
2  2i
4  3i
 4  3i
.
Im
.
.
Re
Review
• A number such as 3i is a purely imaginary
number.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
• 6 + 3i is a complex number.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
• 6 + 3i is a complex number.
• a + bi is the standard form of a complex
number.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
• 6 + 3i is a complex number.
• a + bi is the standard form of a complex
number.
• If a + bi = 6 – 4i then a = 6 and b = -4.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
• 6 + 3i is a complex number.
• a + bi is the standard form of a complex
number.
• If a + bi = 6 – 4i then a = 6 and b = – 4.
• The ‘real part’ of 6 – 4i is 6.
Review
• A number such as 3i is a purely imaginary
number.
• A number such as 6 is a purely real number.
• 6 + 3i is a complex number.
• a + bi is the standard form of a complex
number.
• If a + bi = 6 – 4i then a = 6 and b = – 4.
• The ‘real part’ of 6 – 4i is 6.
• The ‘imaginary part’ of 6 – 4i is -4i.
Assignment
• Sec. 2.4, pg. 137 – 138: #1 – 75 odd