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Complex Numbers
Chapter 12
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Previously, when we encountered an equation like x2 + 4 = 0, we said that there was no solution since solving for x yielded
The Imaginary Number j
2 4
4
x
x
There is no real number that can be squared to produce -4.
Ah… but mathematicians were not satisfied with these so-called unsolvable equations. If the set of real numbers was not up to the task, they would define an expanded system of numbers that could handle the job!
Hence, the development of the set of complex numbers.
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Definition of a Complex Number
The imaginary number j is defined as , where j2 = .
A complex number is a number in the form x+ yj, where x and y are real numbers.
(x is the real part and yj is the imaginary part)
j
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Simplify:
4
9
25
32
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The rectangular Form of a Complex Number
Each complex number can be written in the rectangular form x + yj.
Example Write the complex numbers in rectangular form.
18 11 27 200j
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To add or subtract two complex numbers, add/subtract the real parts and the imaginary parts separately.
Example #1
Addition & Subtraction of Complex Numbers
2 5 3 7j j j
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Example #2
Simplify and write the result in rectangular form.
Addition & Subtraction of Complex Numbers
211 3 6 9 4j j j
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Multiply complex numbers as you would real numbers, using thedistributive property or the FOIL method, as appropriate. Simplify youranswer, keeping in mind that j2 = -1. Always write your final answer in rectangular form, x + yj.
Example #1
Multiplying Complex Numbers
(3 5 )(2 8 )j j
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Example #2
Simplify and write the result in rectangular form.
Multiplying Complex Numbers (continued)
5 (3 7 )j j
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Example #3
Simplify and write the result in rectangular form.
Multiplying Complex Numbers (continued)
29 4 j
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Example #4
Simplify and write the result in rectangular form.
Multiplying with Complex Numbers (continued)
9 144Be careful
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Example #5
Simplify each expression.
Multiplying with Complex Numbers (continued)
( 5)( 3) 5 3
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Example #6
Simplify and write the result in rectangular form.
Multiplying with Complex Numbers (continued)
2
22 4 j
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Complete the following:
Powers of j
1
2
3
4
5
6
7
8
9
j j
j
j
j
j
j
j
j
j
“What pattern do you observe?”
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ExamplesSimplify and write the result in rectangular form.
Powers of j
15 32 451. 2 4j j j 3 22. (2 ) (2 ) (2 )j j j
Note: If a complex expression is in simplest form, then the only power of j that should appear in the expression is j1.
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For a quotient of complex numbers to be in rectangular form, it cannot have j in the denominator.
Scenario 1: The denominator of an expression is in the form yj
Multiply numerator and denominator by j Then use the fact that j2 = -1 to simplify the expression and
write in rectangular form.
Dividing Complex Numbers
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Example #1
Dividing Complex Numbers (continued)
2
3 j
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Example #2
Write the quotient in rectangular form.
Dividing Complex Numbers (continued)
4 2 j
j
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Pairs of complex numbers in the form x + yj and x – yj are called complex conjugates.
These are important because when you multiply the conjugates together (FOIL), the imaginary terms drop out, leaving only x2 + y2.
We will use this idea to simplify a quotient of complex numbers in rectangular form.
Complex Conjugates
. . 5 2 5 2e g j j
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Scenario 2: The denominator of an expression is in the form x+yj
Multiply numerator and denominator by the conjugate of the denominator
Then use the fact that j2 = -1 to simplify the expression and write in rectangular form.
Complex Conjugates (continued)
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Example #1
Complex Conjugates (continued)
8 7
1 2
j
j
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Example #2
Write the quotient in rectangular form.
Complex Conjugates (continued)
2 121
1 9
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A complex number can be represented graphically as a point in the rectangular coordinate system.
For a complex number in the form x + yj, the real part, x, is the x-value and the imaginary part, y, is the y-value.
In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.
Graphical Representation of Complex Numbers
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Graphical Representation of Complex Numbers
Graph the points in the complex plane:
A: -3 + 4j
B: -j
C: 6
D: 2 – 7j
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Polar Coordinates
Earlier, we saw that a point in the plane could be located by polar coordinates, as well as by rectangular coordinates, and we learned to convert between polar and rectangular.
2 2
cos sin
tan
x r y r
yr x y
x
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Polar Form of a Complex Number
Now, we will use a similar technique with complex numbers, converting between rectangular and polar form*.
*The polar form is sometimes called the trigonometric form.
We’ll start by plotting the complex number x + yj, drawing a vector from the origin to the point.
x yj
r
siny r
cosx r real
imaginary
To convert to polar form, we need to know:
2 2
cos sin
tan
x r y r
yr x y
x
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The polar form is found by substituting the values of x and y into the rectangular form.
cos sinx r y r
cos sinr r jx yj
cos sinr jx yj or
A commonly used shortcut notation for the polar form is r
cos sinr jr
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For example,
13 cos 25 sin 25
1.5 241.8
j
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Example
Represent the complex numbers graphically and give the polar form of each.
1) 2 + 3j 2) 4
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Example
Represent the complex numbers graphically and give the polar form of each.
3) 4) 1 2j6 j
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Example
The current in a certain microprocessor circuit is given by
Write this in rectangular form.
3.75 15.0 .A
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The Exponential Form of a Complex Number
The exponential form of a complex number is written as
This form is used commonly in electronics and physics applications, and is convenient for multiplying complex numbers (you simply use the laws of exponents).
Remember, from the chapter on exponential and logarithmic equations, that e is an irrational number that is approximately equal to 2.71828. (It is called the natural base.)
jre where is expres radse ii .d n ans
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The Exponential Form of a Complex Number
cos sinjre r j in radians
known as Euler’s Formula
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Example
Write the complex number in exponential form.
5 cos180 sin180j
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Example
Write the complex number in exponential form.
62.5
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Example
Write the complex number in exponential form.
1 2 j
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Example
Express the complex number in rectangular and polar forms.
35 je
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We have have now used three forms of a complex number:
1) Rectangular:
2) Polar:
3) Exponential:
x yj
cos sinr j r
given in radians. jre
2 2 2
cos sin
tan
j
where
x yj r j r re
yr x y and
x
So we have,
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End of Section
